- Email: [email protected]

1 [email protected],

Institut f¨ ur Mathematik, Universit¨at Z¨ urich

Preface These lecture notes are written for the students of the course “Introduction to mathematical finance” (fall/winter semester 2016). They are a guideline for the study of the course but do not aim to be fully comprehensive on the subject. They are an edited version of the lecture notes I wrote for the students of the last year, made up of (adjusted) parts of the books mentioned below, and some personal notes and considerations. The main reference books for the course are: • “ Introduction to the Mathematics of Finance”, by R.J. Williams; • “PDE and martingale methods in option pricing”, by Andrea Pascucci (chapters 1,2). Other useful books: • “Introduction to mathematical finance”, by Stanley R. Pliska, for a rigorous presentation of mathematical finance and some in-depth analysis on topics that are usually taken for granted; • “Arbitrage theory in continuous time”, by Thomas Bj¨ork, for a practiceoriented, simplified and wide exposition on arbitrage pricing/hedging and interest rates.

2

Chapter 1 Introduction to Financial Markets and Derivatives 1.1

Generalities on Financial Markets

Financial markets can be of two kinds, based on the securities, i.e the tradable assets, that are traded: Primary financial markets consist of basic securities, like: • Stocks, which make up the equity of quoted companies; • Currencies; • Bonds, which are fixed income instruments, providing the owner with a (more or less) deterministic cash flow over a future period; • Commodities, which are economic goods or services produced to satisfy wants or needs (e.g. wheat, oil, metal, ..); • Indexes, which are statistical averages of prices of selected securities. Secondary financial markets consist of more sophisticated securities, called derivatives or contingent claims, which depend on the value of some 3

4

1. Introduction to Financial Markets and Derivatives other security, called the underlying asset, usually from a primary market. That is, based on the type of underlying asset, we have equity, credit, interest rate and FX (foreign exchange) derivatives. Some of the most common derivatives are: • Options; • Futures; • Forwards; • Convertible bonds; • Caps and floors; • Interest rate swaps; • Credit default swaps. Financial markets, both primary and secondary, can be either public exchanges, i.e. trading venues open to all interested parties (sellers and buyers) that use a common technology platform, or over-the-counter (OTC) markets, that consist of direct bilateral contracts carried out on the phone1 . An OTC market typically involves only big financial institutions or corporation, given the considerable amount of credit default risk. OTC markets used to be made of a network of dealers, who also work as market makers, matching the incoming offers and demands but also trading between each other. Beside the counterparty default risk, this structure brought about a remarkable amount of systemic risk, that is the risk of a domino effect of defaults. To lower the financial risks affecting OTC markets, regulators stepped in changing the structure of over-the-counter transactions from a complex non-transparent network of dealers and costumers to a central counterparty (CCP) clearing 1

To get an idea of the size of these two types of derivatives markets, also classified by

currency, country, underlying, etc, you can look at the statistics provided by the Bank for International Settlements on their website http://www.bis.org/statistics/about_ derivatives_stats.htm?m=6%7C32.

1.1 Generalities on Financial Markets system2 . A market is said to be liquid if assets can be easily bought and sold, without impacting the price. To understand the concept of liquidity we may look at the structure of a central limit order book (CLOB), that is a trading system used by most public exchanges, where the customer orders (bids and offers) are matched on a ‘price-time priority’ basis. The best (highest) price among the orders submitted by potential buyers (bids) is called the bid, while the best price (lowest) among the orders submitted by potential sellers (asking prices or offers) is called the ask. A transaction takes place when either a potential buyer is willing to pay the ask, or a potential seller is willing to accept the bid, or else they meet in the middle if both of them change their orders. Within this system, a customer can enter either a market order, which is immediately executed at the best available price (bid or ask), or a limit order, which specifies the price and the amount of security that the customer is willing to buy/sell. Limit orders can also be canceled, while market orders cannot. The bid-ask spread is the gap between the bid and the ask, and makes up the largest part of transaction costs, together with brokerage fees. That is why the bid-ask spread is also considered a measure of the market liquidity. Customers can also see the market depth or the “stack” of the present limit orders, that is the total volume of orders at the correspondent prices. The CLOB is by definition fully transparent, real-time, anonymous and low-cost in execution. It may be described as an order-driven matched bargain market. On the other side, OTC markets are quote-driven markets, where there is a market-maker who charges the customer with a compensation (like the bid-ask spread) for providing liquidity in the asset. The current or spot price of an asset actually refers to the price of the last trade. It is a historical price, but during market hours that’s usually very few seconds ago for very liquid stocks. The investor who buys a financial 2

On this topic, you can read the reports of the International Monetary Fund, “Making

over-the-counter derivatives safer: the role of central counterparties”, and of the Federal Reserve of Chicago, “Central counterparty clearing”.

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6

1. Introduction to Financial Markets and Derivatives

asset is said to have a long position on that asset, while the one who sells the asset is said to have a short position on it.

1.1.1

General Assumptions

In this course, we keep to the very theoretical and simplified assumption of perfect liquidity and zero transaction costs, i.e. zero bid-ask spread. We also assume that the interest rate on deposits is deterministic and constant over time. We use mathematics to set up a model for a physical phenomenon, specifically financial markets, that verifies some physical laws, specifically some economical criteria. The main economical assumption we always want to be satisfied is that of a viable market, that is free of arbitrage opportunities. Roughly speaking, an arbitrage opportunity is a possibility to make money out of nothing without risking any loss. We will define it rigorously in the next chapters. There are other economic criteria of plausibility related to arbitrage, for example the law of one price and the non-existence of dominant strategies. The law of one price is the weaker assumption of the three and already has very significant implications in the pricing of derivatives. It say that if two securities have the same market value at a future time T , they must have the same value at any other time prior to t: XT = YT

⇒

Xt = Yt ∀t ≤ T.

Our focus will be on two problems: the pricing and hedging/replication of derivatives, in particular of options. Determining the right price for a derivative is important in order to preserve the viability of the financial market. On the other hand, a seller of a derivative should be able to hedge the risks associated with the contract by investing in the market in such a way to have enough money to pay the payoff at maturity. This two problems are in fact tightly connected, as we will see along the course. They have been investigated for ages but only in the 0 70s mathematicians and physicists came

1.1 Generalities on Financial Markets

7

into play and solved them by means of rigorous justifications3 .

1.1.2

Interest rates and bonds

In the continuous-time setting, we denote by r the annual risk-free interest rate and any time is expressed in years. In the discrete-time setting, the time is represented by a set of consecutive natural numbers {0, 1, 2, . . . , T } ∈ NT . Two different rules for computing the interests accrued on a deposit (think about a savings account) over a period of time [0, T ] are available. Simple interest over a period of time of length T : the interest on the deposit accrues proportionally to the investment time, i.e. Time Value of deposit

t=0

t=T

x

x(1 + rT )

We also refer to the quantity rT as the simple rate over a period of time of length T . Compound interest is computed in a different way depending on how the time is measured. In discrete time: the interest accrued bewteen any two consecutive dates is summed up with the principal and hence it also accumulates interests over the next period. That is:

3

Time

t=0

Value

x

t=1

t=2

x(1 + r) x(1 + r)(1 + r) = x(1 + r)2

···

t=T

···

x(1 + r)T

See Black-Scholes (1973) and Merton (1973), who first derived closed formulas for call

option prices in a setting where underlying prices are modeled by geometric Brownian motion, that was proposed by Samuelson (1965), following the hint in the PhD thesis of Louis Bachelier (1900). Also see Harrison-Pliska (1981) for the link between continuoustime trading and the theory of stochastic integrals.

8

1. Introduction to Financial Markets and Derivatives

In continuous time: the deposit accrues interest on each infinitesimal sub-period, such that at each time t ∈ [0, T ] the interest over (t − dt, t] is added to the principal value and it also earns interest from then on. The compound formula in continuous time is obtained by dividing the whole time period in a finite number of sub-periods, applying the formula valid in discrete-time, adjusted with the length of subintervals, then taking the limit as the time step goes to zero. Precisely, divide the T time period [0, T ] in N sub-periods of the same length N and assume T T that the simple interest over (k − 1) N ,kN , for every k = 1, . . . , N , is T paid and added to the principal at time k N :

Time

t=0

Value

x

t=

T N

T t = 2N

T T 2 x(1 + r N ) x(1 + r N )

···

t=T

···

T N x(1 + r N )

Finally, by taking the limit as N goes to infinity, we get N T −−−→ xerT . x 1+r N →∞ N So, to resume, the compound interest formula is: Time

t

T xer(T −t)

Value of deposit (continuous-time framework) x Value of deposit (discrete-time framework)

x

x(1 + r)T −t

Bonds Bonds are debt securities, under which the issuer owes the holder a debt, consisting of the principal and an interest: the holder of the bond is the lender (creditor), the issuer of the bond is the borrower (debtor). All kinds of bonds share the property of providing the owner with a deterministic cash flow, thus they are also known as fixed income instruments. They are basically of two types: zero-coupon bonds and coupon bonds.

1.1 Generalities on Financial Markets

Zero-coupon bonds (also known as pure discount bonds) with a given maturity date T (briefly called T -bonds), are contracts that guarantee the holder 1 unit of currency to be paid at time T , with no intermediate payments. The contract value at time t < T is denoted by ZtT and is computed in terms of the risk-free interest rate. Specifically, in the discrete-time setting, the relation between the price of the zero-coupon bond at any time t = 0, 1, 2, . . . , T and the risk-free rate r is the following: ZtT (1 + r)T −t = 1,

i.e. ZtT = (1 + r)−(T −t) .

Coupon bonds with maturity T , are contracts that provide the holder with intermediate interest payments c1 , c2 , . . . , cN , called coupons, at some times T1 < T2 < . . . < TN < T prior to maturity, when both the principal and the coupon cN +1 is paid. The value of the coupons may be predetermined at the time the bond is issued (fixed coupon bonds), or rather reset for every coupon period (floating rate bonds). In the last case, the resetting is often based on some financial benchmark, like a market interest rate, for example the LIBOR rate. There is also another type of bonds, that is convertible bonds, which are issued by quoted companies and allows its holder to convert the bond into equity (stock) in the underlying company. This is actually a derivative, it has all the features of a coupon bond, such as a coupon and a maturity date, but also provides for a conversion ratio of the credit into equity ownership and quotes the price at which the conversion can take place. To compensate for having additional value through the option to convert the bond to stock, a convertible bond typically has a coupon rate lower than that of similar nonconvertible debt. The investor receives the potential upside of conversion into equity while protecting downside with cash flow from the coupon payments and the return of principal upon maturity. Since there exist financial instruments providing a deterministic increase in the principal value of an investment over time, the time itself has an intrinsic monetary value. Thus, when considering a value of x units of money

9

10

1. Introduction to Financial Markets and Derivatives at time T , we refer to the amount x(1 + r)−(T −t) units of money, respectively xe−r(T −t) in the continuous-time setting, as the discounted value of x at time t < T . Conversely, an amount of money of x units of money at time t will have a value at time T equal to x(1 + r)T −t units of money, respectively xer(T −t) in the continuous-time setting.

1.1.3

Derivatives and pricing

The simplest type of derivatives is that of options, that are contracts giving the owner the right, but not the obligation, to buy or sell a given security at a pre-determined price (called the strike price) within a fixed future time period. There are two types of option: an European option can only be exercised at the final time, also called maturity or expiration date, while an American option can be exercised at any time within the given period. We will come back later to see options more in detail. First, let us mention some other basic contracts.

Forward and Futures Contracts Forward and futures contracts are commitments to buy or sell at a fixed future date T a given security at a predetermined price. Forward contracts involve no cash flows up to the delivery date T. At T , the long-position investor receives the underlying asset and pays the forward price ftT , which is determined at time t in such a way that the time-t price of the forward contract equals zero. Forward contracts are over-thecounter instruments. Note that although their price at the time the contract is written is equal to zero, they have a non-zero market value which varies in the interval [t, T ]. Note that because the underlying asset price may fluctuate wildly, there is a huge risk that one of the two parties will not be able to execute her obligations. Proposition 1.1. If the law of one price holds, the forward price at time t

1.1 Generalities on Financial Markets

11

with maturity T of an asset with price process S must be equal to ftT = Proof. If ftT >

St ZtT

St = St (1 + r)T −t . ZtT

, consider the following investment:

- at time t, borrow an amount of money equal to St , buy the asset and offer a forward contract with forward price ftT ; - at time T , deliver the asset to the holder of the forward contract, cash in the forward price ftT and pay back the debt incurred at time t, that is now St (1 + r)T −t . The value of such an investment is zero at time t and ftT − ZSTt > 0 at time T . t

This is a risk-free profit and is not allowed by the law of one price. Indeed, if we consider another investment that consists in putting an amount of money equal to (ftT −

St ZtT

)(1 + r)−(T −t) > 0 in a savings account at time t, then its

value at time T will be the same as the first investment described, even if its value at time t is different. The proof in the case of ftT <

St ZtT

is symmetric and we leave it as an

exercise. Futures contracts, unlike forward contracts, are traded on public exchanges. Apart from a higher level of regulation and the presence of a third party, the exchange, in the role of intermediary and guarantor, futures mitigate the credit risk involved by means of the following measures: (i) standardization of the contract, at the level of quality and quantity of the asset exchanged, to ensure liquidity of futures contract, allowing to get out of positions more easily; (ii) the use of margins, specifically initial and maintenance margins; (iii) a peculiar settlement procedure called marking-to-market where the deposit is adjusted daily;

12

1. Introduction to Financial Markets and Derivatives

(iv) price limits, that once reached trigger a suspension of trading on the futures, to avoid panic trading based on faulty information or overreaction to real information. At the date the contract is signed, each counterparty must deposit the initial margin, usually a percentage varying from 2% to 20% of the value of the contract, in a specific account. At each trading date prior to maturity, the contract is marked to market, which means that profits and losses from movements of the forward price are transferred between the two margin accounts: at time t, if the forward price has increased since last time t − 1, an T amount equal to ftT − ft−1 is taken form the margin account of the seller of the future contract and deposited into the margin account of the buyer; if the forward price has decreased, then the same amount is taken from the margin account of the buyer of the future contract and deposited into the margin account of the seller. Each investor is allowed to withdraw from its margin account whatever is left over the initial margin. When losses use up the margin deposit to below a pre-determined level (lower than the initial margin), called the maintenance margin, the investor receives a margin call to replenish the margin deposit. Roughly speaking, it is like every day the two counterparties cancel the futures contract they entered into, and replace it by a new futures contract with the same delivery date but a new price, accordingly to the current market forward price for the asset. Over time, the forward price moves towards the spot price of the asset, that is reached at maturity, i.e. fTT = ST . Marking to market eliminates, or better reduces, the counterparty risk because profits and losses on future positions are paid over at the end of every trading day, reducing any financial incentives for not making delivery. So, at maturity, the cumulative cash flow for the buyer of the futures contract resulting from the marking-to-market adjustments is T X t=1

T (ftT − ft−1 ) = fTT − f0T = ST − f0T .

1.1 Generalities on Financial Markets

13

In order to conclude the contract, the buyer will then pay the market price ST of the underlying asset, that is delivered by the seller of the contract. This gives a final cumulative cash flow of value −f0T , and a final value of ST − f0T , for the buyer, exactly like in the forward contract. Example 1.2. Consider a tofu manufacturer who needs to buy 1000 tons of soybeans in a three months time and wants to be protected against a steep rise in the asset price, the market price of soy beans. The current price of soybeans is 160 CHF/ton. We consider the two situations where he buys a forward and a futures contract on the the same underlying asset with a forward price equal to 165 CHF/ton. Supposing that the price of soybeans in three months drops to 100 CHF/ton and that the quoted forward price evolves according to Month Futures price (CHF/ton)

1

2

3

150

120

100

let us analyze the two contracts: 1. (Forward) There is no cash flow between the value date (the date at which the contract is written) and the settlement date. At the settlement date, the long position investor turns out to be in a very unfavorable position, having to pay for the asset under contract a total amount of 65 000 CHF more than the actual market price. If the buyer is not able to pay that cash difference, he defaults and seller faces a loss. 2. (Futures) On every month, the long position investor pays the drop in the market futures price from the previous month, that is

Month

Buyer’s cash flow (CHF)

Seller’s cash flow (CHF)

1

-15

+15

2

-30

+30

3

-20

+20

-100

+100

-165

+165

Total cash flow

14

1. Introduction to Financial Markets and Derivatives

The cash flow in the table above refer to a quantity of 1 ton of soybeans, so it has to be multiplied for the principal amount of the contract, which is 1000 tons. Since the movements of the futures price are paid month by month, it is unlikely that one of the two counterparties withdraws from the contract before the settlement date. An opposite scenario where the price rises to 225 CHF/ton would see the seller in trouble to pay the cash difference and the buyer facing a risk of his counterparty’s default. European Options: Call and Put Definition 1.3. A European call option on a certain underlying security with strike price K and maturity (exercise date) T is a contract that gives the owner the right to buy the underlying security at time T at the price K. If we denote by St and Ct respectively the price of the underlying security and of the call option at any time t ≤ T , then the value of the option at maturity is CT = (ST − K)+ = max{ST − K, 0}. Definition 1.4. A European put option on the underlying security with strike priceK and maturity T is a contract that gives the holder the right to sell the underlying security at time T at the price K. If we denote by Pt the price of the put option at any time t ≤ T , then the value of the option at maturity is PT = (K − ST )+ = max{K − ST , 0}. Based on arbitrage arguments, we can obtain the following formula relating the prices of call and put options. Proposition 1.5 (Put-Call parity). Let C, P denote respectively the prices of a call and a put, both of European type, on the same underlying asset with price S and with the same strike price K and maturity T , then: Ct − Pt = St − K(1 + r)−(T −t) ,

∀t ≤ T.

(1.1)

1.1 Generalities on Financial Markets

Proof. Consider the two investment X and Y , which at time t have value Xt = Ct + K(1 + r)−(T −t) ,

and Yt = St + Pt ,

respectively. This amounts of buying a call option and putting K(1+r)−(T −t) units of currency in a saving account for the investment X, and buying both the underlying asset and a put option for the investment Y . It is easy to check that at time T both investments will have the same value XT = YT = max{ST , K}. Thus, the put-call parity formula (1.1) follows by the law of one price. Derivatives may be used with two objectives: speculation and hedging. The speculator who expects a market crash or a downward inversion in the price of a certain security, may profit substantially by investing in put options for that security. On the other hand, hedging means protecting oneself against the financial risk associated with one’s own portfolio holdings, by reducing the exposure of the portfolio to the random fluctuations of market prices. For instance, a investor holding a considerable amount of a stock, may reduce the exposure of the long position to a possible dramatic decrease in the stock price by buying put options.

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Chapter 2 Single Period Market Models We will present two fundamental problems in the modeling of financial markets, starting from very simplified concrete examples in Section 2.1. Then, in Section 2.2 we formally set up a first discrete market model living on a single period, which allows us to illustrate important economic principles without resorting to any sophisticated probabilistic and mathematical knowledge.

2.1

Risk-Neutral and Arbitrage Pricing

Let us consider a simple example of a financial market model consisting in one risky security with price S and a savings account with value B, which accrues at the constant interest rate r, and there are only two trading dates, the initial date t = 0 and the final date T = 1. For simplicity, we will refer to the savings account as the bond 1 Assume that the final value of the security S1 may take two different values S + , S− depending on the outcome of some event, for example the rolling of a die: if the outcome is in {1, 2}, then the price will be S + , while if the outcome is in {3, 4, 5, 6}, the price will be S − . 1

This should not be misunderstood with the instruments defined in Section 1.1.3, the

zero-coupon bond and the coupon bond. It is just a notation, used here as it is in most of the literature, and it refers to the fact that both savings accounts and bonds accumulate interest according to the market interest rate.

16

2.1 Risk-Neutral and Arbitrage Pricing

17

Risk-Neutral Probability Now, let us assume that the initial price S0 is given by the market quotation, but we are not able to determine the probabilities of the random events driving the price of the security. Time

t=0

Bond

1

Risky security

S0

t=1

S1 =

(1 + r) ( S + , if E1 S − , if E2

We can extract from the data the unique probabilities q, 1 − q to be assigned to the events E1 , E2 , respectively, so that S0 is a risk-neutral price. This is called the risk-neutral probability and is obtained by solving S0 = (1 + r)−1 (qS + + (1 − q)S − )

⇒

q=

S0 (1 + r) − S − . S+ − S−

Note that q ∈ (0, 1) if and only if S − < (1 + r)S0 < S + .

(2.1)

On the other hand, if one of the two strict inequalities is not satisfied, the market admits an arbitrage opportunity (prove it as an exercise). Thus in a reasonable model from the economic point of view, the values (q, 1 − q) determines a probability measure on the family of events {E1 , E2 }. This does not depend on a subjective estimations but it is implicitly contained in the market quotations. Arbitrage Price Let us add another risky security to the model, with price C, which only depends on S, i.e. on the events E1 , E2 . We think of C as the price of a derivative written on the underlying security with price S. Assuming that price of the underlying security is quoted on the market, we are interested in determining the right price for the derivative

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2. Single Period Market Models

Time

t=0

Bond

1

Underlying security

S0

t=1

S1 =

(1 + r) ( S + , if E1 S − , if E2 (

Derivative security

?

C1 =

C + , if E1 C − , if E2

A first approach is to compute the risk-neutral probability for the underlying prices, and then use it to compute the risk-neutral price C0 of the derivative. So, if q is computed as above, we get C0 = (1 + r)−1 (qC + + (1 − q)C − ). A different approach comes directly from the other fundamental problem of the theory and practice of derivatives, i.e. the replication/hedging. Indeed, since we are assuming that the law of one price holds, if we find a strategy of investment whose final value V1 coincides with the final value of the derivative C1 , then the initial investment V0 must be exactly the price of the derivative at time t = 0. The initial value V0 of the replicating portfolio is called the arbitrage-free price for the derivative, in the sense that it is the only price satisfying the law of one price, hence any other price would allow for an arbitrage opportunity to appear. A question immediately arises: how is the arbitrage-free price related to the risk-neutral price? Obviously we hope they coincide.... In our specific example, we want to construct an investment on the underlying security and the bond: let α and β be respectively the units of security and bond held in the portfolio from time t = 0 to the final time t = 1, the value of the portfolio is given by V = αS + βB. We want our portfolio to reach at the final time the same value of the deriva-

2.1 Risk-Neutral and Arbitrage Pricing

19

tive in every possible scenario, i.e. (

V1 = αS + + β(1 + r) = C + , if E1 , V1 = αS − + β(1 + r) = C − , if E2

We have here a linear system of two equations in two unknown variables2 that admits a unique solution given by α=

C+ − C− , S+ − S−

β = (1 + r)−1

S +C − − C +S − . S+ − S−

Therefore, the arbitrage price for the derivative is V0 = αS0 + β = S0

+ − + − C+ − C− −1 S C − C S + (1 + r) . S+ − S− S+ − S−

By rearranging the terms in a linear combination of C + and C − and recalling the expression of q obtained before, we get V0 = C0 = (1 + r)−1 (qC + + (1 − q)C − ), as desired. This situation, where the arbitrage price coincides with the riskneutral price and only depends on the quoted prices at the initial time, was made possible by the assumption that the market is free of arbitrage opportunities and that the derivative is replicable.

Incomplete markets Now suppose that the price C of the second risky security is not completely determined by the price S of the first risky security, for instance they evolve according to the following outcomes of the die rolling: 2

Note that we must have S + 6= S − as already observed in (2.1) to avoid arbitrage

opportunities.

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2. Single Period Market Models

Time

t=0

Bond

1

t=1 (1 + r) (

Risky security S

S0

S1 =

S − , if {3, 4, 5, 6} (

Risky security C

?

C1 =

S + , if {1, 2}

C + , if {1, 2, 3, 4} C − , if {5, 6}

The sample space of this model is made up of three elementary events, to which we want to assign a probability in order to be able to price the derivative: {1, 2}, {3, 4} and {5, 6}. Given the quoted price S0 , we look for a risk-neutral probability Q, specified by Q({3, 4}) = q1 ,

Q({5, 6}) = q2 ,

Q({1, 2}) = 1 − q1 − q2 ,

such that q1 , q2 , q1 + q2 ∈ (0, 1). Analogously to the previous case, we have to solve S0 = (1 + r)−1 ((1 − q1 − q2 )S + + q1 S − + q2 S − ). Unfortunately, this means that there exist infinitely many risk-neutral probabilities for this model, and consequently infinitely many arbitrage prices for the risky security C. Note that if C0 was also observable on the market, than we would have two linear equations and two unknown parameters q1 , q2 , thereby ensuring the existence of a unique risk-neutral probability and a unique risk-neutral price. If we try to find the arbitrage-free price of the derivative, by imposing the replication condition we get the system + + αS + β(1 + r) = C , if {1, 2}, αS − + β(1 + r) = C + , if {3, 4}, αS − + β(1 + r) = C − , if {5, 6},

which is not solvable. Therefore the contingent claim is not replicable and it is not possible to define an arbitrage-free price. Because of this, the market described here is said to be an incomplete market. Although the pricing problem is not unequivocally solvable and we cannot invest in the market

2.2 The Formal Set-Up

21

in a way that assures us a perfect replication of the claim at the final date, we can solve an interesting problem from the point of view of the seller. Precisely, imagine a bank that sells the derivative C at time t = 0: even if they are aware that there is no replicating strategy, they can operate in a way to assure themselves that they will accumulate (at least) enough money to pay out the claim at time T = 1. Indeed, if we eliminate one of the three equations in the system above, we can at least partially hedge the risks of the sale, that is assure the replication in two out of three scenarios. In particular, if we eliminate the third equation and solve for the first two, we obtain a portfolio whose final value is + C , if {1, 2}, V1 =

C + , if {3, 4}, C + , if {5, 6},

This strategy is thus good for the bank, because she is able to replicate the derivative in case one of the events {1, 2}, {3, 4} is realized, and she accumulates even more money than necessary if the event {5, 6} is realized. This is what is called a super-replication.

2.2

The Formal Set-Up

Let us formally define the simple case of a single-period model consisting only of a risk-free security, called the bond, and a risky security, called the stock. We then illustrate the economic principles that will also be considered later on when dealing with more complex models. The model is specified by the following elements: • Two trading dates, the initial date t = 0 and the final date T = 1. • A finite sample space Ω = {ω1 , ω2 , . . . , ωn } with n < ∞. Each elementary event ωi , i = 1, . . . , n, represents a ‘state of the world’, also called scenario; the investors at time t = 0 have no information on the

22

2. Single Period Market Models

outcome, while at time T = 1 they know exactly which scenario has materialized. • A probability measure P on (Ω, P(Ω)), where P(Ω) is the powerset of the sample space, such that P(ωi ) := P({ωi }) > 03 for all i = 1, . . . , n. • A bank account, called the ‘bond’, with initial value B0 = 1 and deterministic final value B1 = 1 + r, where r is the risk-free market interest rate, fixed at time t = 0. • A risky security, called the ‘stock’, whose initial value S0 is known (quoted on the market) and whose final value S1 is a random variable on (Ω, P(Ω)). Definition 2.1. In a single-period market model, a trading strategy is a triple of real constants (V0 , α, β), where V0 denotes the value of the initial investment, while α and β denote the number of shares of the stock and the units of the bond, respectively, to be held in the portfolio over the time interval (0, 1]. The value of the trading strategy, also called portfolio value, at time T = 1 is V1 = αS1 + βB1 .

(2.2)

Note that the investor decides the trading strategy at the initial date, then they apply it over the next time period and wait for the outcome at the final date. We will restrict our consideration to the strategies involving niether cash inflow nor outflow aside the initial investment, therefore the following relation must be satisfied: V0 = αS0 + βB0 = αS0 + β.

(2.3)

This means that the whole and only amount of money invested in the bond and stock over the time interval (0, 1] is exactly given by the initial investment 3

By choosing a non-degenerate probability measure we avoid ambiguities when dealing

with equalities of random variables: if X, Y are two random variables on (Ω, P(Ω), P), usually X = Y means equality P-almost surely, while in our setting it means X(ω) = Y (ω) for all ω ∈ Ω.

2.2 The Formal Set-Up

23

of the strategy. Such trading strategies are called self-financing. Since selffinancing trading strategies are univocally specified by the stock and bond investments α, β, or equivalently by the initial investment V0 and either α or β, henceforth we will shorter the notation and write just (α, β) for a self-financing trading strategy. Definition 2.2. The gain of a trading strategy (V0 , α, β) over the time period (0, 1] is a random variable on (Ω, P(Ω)), defined by G := βr + α(S1 − S0 ).

(2.4)

By simple rearrangement of the equations (2.2),(2.4), we get the following self-financing condition, equivalent to (2.3), that relates the final value to the gain of a trading strategy: V1 = V0 + G.

(2.5)

In order to compare prices at different times, it is convenient to normalize the market values with respect to the bond. Thus, we define the discounted stock price S ∗ as St∗ :=

St , Bt

t = 0, 1,

the discounted value V ∗ of a trading strategy (α, β) as Vt∗ := β + αSt∗ =

Vt , Bt

t = 0, 1,

and the discounted gain G∗ of a trading strategy (α, β) as G∗ := α(S1∗ − S0∗ ). The self-financing condition for the discounted market is the same as for the original market: V1∗ = V0 + G∗ .

(2.6)

24

2. Single Period Market Models

2.3

Economic Considerations and the No-Arbitrage condition

As anticipated in Lecture 1, it is desirable that the mathematical model of a financial market satisfies some reasonable economic principles. In the following, we see three related criteria, the strongest of which will be taken as assumption for the models we study. Definition 2.3. A trading strategy with value V is said to be dominant if there exists another trading strategy, with value V˜ , such that V0 = V˜0 and V1 > V˜1 . Note that by equalities and inequalities of random variables, we mean that they hold in every scenario, for instance V1 (ωi ) > V˜1 (ωi ) for all i = 1, . . . , n in Definition 2.3. The existence of dominant strategies in the market has a few equivalent conditions. Lemma 2.4. There exists a dominant strategy if and only if there exists a trading strategy, with value V , satisfying V0 = 0 and V1 > 0. Proof. If a strategy as in the statement of Lemma 2.4 exists, it is also dominant, since it dominates the strategy that starts with zero money and does not invest in anything. Conversely, if a strategy (α, β) dominates a strategy ˜ then by defining (¯ ¯ = (α, β) − (˜ ˜ we get a strategy as in the (˜ α, β), α, β) α, β), statement of Lemma 2.4. Lemma 2.4 describes the situation where an investor starts with zero money and is able to end up with a sure strictly positive gain. Obviously this situation is neither realistic nor reasonable and must be avoided. An analogous reasoning is valid for a strategy as in Lemma 2.5. Lemma 2.5. There exists a dominant strategy if and only if there exists a trading strategy, with value V , satisfying V0 < 0 and V1 ≥ 0. Proof. Prove it as an exercise.

2.3 Economic Considerations and the NA condition

25

If we interpret the final value of an investment strategy as a contingent claim, the initial value of such investment would be interpreted as the price of such claim. This implies that the existence of a dominating strategy in the market also leads to illogical pricing, because two claims with a surely different value would have the same price at time t = 0. Instead, the pricing of contingent claims would be logically consistent if the price of any claim was unique and a claim that pays more than another one in every state of the world had a higher price at time t = 0. There is a mathematical formalization of these properties. Definition 2.6. A linear pricing measure is a non-negative vector π = (π1 , . . . , πn ) ∈ Rn+ such that the value V of any self-financing trading strategy satisfies V0 =

n X

πi V1∗ (ωi ).

i=1

If there is a linear pricing measure, then the inconsistency disappears. An equivalent way to characterize linear pricing measures is shown in the following lemma and it also gives an explanation for the appellation “measure”. Lemma 2.7. π is a linear pricing measure if and only if it is a probability measure on (Ω, P(Ω)) and satisfies S0∗

=

n X

πi S1∗ (ωi ).

(2.7)

i=1

Proof. If π is a linear pricing measure, by Definition 2.6 we have β+

αS0∗

By taking α = 0, we get

Pn

=

n X

πi (β + αS1∗ (ωi )) .

(2.8)

i=1

i=1

πi = 1, which makes π a probability mea-

sure; then by taking α = 1 we get the equation (2.7). Conversely, if π is a probability measure satisfying (2.7), then (2.8) holds by substitution. Note that, if a linear pricing measure exists, by (2.7) and Definition 2.6, both the initial stock price and the initial value of any strategy are equal to

26

2. Single Period Market Models

the expectation under such probability measure of the final discounted stock price and of the final discounted value of the strategy, respectively. This reminds us of the risk-neutral probability measure seen in the introductory examples of Lecture 2. However, the two objects do not coincide, but we will see that the risk-neutral probability is just a special kind of linear pricing measure. Lemma 2.8 shows that models which exclude dominant strategies can be economically reasonable, in that they allow for a logically consistent pricing of claims. Lemma 2.8. There exists a linear pricing measure if and only if there are no dominant strategies. The proof pivots on the linear programming duality theory and we do not show it. The assumption of non-existence of dominant strategies is stronger than the assumption mentioned in Lecture 1 and formalized as follows: Definition 2.9. The law of one price holds if there do not exist two trading strategies, with value V and V˜ respectively, such that V1 = V˜1 but V0 > V˜0 . It is clear how this law is crucial to avoid ambiguity about the initial price of any claim. Lemma 2.10. If there are no dominant strategies, then the law of one price holds. Proof. If the law of one price does not hold, then by definition there exist ¯ and (˜ ˜ such that V¯1 = V˜1 but V¯0 > V˜0 . This two trading strategies (¯ α, β) α, β) ˜ > G. ¯ By defining a new strategy α = α implies that G ˜−α ¯ , β = −αS0 , this has value V0 = 0, V1 > 0. Thus a dominant strategy exists. Remark 2.11. The converse of Lemma 2.10 is not true. Indeed, you can find a dominant strategy even in a model that satisfies the law of one price. Example 2.12 presents a situation of this kind.

2.3 Economic Considerations and the NA condition

Example 2.12. Consider a single-period market model specified by the following data: the sample space contains only two elementary events, Ω = {ω1 , ω2 }, the interest rate is r = 1, the stock price at time 0 is S0 = 10 and takes the following values at time T = 1, depending of the ‘state of the world’: S1 (ω1 ) = 12, S1 (ω2 ) = 8. In this model, the law of one price is satisfied: for all X = (X1 , X2 ) ∈ R2 , there exists a unique trading strategy (α, β) such that V1 (ω1 ) = αS1 (ω1 ) + β(1 + r) = X1 ,

V1 (ω2 ) = αS1 (ω2 ) + β(1 + r) = X2 .

However, we can define a dominant strategy by taking α = −1 and β = 10, whose value at time 0 is V0 = 0 and at time T = 1 is V1 (ω1 ) = 8 > 0, V1 (ω2 ) = 12 > 0. Once we agreed upon restricting our attention to market models that do not allow for the existence of dominant trading strategies, a further restriction seems necessary: would you allow for trading strategies that start with a null initial investment, do not lose any money and end up with a strictly positive gain in at least one (but not all) scenario? If you care for the equilibrium of the market model, certainly not. This situation corresponds to the concept of arbitrage opportunity mentioned in Lecture 1 and formalized below. Definition 2.13. An arbitrage opportunity is a (self-financing) trading strategy with value V such that: (i) V0 = 0, (ii) V1 ≥ 0, (iii) EP [V1 ] > 0, i.e. P(V1 > 0) > 0. Lemma 2.14. If there exists a dominant strategy, then there exists an arbitrage opportunity. The statement is trivially true, by definition.

27

28

2. Single Period Market Models

Remark 2.15. The converse of Lemma 2.14 is not true. Indeed, you may find an arbitrage opportunity in a model excluding dominant strategies. Example 2.16 presents a situation of this kind. Example 2.16. Consider a single-period market model specified by the following data: the sample space contains only two elementary events, Ω = {ω1 , ω2 }, the interest rate is r = 0, the stock price at time 0 is S0 = 10 and takes the following values at time T = 1, depending of the ‘state of the world’: S1 (ω1 ) = 12, S1 (ω2 ) = 10. The strategy defined by α = 1, β = −10 is an arbitrage opportunity, because: V0 = 0 and V1 (ω1 ) = αS1 (ω1 ) + β(1 + r) = 12 − 10 = 2, V1 (ω2 ) = αS1 (ω2 ) + β(1 + r) = 10 − 10 = 0. Nevertheless, we can prove that there are no dominant strategies in the market. Indeed: π = (0, 1) is a linear pricing measure, by Lemma 2.7, then Lemma 2.8 concludes the proof. The set of conditions in Definition 2.13 can be equivalently rewritten in terms of the gain of the trading strategy. Lemma 2.17. (α, β) is an arbitrage opportunity if and only if it satisfies (i) and its gain G satisfies: (a) G ≥ 0, (b) EP [G] > 0. Proof. Suppose that (α, β) is an arbitrage opportunity. Then, by definition, G = V1 − V0 ≥ 0 and EP [G] = EP [V1 ] − V0 = EP [V1 ] > 0. Conversely, suppose that (i), (a) and (b) are satisfied by some trading strategy (α, β). Then V1 = V0 + G = G ≥ 0 and EP [V1 ] = EP [G] > 0. We can resume the relation between the models satisfying the conditions explained in this section as: NA ⊂ No dominant strategies ⊂ law of one price,

2.4 The Binomial Model on a Single Period

where we denote by NA the no-arbitrage condition, i.e. the non-existence of arbitrage opportunities. This means that market models can be divided in four categories: there are no arbitrage opportunities, there are arbitrage opportunities but no dominant strategies, there are dominant strategies but the law of one price holds, and the law of one price does not hold. An example of model that does not even satisfy the law of one price is given in Example 2.18. Example 2.18. Consider the market model specified in Example 2.12 but change the specification of the stock price at time T = 1 in the state ω2 as S1 (ω2 ) = S1 (ω1 ) = 12. Then, for any given real constant X ∈ R, there exist infinitely many trading strategies whose final value is V1 = X. That is because we have to solve one equation in two variables: V1 = αS1 + β(1 + r) = X. We want to work with models of the first category, that is satisfying the NA condition. This is of course a theoretical assumption, because in real markets much more frictions are involved and the markets are far from being ideal. Nevertheless, in very liquid markets any arbitrage possibility that arises is quickly eliminated (by all the investors taking advantage of it) and the market equilibrium is restored. So our assumption is reasonable. The NA condition is not easy to be checked, but we will see in future chapters a fundamental theorem that provides us with an equivalent condition involving a special probability measure.

2.4

The Binomial Model on a Single Period

A special case of this simple one period model is given by the so called Binomial or Cox-Ross-Rubinstein (CRR) model, from the names of the authors who introduced it in 1979. We will consider here the single-period case of the model, which is in general a multi-period model. We will study in detail the general setting of the binomial model in the next chapter.

29

30

2. Single Period Market Models

Assumption 2.4.1 (Single-period binomial model). The initial stock price S0 is strictly positive and the stock price at the final date T = 1 is proportional to the initial stock price, in the following way: S1 = S0 ξ, where ξ is a Bernoulli-like random variable, taking only two values u, d, u > d > 0, with probabilities P(ξ = u) = p,

P(ξ = d) = 1 − p.

The sample space is the path space, that is the set of possible trajectories for the stock price: Ω = {(S0 , S0 u), (S0 , S0 d)}. Definition 2.19. A European contingent claim (ECC) is represented by a random variable X on (Ω, P(Ω)) and it is said to be attainable if there exists a self-financing trading strategy with final value V1 = X. Such trading strategy, if it exists, is called a replicating, or hedging, strategy for X. Now, given a European claim X, consider the problem of hedging, that is to find the hedging strategy for X, if it exists. Since the claim is a random variable on the path space, it can take only two values, which we denote by X u in the scenario (S0 , S0 u) and X d in the scenario (S0 , S0 d). By imposing the replicating condition, we get: (

αS0 u + β(1 + r) = X u αS0 d + β(1 + r) = X d

Solving for α, β we find Xu − Xd , (u − d)S0 uX d − dX u β = (1 + r)−1 u−d

α =

(2.9) (2.10)

2.4 The Binomial Model on a Single Period

31

Thus the initial investment needed to replicate the derivative is V0 = αS0 + β 1 (1 + r − d)X u + (u − (1 + r))X d = (1 + r)(u − d) = (1 + r)−1 qX u + (1 − q)X d where 1+r−d . (2.11) u−d Note that q ∈ (0, 1) if and only if d < 1 + r < u. In this case, we can define q=

a probability measure Q by Q(ξ = u) = q,

Q(ξ = d) = 1 − q,

(2.12)

and the initial value of the hedging strategy can be rewritten as V0 = EQ [X ∗ ] ,

(2.13)

where X ∗ = (1 + r)−1 X is the discounted claim. Furthermore, the initial stock price can also be rewritten as the expectation under Q of the discounted final stock value: S0 = EQ [S1∗ ]. Thus, Q is the unique risk-neutral probability of the model. The appellation “risk-neutral” refers to the fact that, under Q, a risk-neutral investor, i.e. an investor who has no preference between the expected value of a payoff and the random payoff itself, is indifferent to investing either on shares of the stock or on bonds, since they have the same expected payoff. Note that the risk-neutral probability is only a theoretical object and it is unrelated to the market (or real-world, or objective) probability P that specifies the dynamics of the stock price in the model. We remark that the only property shared by P and Q is that they both assign a strictly positive value to every possible event. Theorem 2.20. The single-period binomial model satisfies the NA condition if and only if d < 1 + r < u.

32

2. Single Period Market Models

Proof. We leave the proof of the implication ⇒ of the statement for homework. (⇐) If d < 1+r < u, we have already proved that we can define a probability measure Q by (2.11) and that under this measure the initial stock price S0 is equal to the expected value of the discounted stock price at time T = 1. Suppose that an arbitrage opportunity (α, β) exists, by Definition 2.13, it must satisfy the conditions (i)-(iii). Now, (i) translates to V0 = αS0 + β = 0 and (ii) to V1 = αS1 + β(1 + r) ≥ 0. By taking the expectation under Q, we get EQ [V1 ] = αEQ [S1 ] + β(1 + r) = α(1 + r)S0 + β(1 + r) = (1 + r)V0 = 0. Thus, condition (iii) cannot be satisfied: we found a contradiction. Henceforth we assume that the model satisfies the NA condition: Assumption 2.4.2. We consider a single-period binomial model with d < 1 + r < u. We have also seen that for any given European claim X there exists a hedging strategy, defined in (2.9)-(3.4). By the lemmas proved in Section 2.3, the law of one price holds, hence the time 0 value of the hedging strategy determines the price of the derivative. At this point, another concern arises: by introducing the derivative security X in the market, with price determined by (2.13), are we introducing a possibility for arbitrage opportunities? We would like that the price of the derivative is such that also the secondary market composed of the bond, the stock, and the derivative itself, is free of arbitrage opportunities. In this case, the price of the derivative is called the arbitrage-free price.

2.4 The Binomial Model on a Single Period

Fortunately, this is true for the initial value of the hedging strategy: Theorem 2.21 (Thm 2.2.1 in [5]). V0 = EQ [X ∗ ] is the unique arbitrage-free price for the European contingent claim X. We defer the proof to the multi period case.

33

Chapter 3 Finite Market Models The single period market model set up in Section 2.2 can be extended to the multi-period case. Before considering the general model set-up, we will focus on the most well-known model in this family, that is the Binomial (or CRR) Model.

3.1

The (Multi-Period) Binomial Model

The general binomial model is the extension of the single period case presented in Section 2.4. Assumption 3.1.1 (Binomial model). The model is specified by the following elements: • Finitely many trading dates, to which we assign successive integers t = 0, 1, 2, . . . , T,

T ∈ N.

• The bond, whose value dynamics is described by the stochastic process B = {Bt }t=0,1,...,T , which is in fact deterministic, given by B0 = 1 and

Bt = (1 + r)t , t = 1, 2, . . . , T,

where r > 0 is the constant risk-free interest rate on a unitary trading period. 34

3.1 The (Multi-Period) Binomial Model

35

• The stock, whose price dynamics is described by the stochastic process S = {St }t=0,1,...,T driven by a family of idependent and identically distributed Bernoulli-like random variables {ξt , t = 1, 2, . . . , T }. Precisely, the initial stock price S0 is a strictly positive constant (quoted on the market) and the price at any time t = 1, 2, . . . , T is proportional to the price at the previous date: St = St−1 ξt ,

where P ξt = pδu + (1 − p)δd ,

where 0 < d < 1 + r < u in order to make the model arbitrage-free, analogously to the single-period case (see Theorem 2.20). • The probability space (Ω, F, P), composed of the path space ! ) ( T Y ξt , ξt = u, d, t = 1, 2, . . . , T , Ω= S0 , S0 ξ1 , . . . , S0 t=1

containing all possible realizations of the stock price trajectory (S0 , S1 , . . . , ST ), the σ-algebra F = P(Ω), and the probability measure P associated to the distribution P ξ , i.e. P (S0 , S0 u, S0 u2 , . . . , S0 uT ) = pT , P (S0 , S0 d, S0 du, . . . , S0 duT −1 ) = (1 − p)pT −1 , and so on. • The natural filtration F = (Ft )t=0,...,T of the stock price process, i.e. Ft = σ ({S0 , . . . , St }) ,

t = 0, 1, . . . , T,

representing the flow of information available to the investor at each time. Note that, in particular, F0 = {∅, Ω} is the trivial σ-algebra, and FT = F. We have to define trading strategies in this context. Recalling Definition 3.21 in the single-period case, a trading strategy is specified by an initial investment and the quantity of securities to be held in the portfolio over

36

3. Finite Market Models

each trading period. In the multi-period case, we have T + 1 trading dates, i.e. T periods; we need stochastic processes in order to model such holdings. A first classification is based on which securities we consider available for trading: a trading strategy in the primary market can include in the portfolio only the bond and the stock; a trading strategy in the secondary market can also include a derivative. This second type of strategies is defined as a straightforward extension of the first type and is considered when looking for the arbitrage price of the derivative itself. When not otherwise specified, we consider trading strategies in the primary market. Definition 3.1. A trading strategy is a triple (V0 , α, β), where V0 ∈ R denotes the value of the initial investment, and α = (αt )t=1,...,T , β = (βt )t=1,...,T are real-valued F-predictable stochastic processes where, for all t = 1, . . . , T , αt , βt denote respectively the number of shares of the stock and the units of the bond to be held in the portfolio over the time interval (t−1, t]. α and β are respectively called the stock holding process and bond holding process. The portfolio value process V α,β = (Vtα,β )t=1,...,T of the trading strategy (V0 , α, β) is an adapted stochastic process on (Ω, F, F, P), defined by Vtα,β = αt St + βt Bt ,

t = 1, . . . , T.

(3.1)

Note that the predictability of the holding processes with respect to the filtration generated by the past (and present) stock prices is a natural assumption: the investor can only decide based on the market information up to that time, and an investment decision taken and implemented at a time t, has to be preserved until the next trading date t + 1. As we did in Section 2.2, we concentrate on those trading strategies involving no cash inflow nor outflow aside the initial investment, defined as follows. Definition 3.2. A trading strategy (V0 , α, β) is said to be self-financing if it satisfies V0 = α1 S0 + β1 ,

(3.2)

3.1 The (Multi-Period) Binomial Model

37

and αt St + βt Bt = αt+1 St + βt+1 Bt ,

t = 1, . . . , T − 1

(3.3)

The self-financing property (3.3) means that at any trading date t the portfolio holdings are rebalanced, i.e. αt , βt are changed into αt+1 , βt+1 in a way that no external money is needed and no money is withdrawn. Remark 3.3. A self-financing trading strategy is univocally specified by the stock and bond holding processes α, β, or equivalently by the initial investment V0 and either α or β. Indeed: (α,β)

• Given α, β, V0 is specified by (3.2). We denote it by V0

.

• Given α, V0 , the equation (3.3) implies that, for all t = 1, . . . , T − 1, βt+1 = βt −

St (αt+1 − αt ) Bt

= ··· t X Sn (αn+1 − αn ), = V0 − α1 S0 − B n n=1

(3.4)

which specifies β. Viceversa, the specification (3.4) implies (3.3), that is makes the strategy self-financing. The self-financing property (3.3) means that at any trading date t the number of units of the bond sold (respectively bought) is determined by the amount of money needed to change (respectively obtained by changing) the number of shares of the stock held in the portfolio from αt into αt+1 . Since we only work with self-financing trading strategies, henceforth we will denote by just (α, β) the self-financing trading strategy with initial value V0α,β given by (3.2), according to Remark 3.3. Definition 2.13 is trivially extended to the multi-period case. Definition 3.4. An arbitrage opportunity is a self-financing trading strategy (α, β) such that:

38

3. Finite Market Models (i) V0α,β = 0, (ii) VTα,β ≥ 0, (iii) EP [VTα,β ] > 0, i.e. P(VTα,β > 0) > 0. The definition of the normalized market with respect to the bond defined in Section 2.2 extends to the multi-period model: we call ! α,β S V t t S∗ = , and V ∗ = Bt t=0,...,T Bt

t=0,...,T

the discounted stock price process and the discounted value of the trading strategy (α, β), respectively. The self-financing conditions (3.2)(3.3) have an equivalent for the discounted market. Remark 3.5. A trading strategy (V0 , α, β) is self-financing if and only if the correspondent discounted value process V ∗ satisfies (3.2) and Vt∗ = αt+1 St∗ + βt+1

∀t = 1, . . . , T − 1.

(3.5)

Indeed, for all t = 1, . . . , T − 1: αt+1 St∗ + βt+1 = (1 + r)−t (αt+1 St + βt+1 Bt ) = (1 + r)−t (αt St + βt Bt )

by (3.3)

= αt St∗ + βt , and viceversa. In the single-period case, we showed that every European contingent claim is attainable, by finding the replicating trading strategy. Is that also true in the multi-period case? We proceed in the same way: given a ECC X with maturity T , we seek a self-financing trading strategy (α, β) such that the correspondent portfolio replicates the claim at maturity, i.e. VTα,β = X. Remark 3.6. Note that, by the extension of Lemma A.5 to a finite number of random variables, being measurable with respect to FT = σ(S0 , S1 , . . . , ST ), X can be written as a function of the stock prices up to maturity, i.e. X = fT (S0 , S1 , . . . , ST ),

fT : RT +1 → R measurable.

3.1 The (Multi-Period) Binomial Model

39

This time, the trading strategy is not completely determined at the initial time t = 0, but is instead a predictable stochastic process. Thus, we have to work backward on the nodes of the binary tree in order to find at each time the stock and bond allocations that guarantee the replication of the claim and satisfy the self-financing condition. First, we assume to be at time T − 1, hence to know the stock prices S0 , S1 , . . . , ST −1 , and we impose the replicating condition in the two possible scenarios ST = ST −1 u and ST = ST −1 d to find the allocations on the last time period (T − 1, T ]: ( αT ST −1 u + βT (1 + r)T = X u αT ST −1 d + βT (1 + r)T = X d

,

where X u := fT (S0 , S1 , . . . , ST −1 , ST −1 u), X d := fT (S0 , S1 , . . . , ST −1 , ST −1 d). Solving for αT , βT we find Xu − Xd , (u − d)ST −1 uX d − dX u . = (1 + r)−T u−d

αT = βT

The capital required at time T − 1 in order to hedge the claim with a self-financing investment is T −1 VTα,β −1 = αT ST −1 + βT (1 + r)

= (1 + r)−1 qX u + (1 − q)X d

= (1 + r)−1 EQ [X|FT −1 ] where q =

1+r−d u−d

∈ (0, 1) and Q is the probability measure defined by it, as

in (2.12). Then, we proceed backward in time using the same reasoning: for all t = T − 1, . . . , 1, we assume to be at time t − 1, i.e. we condition our information on Ft−1 , and impose the replicating condition for Vtα,β , computed at the previous step as Vtα,β = αt+1 St + βt+1 (1 + r)t =

1 EQ [X|Ft ]. T −t (1 + r)

40

3. Finite Market Models Always according to Remark 3.6, since Vtα,β is an Ft -measurable random variable, we can rewrite it terms of a measurable function of the first t stock prices: Vtα,β = ft (S0 , . . . , St ),

ft : Rt+1 → R measurable.

Then we can find the allocations on the time period (t − 1, t]: ( αt St−1 u + βt (1 + r)t = Vtu := ft (S0 , . . . , St−1 u), αt St−1 d + βt (1 + r)t = Vtd := ft (S0 , . . . , St−1 d). The system is solved by Vtu − Vtd = , (u − d)St−1 uV d − dVtu = (1 + r)−t t , u−d

αt βt

(3.6) (3.7)

which give a value equal to α,β Vt−1 = αt St−1 + βt (1 + r)t−1 1 EQ [Vtα,β |Ft−1 ] = 1+r Q 1 Q = E E [X|F ]|F t t−1 (1 + r)T −t+1 1 EQ [X|Ft−1 ]. = (1 + r)T −t+1

(3.8)

at time t − 1. Thus, by induction, the formula (3.8) holds for every t = T, . . . , 1. To resume, we proved that any European contingent claim X is attainable, and we computed the hedging strategy and the correspondent portfolio value at all trading times. In particular, since F0 is the trivial σ-algebra and hence EQ [X|F0 ] = EQ [X], the initial investment of the hedging strategy is V0α,β = (1 + r)−T EQ [X], = EQ [X ∗ ], where X ∗ = (1 + r)−T X is the discounted value of the contingent claim.

3.1 The (Multi-Period) Binomial Model

41

Note that, by the law of one price, the only candidate for the price of the derivative is V0α,β . As in Section 2.4, when pricing a European derivative, we not only want to respect the law of one price, but we also want not to introduce arbitrage opportunities in the market by issuing the derivative instrument, that is we want an arbitrage-free price for X. In order to prove the analogue of Theorem 2.21, we need to define trading strategies and arbitrage opportunities in the secondary market. Definition 3.7. A (semi-static) trading strategy in the secondary market (S, B, X) is a collection (V0 , α, β, γ), where V0 ∈ R denotes the value of the initial investment, α = (αt )t=1,...,T and β = (βt )t=1,...,T are R-valued Fpredictable stochastic processes representing the stock and bond holdings respectively, and γ ∈ R represents the number of derivatives to be held in the portfolio over the entire time period (0, T ]. The portfolio value at time t is given by Vtα,β,γ = γCt + αt St + βt Bt ,

t = 1, . . . , T

(3.9)

where Ct is the price of the derivative X at time t, for t = 1, . . . , T , and in particular CT = X. A strategy (V0 , α, β, γ) in the secondary market (S, B, X) is self-financing if and only if (α, β) satisfies (3.3) and V0 = γC0 + α1 S0 + β1 .

(3.10)

As usual, we identify a self-financing trading strategy with its holding processes only, since the initial investment is univocally determined by (3.10): V0α,β,γ = γC0 + α1 S0 + β1 . Arbitrage opportunities in the secondary market are defined as in Definition 3.4, replacing V α,β with V α,β,γ . As in the single-period binomial model, it turns out that the initial value of the hedging strategy is the fair price for the contingent claim, because it is the only one that satisfies the law of one price and does not introduce arbitrage opportunities in the secondary market.

42

3. Finite Market Models

Theorem 3.8 (Thm 2.2.4 in [5]). C0 = EQ [X ∗ ]

(3.11)

is the unique arbitrage-free price at time 0 for the ECC X. In order to prove the theorem, we need the following fundamental property that characterizes all discounted processes in the market. Lemma 3.9. The discounted stock price process S ∗ and the discounted value process of any self-financing trading strategy are (F, Q)-martingales. Proof. Both S ∗ and V ∗ are clearly adapted to F and Q-integrable, by definition. For all t = 1, . . . , T , by independence of ξt and the stock prices {S0 , S1 , . . . , St−1 }, we have EQ [St∗ |Ft−1 ] = (1 + r)−t EQ [St−1 ξt |Ft−1 ] = (1 + r)−t St−1 EQ [ξt |Ft−1 ] ∗ = (1 + r)−1 St−1 EQ [ξt ] u−1−r 1+r−d −1 ∗ +d = (1 + r) St−1 u u−d u−d ∗ = St−1 .

Then, by using the martingale property for S ∗ just proved, for all t = 1, . . . , T we have EQ [Vt∗ |Ft−1 ] = αt EQ [St∗ |Ft−1 ] + βt ∗ = αt St−1 + βt ∗ = Vt−1 ,

by (3.5)

which completes the proof. Proof of Theorem 3.8. (Uniqueness) We show that if the price of the claim C0 was different from (3.11), then arbitrage opprotunities in the secondary market (S, B, X) would exist. Let us denote (α, β) the hedging strategy for X as defined previously

3.1 The (Multi-Period) Binomial Model via the backward procedure. Suppose that C0 > V0α,β . Consider the following investment: at time 0 sell one contingent claim at the price C0 , invest in the primary market according to (α, β), and buy C0 − V0 additional units of the bond, to be kept in the portfolio over the entire period (0, T ]. At each time t = 1, . . . , T − 1, trade according to (α, β). The one just described is a ˜ γ) in the secondary market self-financing (semi-static) trading strategy (˜ α, β, (S, B, X) , defined by α ˜ = α, β˜ = β + C0 − V0 , and γ = −1. Its initial value is ˜

˜ β,γ = −C0 + α1 S0 + β1 + C0 − V0 = 0, V0α,

and its final value at time T is ˜

˜ β,γ VTα, = − X + αT ST + (βT + C0 − V0 )BT

= − X + X + (C0 − V0 )BT > 0, thanks to the fact that (α, β) replicates X. This is an arbitrage opportunity. Now, suppose that C0 < V0α,β . The self-financing (semi-static) trading ˜ −γ) in the secondary market (X, S, B) is clearly an arbistrategy (−˜ α, −β, trage. So, whenever the price is different from (3.11), we found an arbitrage opportunity. (Arbitrage-free price) Finally, we have to show that if C0 = V0α,β , there ¯ γ¯ ) be are no arbitrage opportunities in the secondary market. Let (¯ α, β, a self-financing (semi-static) trading strategy in (S, B, X) with null initial investment, i.e. ¯

¯ β,¯ γ V0α, = γ¯ C0 + α ¯ 1 S0 + β¯1 = 0,

and non-negative final value, i.e. ¯γ ¯ β,¯ VTα, = γ¯ X + α ¯ T ST + β¯T BT ≥ 0.

¯ is a self-financing strategy in the primary market We remark that (¯ α, β) (S, B), thus, by Lemma 3.9, its discounted value process is a martingale

43

44

3. Finite Market Models

under the probability measure Q. Then: h i ¯γ ¯ β,¯ ¯ β¯ EQ [VTα, ] = EQ VTα, + γ¯ X α, ¯ β¯ T Q ∗ = (1 + r) V0 + γ¯ E [X ] ¯ 1 S0 + β¯1 + γ¯ C0 = (1 + r)T α ¯

¯ β,¯ γ = (1 + r)T V0α,

= 0.

Corollary 3.10. For any t ∈ {0, 1, . . . , T }, Ct =

1 EQ [X|Ft ] (1 + r)T −t

is the unique arbitrage-free price at time t for the ECC X.

3.1.1

Hedging and Pricing American Contingent Claims

American derivatives are characterized by giving the owner the possibility of exercise, i.e. cash in the payoff of the claim, at any time between the purchase and the maturity of the contract. Definition 3.11. An American contingent claim (ACC) with maturity T ∈ N is a R-valued F-adapted stochastic process. Any F-stopping time with values in {0, 1, . . . , T } is called an exercise time. For all what follows, let us consider an ACC Y = (Yt )t=0,...,T . For any t ∈ {0, 1, . . . , T }, the random variable Yt is the payoff of the claim if the owner of the contract decides to exercise the claim at time t. Intuitively, the measurability condition depends on the fact that the payoff Yt is known given the information available at time t. In the same way, the decision to exercise the claim at a certain time t, can only be taken based on the market information available at that time, given by Ft . That justifies the fact that the exercise time, i.e. the moment in which the owner decides to exercise

3.1 The (Multi-Period) Binomial Model

the claim, is required to be a stopping time with respect to the filtration F. Moreover, taking any exercise time τ only finite values, the random variable Yτ is well defined and represents the payoff of the ACC relative to τ . For any s, t ∈ {0, 1, . . . , T }, s ≤ t, we denote by T[s,t] the set of exercise times taking values in the interval [s, t], that is T[s,t] := {τ F − stopping time : τ ∈ {s, s + 1, . . . , t}}. In general, unlike European claims, an American claim is not attainable, that is there does not exist a self-financing trading strategy whose value exactly replicates the payoff relative to any exercise time. However, the seller of an ACC can seek a super-hedging strategy to cover the risks coming from all possibile exercise times chosen by the long position investor. Definition 3.12. For any t ∈ {0, 1, . . . , T }, we denote by Ut the minimum wealth that the seller needs to have in the portfolio in order to be able to replicate the payoff relative to any exercise time in the remaining time to maturity, that is Yτ for all τ ∈ T[t,T ] . We call U = (Ut )t=0,...,T the minimum wealth process for Y . A super-hedging (or super-replicating) strategy for Y is a self-financing trading strategy (α, β) whose value is at least as great as the minimum wealth U at any time, that is Vtα,β ≥ Ut

∀t ∈ {0, 1, . . . , T }.

Note that, from the definition of the minimum wealth process, U is an adapted process such that UT = YT . The procedure to find the super-hedging strategy for the ACC Y is similar to the one to find the replicating strategy for a European claim and takes into account the arguments presented above. We proceed backwards on the nodes of the binary tree as follows. First, conditioned on FT −1 , we only have two payoffs to cover, the one relative to the exercise time T − 1, YT −1 and the final payoff YT = UT correspondent to the situation where the buyer waits until maturity to cash in the claim. Since YT −1 is known at time T − 1, it remains to compute the portfolio allocations

45

46

3. Finite Market Models

that allow to replicate the final payoff, exactly as we were dealing with an European claim. So, we impose the replicating condition in the two possible scenarios {ST = ST −1 u} and {ST = ST −1 d}: (

α ˜ T ST −1 u + β˜T (1 + r)T = UTu := fT (S0 , . . . , ST −1 , ST −1 u) , α ˜ T ST −1 d + β˜T (1 + r)T = UTd := fT (S0 , . . . , ST −1 , ST −1 d)

where YT = fT (S0 , . . . , ST −1 , ST ),

fT : RT +1 → R measurable.

Solving for α ˜ T , β˜T we find the allocations UTu − UTd , (u − d)ST −1 uU d − dUTu = (1 + r)−T T , u−d

α ˜T = β˜T

which are financed at time T − 1 by a capital α ˜ T ST −1 + β˜T BT −1 = (1 + r)−1 EQ [UT |FT −1 ] where Q is the risk-neutral probability defined in (2.12). So, the minimum wealth needed by the seller at time T − 1 in order to be protected in the two possible situations of the buyer exercising either at time T − 1 or at maturity T is UT −1 = max YT −1 , (1 + r)−1 EQ [UT |FT −1 ] . Moreover, let us denote δ˜T := UT −1 − (1 + r)−1 EQ [UT |FT −1 ] the excess wealth in UT −1 over what is needed to cover the payoff at maturity. Then, we analogously proceed backward in time. Precisely, for all t = T − 1, . . . , 1, conditioned on Ft−1 , we impose the replicating condition for Ut , computed at the previous step as Ut = max Yt , (1 + r)−1 EQ [Ut+1 |Ft ] ,

3.1 The (Multi-Period) Binomial Model

47

in order to find the allocations in stock and bond over the period (t − 1, t] needed to cover the payoff relative to all exercise times in T[t,T ] , i.e. Utu − Utd , (u − d)St−1 uU d − dUtu = (1 + r)−t t , u−d

α ˜t =

(3.12)

β˜t

(3.13)

where Utu , Utd are respectively the values of Ut in the scenarios {St = St−1 u}, {St = St−1 d}. Then, we compare the wealth needed at time t − 1 to finance such strategy over the period (t − 1, t], i.e. α ˜ t St−1 + β˜t Bt−1 = (1 + r)−1 EQ [Ut |Ft−1 ], with the payoff at time t − 1 of the ACC and we take the maximum among the two of them: Ut−1 = max Yt−1 , (1 + r)−1 EQ [Ut |Ft−1 ] . Finally, we define the excess wealth in Ut−1 over what is needed to finance ˜ over (t − 1, t], i.e. the strategy (˜ α, β) δ˜t := Ut−1 − (1 + r)−1 EQ [Ut |Ft−1 ].

(3.14)

By induction, the above formulas hold for every t = T, . . . , 1. To resume, we have iteratively defined the minimum wealth process U for the seller of an ACC Y : UT = YT U = max Y , (1 + r)−1 EQ [U |F ] , t = 1, . . . , T, t−1 t−1 t t−1

(3.15)

˜ replicating U . Note that (˜ ˜ is and we found a trading strategy (˜ α, β) α, β) however not self-financing, because ∀t = 1, . . . , T − 1,

α ˜ t St + β˜t Bt = Ut α ˜ t+1 St + β˜t+1 Bt = (1 + r)−1 EQ [Ut+1 |Ft ]

We also defined the excess wealth process δ˜ = (δ˜t )t=1,...,T in (3.14)

(3.16)

48

3. Finite Market Models

Our objective is to find a super-hedging strategy for Y . Unfortunately ˜ is not self-financing. So, we now define another trading strategy (˜ α, β) ˜ and, addi(α∗ , β ∗ ) that trades in the stock and bond according to (˜ α, β) tionally, it invests the excess wealth at each time in the bond and keeps that position fixed until maturity, i.e. αt∗ = α ˜t,

(3.17)

βt∗ = β˜t +

t X n=1

δ˜n , Bn−1

∀t = 1, . . . , T.

(3.18)

The following lemma shows the properties of the trading strategy defined in (3.17)-(3.18) and of the minimum wealth process defined in (3.15). Lemma 3.13. The trading strategy (U0 , α∗ , β ∗ ) is a super-hedging strategy for the ACC Y . Moreover, the random time τ0∗ := min{t ≥ 0 : Ut = Yt } is an F-stopping time and Vtα

∗ ,β ∗

= Ut

∀t : 0 ≤ t ≤ τ0∗ .

(3.19)

Proof. First, we have to check the self-financing condition. For all t = 1, . . . , T − 1, ∗ ∗ αt+1 St + βt+1 Bt = α ˜ t+1 St + β˜t+1 Bt +

t+1 X δ˜n Bt B n−1 n=1

= (1 + r)−1 EQ [Ut+1 |Ft ] + = Ut +

t X δ˜n Bt + δ˜t+1 B n−1 n=1

t X δ˜n Bt B n−1 n=1

=α ˜ t St + β˜t Bt + = αt∗ St + βt∗ Bt .

t X δ˜n Bt B n−1 n=1

by (3.16) by (3.14) by (3.16)

3.1 The (Multi-Period) Binomial Model

49

It remains to prove the condition at time t = 0: ˜ 1 S0 + β˜1 + δ˜1 α1∗ S0 + β1∗ = α =α ˜ 1 S0 + β˜1 + U0 − EQ [U1 ] = U0 . Here, we used the fact that F0 is the trivial σ-algebra to write EQ [U1 |F0 ] = EQ [U1 ]. The value of the self-financing strategy (α∗ , β ∗ ) at any time t = 1, . . . , T is ∗ ∗ Vtα ,β

=

αt∗ St

+

βt∗ Bt

=α ˜ t St + β˜t Bt +

t X δ˜n Bt ≥ Ut , Bn−1 n=1

(3.20)

because δ˜t ≥ 0 for all t = 1, . . . , T . Thus (α∗ , β ∗ ) is a super-hedging strategy for the ACC Y . Then, τ0∗ ∈ T[0,T ] , because 0 ≤ τ0∗ ≤ T by definition and since UT = YT , and for all t = 1, . . . , T {τ0∗ = t} =

t−1 T

{Yn < Un } ∩ {Yt = Ut } ∈ Ft .

n=0

Finally, for all t = 0, . . . , τ0∗ − 1, Yt < Ut , hence δ˜t+1 = 0. This implies Vtα

∗ ,β ∗

= Ut for all t = 0, . . . , τ0∗ , by (3.20).

Before going on, let us recall a fundamental theorem in Probability theory that is a very useful tool when dealing with stopped processes, as in the case of American contingent claims. Theorem 3.14 (Doob’s Optional Sampling). Let τ, ν be two bounded Fstopping times on a probability space (Ω, F, P), such that τ ≤ ν ≤ T P-almost surely for some T ∈ N. If X is a sub-martingale, then Xτ ≤ E[Xν |Fτ ]. In particular, if X is a martingale, then Xτ = E[Xν |Fτ ].

50

3. Finite Market Models Let Y ∗ = {Yt∗ }t=0...,T , U ∗ = {Ut∗ }t=0...,T denote the discounted payoff and minimum wealth for the seller, respectively, that is Yt∗ = (1 + r)−t Yt ,

Ut∗ = (1 + r)−t Ut ,

∀t = 0, . . . , T.

Note that UT∗ = YT∗ and ∗ |Ft ] , Ut∗ = max Yt∗ , EQ [Ut+1

∀t = 0, . . . , T − 1.

(3.21)

The process U ∗ is called the Snell envelope of Y ∗ . Lemma 3.15. The Snell envelope U ∗ of Y ∗ has the following properties. (i) U ∗ is the smallest (F, Q)-super-martingale that dominates Y ∗ , i.e. such that U ∗ ≥ Y ∗ . (ii) For all t = 0, 1, . . . , T , the random time τt∗ := min{s ≥ t : Us = Ys }

(3.22)

is an element of T[t,T ] . (iii) For all t = 0, 1, . . . , T , the stopped process

Uτ∗t∗ ∧s

, is an s=t,t+1,...,T

(F, Q)-martingale. (iv) For all t = 0, 1, . . . , T , Ut∗ = max EQ [Yτ∗ |Ft ] = EQ [Yτ∗t∗ |Ft ]. τ ∈T[t,T ]

(3.23)

The financial interpretation of (3.23) is that the Snell envelope is the risk-neutral (conditional) expectation of the discounted payoff relative to the first future optimal exercise time. A detailed discussion on the meaning of optimal exercise of ACCs concludes the present sub-section. Proof. (i). The proof is part of an exercise sheet. (ii). We have already proved it for t = 0, and the proof for t = 1, . . . , T is analogous.

3.1 The (Multi-Period) Binomial Model

(iii). Let us prove it for t = 0, for t = 1, . . . , T being analogous. First note that Uτ∗0∗ ∧· is integrable, since the probability space is finite; then, for all t = 1, . . . , T − 1, ∗ − Ut∗ ) Uτ∗0∗ ∧(t+1) − Uτ∗0∗ ∧t = 1{τ0∗ >t} (Ut+1

∗ ∗ = 1{τ0∗ >t} Ut+1 − EQ [Ut+1 |Ft ] , ∗ since Ut∗ = EQ [Ut+1 |Ft ] for all t ≤ τ0∗ by (3.21) and the definition of τ0∗ . Thus,

taking the conditional expectation with respect to Ft , we get ∗ ∗ − Ut+1 |Ft = 0, EQ [Uτ∗0∗ ∧(t+1) − Uτ∗0∗ ∧t |Ft ] = 1{τ0∗ >t} EQ Ut+1 since {τ0∗ > t} ∈ Ft by Remark A.39. (iv). Let t ∈ {0, 1 . . . , T } and τ ∈ T[t,T ] arbitrary. By (i) and Remark A.43, (Uτ∗∧s )s=t,t+1,...,T , is a (F, Q)-super-martingale, hence Ut∗ = Uτ∗∧t ≥ EQ [Uτ∗∧T |Ft ] = EQ [Uτ∗ |Ft ] ≥ EQ [Yτ∗ |Ft ]. Since τ was arbitrarily chosen, we conclude that Ut∗ ≥ maxτ ∈T[t,T ] EQ [Yτ∗ |Ft ]. Then, we have to prove that there exists an exercise time achieving the equality, in particular that τt∗ does so. By (iii) and Theorem 3.14, we have Ut∗ = Uτ∗t∗ ∧t = EQ [Uτ∗t∗ ∧T |Ft ] = EQ [Uτ∗t∗ |Ft ] = EQ [Yτ∗t∗ |Ft ]. This ends the proof. It remains to determine, if it exists, the arbitrage-free price for the ACC, that is a price which does not allow for abritrage opportunities in the market where trading in the stock and bond is allowed at all trading dates t = 0, 1, . . . , T and the ACC is available to buy/sell only at time t = 0. We already have a candidate, that is the initial value U0 of the minimum superreplicating strategy for the seller. Differently from the case of European contingent claims, when defining arbitrage opportunities in the extended market we have now to distiguish the long and the short position in the ACC, because the first has the possibility

51

52

3. Finite Market Models

to exercise the claim at any time and the existence of one exercise time satisfying certain conditions is enough to give an arbitrage, while the latter must satisfy the arbitrage conditions for all possible exercise times chosen by the long counterparty. Precisely, we consider the following definitions. Definition 3.16. Let Y = (Yt )t=0,...,T be the payoff process of an ACC with price C0 at time 0. An arbitrage opportunity for the seller is a self-financing trading strategy (αs , β s ) in the stock and bond such that V0α

s ,β s

= C0 and,

for all exercise times τ ∈ T[0,T ] , Vτα

s ,β s

− Yτ ≥ 0,

and EQ [Vτα

s ,β s

− Yτ ] > 0.

(3.24)

An arbitrage opportunity for the buyer is a self-financing trading strategy (αb , β b ) in the stock and bond such that V0α

b ,β b

= −C0 and there exists an

exercise time τ ∈ T[0,T ] such that Vτα

b ,β b

+ Yτ ≥ 0,

and EQ [Vτα

b ,β b

+ Yτ ] > 0.

(3.25)

C0 is an arbitrage-free price for Y if it does not allow for any arbitrage opportunity for either the seller or the buyer. To better understand this definition, let us consider an investor who sells the ACC at price C0 and invests in the stock and bond according to an arbitrage strategy (αs , β s ) for the seller. At any time τ his/her counterparty decides to exercise the claim, he/she would pay off Yτ and invest the resultant wealth Vτα

s ,β s

−Yτ at time τ in the bond, fixed until maturity. The final value

at time T of the described investment is then (Vτα

s ,β s

− Yτ )(1 + r)T −τ , which

is non-negative and strictly positive with positive probability. A symmetric investment could be described for an arbitrage strategy for a buyer of the ACC. Before the main result, let us remark on a property of the discounted value process.

3.1 The (Multi-Period) Binomial Model

53

Lemma 3.17. The discounted value process of a self-financing trading strategy (α, β) in the stock and bond, V ∗ = (Vt∗ )t=0,...,T ,

Vt∗ =

Vtα,β , (1 + r)t

satisfies: ∀τ ∈ T[0,T ]

EQ [Vτ∗ ] = V0∗ .

Proof. The claim follows directly from Lemma 3.9 and Theorem 3.14. Theorem 3.18 (Thm 2.3.4. in [5]). The unique arbitrage-free price at time 0 for the ACC Y is C0 = U0 . Proof. (Uniqueness). We have to prove that if C0 6= U0 , then there are arbitrage opportunities either for the seller or for the buyer of the ACC. Suppose first that C0 > U0 . Consider the strategy (αs , β s ) that trades in the stock and bond according to the super-hedging strategy (α∗ , β ∗ ) and invest an amount of money equal to C0 − U0 in the bond over the whole time period (0, T ]. Precisely: for all t = 1, . . . , T , αts = αt∗ and βts = βt∗ + C0 − U0 . The initial value of such strategy, by self-financing, is V0α

s ,β s

= α1∗ S0 + β1∗ + C0 − U0 = C0 ,

and, for all exercise times τ ∈ T[0,T ] , we have Vτα

s ,β s

− Yτ = Vτα

∗ ,β ∗

+ (C0 − U0 )Bτ − Yτ

≥ Uτ + (C0 − U0 )Bτ − Yτ ≥ (C0 − U0 )Bτ , which is strictly positive. Thus, (αs , β s ) is an arbitrage opportunity for the seller. Then, suppose that C0 < U0 . Consider the strategy (αb , β b ) that trades in the stock and bond according to (−α∗ , −β ∗ ) and invest an amount of money equal to U0 − C0 in the bond over the whole time period [0, T ]. Precisely:

54

3. Finite Market Models for all t = 1, . . . , T , αtb = −αt∗ and βtb = −βt∗ + U0 − C0 . The initial value of such strategy, by self-financing, is V0α

b ,β b

= −α1∗ S0 − β1∗ + U0 − C0 = −V0α

∗ ,β ∗

+ U0 − C0 = −C0 .

Now, choosing the exercise time τ0∗ defined in (3.22), we get b

∗ ,β ∗

b

Vτα0∗ ,β + Yτ0∗ = − Vτα0∗

+ (U0 − C0 )Bτ0∗ + Yτ0∗

= − Uτ0∗ + (U0 − C0 )Bτ0∗ + Yτ0∗

by (3.19)

= (U0 − C0 )Bτ0∗ , which is strictly positive. Thus, (αs , β s ) is an arbitrage opportunity for the buyer. (Arbitrage-free). It remains to prove that if the price at time 0 for the ACC is set to C0 = U0 , then no arbitrage opportunities exist. We proceed by contradiction. First, suppose there exists an arbitrage for the seller, that is a trading strategy (αs , β s ) such that V0α

s ,β s

= U0 and (3.24) holds for all

τ ∈ T[0,T ] . Taking the particular exercise time τ0∗ ∈ T[0,T ] and discounting to time 0, it follows that EQ [Vτ∗0∗ − Yτ∗0∗ ] > 0, where V ∗ denotes the discounted value process of (αs , β s ). On the other hand, by Lemma 3.17, we have s ,β s

EQ [Vτ∗0∗ ] = V0∗ = V0α

= U0 ,

and, by (3.23), EQ [Yτ∗0∗ ] = U0 , hence EQ [Vτ∗0∗ − Yτ∗0∗ ] = 0, which gives a contradiction. Then, suppose there exists an arbitrage for the buyer, that is a trading strategy (αb , β b ) such that V0α

b ,β b

= −U0 and (3.25) holds for some exercise

time τ ∈ T[0,T ] . Again, by Lemma 3.17, denoted V ∗ the discounted value process of (αb , β b ), we have EQ [Vτ∗ ] = V0∗ = V0α

b ,β b

= −U0 ,

3.1 The (Multi-Period) Binomial Model and, by (3.23), EQ [Yτ∗ ] ≤ U0 , hence EQ [Vτ∗ + Yτ∗ ] ≤ 0, which gives a contradiction. Therefore, there are no arbitrage opportunities, either for the seller or for the buyer. As in the case of European claims, following the same procedure starting at each time t ∈ {0, 1, . . . , T }, we obtain the following result, showing that the unique arbitrage-free price for the ACC is given by the (discounted) expectation of the payoff relative to the next optimal exercise time. Corollary 3.19. The unique arbitrage-free price at time t ∈ {0, 1, . . . , T } for the ACC Y is Q

Ct = Ut = E

Yτt∗ |Ft . ∗ (1 + r)τt −t

Optimal exercise times Let us explain why, from the point of view of the buyer of the ACC, it is optimal to exercise the claim at time τ0∗ and not before. Clearly, there is no point in exercising at time t if Ut > Yt , because it equates to leaving an asset worth Ut (price of the ACC at time t) for an amount of money equal to Yt (current payoff). On the other hand, since U ≥ Y , it is profitable to exercise the claim at a (random) time τ such that Uτ = Yτ , where the randomness comes from taking into consideration that the equality can be achieved at different times in different scenarios. In particular, we denoted by τ0∗ ∈ T[0,T ] the first time fulfilling such condition; it is a stopping time because at any time t we know whether or not τ0∗ takes value in the past/present, that is in [0, t], and we only know its value if this is in [0, t]. Then, one may ask: should the buyer of the ACC exercise as soon as τ0∗ ,

or should they better wait for some other future time τ also satisfying

Uτ = Yτ ? Note that there could be many occasions where U = Y , and our stopping time τ0∗ is just the first occasion this happens. This entails

55

56

3. Finite Market Models discussing whether τ0∗ is the only optimal exercise time or one of the optimal exercise times. We show that there is no point either in exercising after the (stopping) time τmax := min{t ∈ {0, . . . , T − 1} : δ˜t+1 6= 0} ∧ T 1 Q = min t ∈ {0, . . . , T − 1} : Yt > E [Ut+1 |Ft ] ∧ T, 1+r where min ∅ := ∞. Indeed, let us consider the following two options. • If the long-position investor exercises the claim at time τmax , they get (from the seller of the claim) a wealth Yτmax = Uτmax , which can be invested in the portfolio (α∗ , β ∗ ) over the time period (τmax , T ], giving a strictly higher value than the ACC value at any time in {τmax + 1, τmax + 2, . . . , T }. Precisely: - at time τmax , the long position investor exercises the option, receiving Uτmax , and invest such proceeds in the portfolio (α∗ , β ∗ ), whose value at time τmax is exactly Uτmax , as Vtα

∗ ,β ∗

˜

˜β = Vtα, = Ut

∀t, 0 ≤ t ≤ τmax ;

- at any time s ∈ {τmax + 1, τmax + 2, . . . , T }, the same investor continues trading in the stock and bond according to (α∗ , β ∗ ), so that their portfolio value is given by ∗ ∗ Vsα ,β

=

αs∗ Ss

+

βs∗ Bs

=α ˜ s Ss + β˜s Bs +

s X k=τmax

= Us +

δ˜k Bk−1 +1

s X

δ˜k > Us , Bk−1 k=τmax +1 | {z } >0

because at least one of the terms in the sum is positive, that is δ˜τmax +1 > 0.

3.1 The (Multi-Period) Binomial Model

57

• On the contrary, if the long position investor waits longer than τmax , say they exercise at some time τ > τmax , then the proceeds they get from exercising the claim will not be enough to invest in the portfolio (α∗ , β ∗ ) because, for all τ ∈ T[τmax +1,T ] , Vτα

∗ ,β ∗

τ X

=α ˜ t Sτ + β˜τ Bτ +

k=τmax

= Uτ +

δ˜k Bk−1 +1

τ X

δ˜k > Uτ ≥ Yτ . B k−1 k=τ +1 | max {z } >0

Therefore, it is optimal to exercise at any time in [τ0∗ , τmax ] and not later. We call optimal exercise time any exercise time τ ∈ T[0,T ] such that τ0∗ ≤ τ ≤ τmax . Note that in general τmax ≥ τ0∗ , which means that either τ0∗ = τmax or P(τmax > τ0∗ ) > 0 that is strict inequality holds in at least one scenario (remember that as they are random times they take the same value on some scenario and different values in other scenarios). However, if we are at time τ0∗ and we condition on the available information, that is on Fτ0∗ , we can only have either τ0∗ = τmax or τ0∗ < τmax in the whole set of possible scenarios, which are all scenarios where the stock price trajectory up to time t coincides with the observed one. If the two stopping times coincide, there are no other optimal exercise times after τ0∗ and we have no other optimal choice than exercising right now. On the other side, if τmax > τ0∗ , then we can wonder whether it is more profitable to exercise at τ0∗ or at another stopping time τ ∈ [τ0∗ + 1, τmax ]. Remember that this decision of exercising or not at time τ0∗ has to be made based on the information available up to time τ0∗ . Assume, for simplicity, that we have a strict inequality τmax > τ0∗ . • If we exercise at time τ0∗ , we get a payoff Yτ0∗ = Uτ0∗ =

1 EQ Uτ0∗ +1 |Fτ0∗ , 1+r

where the second equality comes from the fact that τmax > τ0∗ and so δ˜τ ∗ +1 = 0. 0

58

3. Finite Market Models • If we wait and exercise for instance at time τ0∗ +1, which is still optimal, we get a payoff Yτ0∗ +1 = Uτ0∗ +1 , which has a discounted expected value under Q at time τ0∗ equal to 1 EQ Uτ0∗ +1 |Fτ0∗ . 1+r

Therefore, a risk-neutral investor, that is an investor who uses the risk-neutral measure Q to assign a probability to the possible sets of scenarios, would indeed be neutral to the choice (made at time τ0∗ ) of exercising at time τ0∗ or at the next trading date τ0∗ + 1. We could further discuss as to whether using the real world probability P instead of Q influences or not the choice to exercise. Summarizing, there may be other stopping times when exercising is optimal for the buyer of the ACC, and these ones are all in between τ0∗ and τmax (ends included). For this reason, we refer to τ0∗ as the first (instead of the) optimal exercise time. Note that this conclusion does not imply that on a specific trajectory ω ∈ Ω we cannot have Uτ0∗ (ω) >

1 U ∗ (ω). 1+r τ0 +1

For instance, look at the following example of a three-period binomial model, where we assume r = 0, d = 1/2 and u = 2. As usual, we work on the path space Ω = {ω1 , ω2 , . . . , ω8 } and we enumerate the trajectories going from left to right through the branches of the tree chosing first the up movement at each node, i.e. ω1 = (S0 u, S0 uu, S0 uuu), ω2 = (S0 u, S0 uu, S0 uud), ···

···

ω7 = (S0 d, S0 dd, S0 ddu),

ω8 = (S0 d, S0 dd, S0 ddd).

We compute the minimal wealth process for the seller in the different scenar-

3.1 The (Multi-Period) Binomial Model

59

ios, and use the usual notation: U0 =

59 , 81

U1u := U1 ({ω1 , ω2 , ω3 , ω4 }) = 53 , U1d := U1 ({ω5 , ω6 , ω7 , ω8 }) = U2uu := U2 ({ω1 , ω2 }) = 0,

U2ud := U2 ({ω3 , ω4 }) = 32 ,

U2du := U2 ({ω5 , ω6 }) = 23 ,

U2dd := U2 ({ω7 , ω8 }) =

7 , 27

1 . 18

We can then compute the first optimal exercise time τ0∗ and the last one τmax : τ0∗ ({ω1 , ω2 , ω3 , ω4 }) = 1, τ0∗ ({ω5 , ω6 }) = 2, τ0∗ ({ω7 , ω8 }) = 3, τmax ({ω1 , ω2 , ω3 , ω4 }) = 1, τmax ({ω5 , ω6 , ω7 , ω8 }) = 3. If we are at time t = 2 and the realized scenario is one in the event {S1 = 1 , S2 2

= 1} = {ω5 , ω6 }, we know that today is the first optimal time to

exercise the claim, because τ0∗ ({ω5 , ω6 }) = 2, but we also know that we can indifferently wait until tomorrow, because τmax ({ω5 , ω6 }) = 3. However, this neutrality in whether or not exercising today, does not prevent the possibility that in one of the possible scenarios ω5 , ω6 the payoff at maturity turns out to be strictly greater/smaller than the profit realized by exercising the option today (t = 2). In particular, if we exercise at time t = 2, we get U2du = 23 ,

60

3. Finite Market Models

while if we wait until the next and final trading date T = 3, in the scenario ω5 = {S1 = 21 , S2 = 1, S3 = 2} we receive a payoff X3duu = 53 , which is strictly 1 , S2 = 1, S3 = 12 } 2 U2du . However, at time

greater than U2du , while in the scenario ω6 = {S1 = we receive X3dud = 61 , which is strictly smaller than

t = 2, we can only compare the profit of exercising now with the (discounted) expected profit of exercising at maturity, because we don’t know which is the true state of the world between ω5 and ω6 (we only observe the event {ω5 , ω6 }).

3.2

Finite Market Models: General Setting

3.2.1

Model specification and the role of filtration

The binomial model is just a particular example of a large class of models which go under the name of “finite market models”. In the present section, we give the general definition of a model in this class, we analyse the role of filtrations in modeling the market information flow, and we prove the two fundamental theorems of asset pricing, putting in relation the absence of arbitrage opportunities and the existence of a replicating strategy with the existence and uniqueness of a risk-neutral measure. In models which are both arbitrage-free and complete, the unique arbitrage-free price of any ECC is well determined. A finite market model is specified by the following elements: • Finitely many trading dates, to which we assign successive integers t = 0, 1, 2, . . . , T,

T ∈ N.

˜ where the sample space is finite, i.e. ˜ F, ˜ P), • A probability space (Ω, ˜ < ∞, the σ-algebra is the power set of the sample space, i.e. #Ω ˜ is strictly positive, i.e. ˜ and the probability measure P F˜ = P(Ω), ˜ ˜ P({ω}) > 0 for all ω ∈ Ω.

3.2 Finite Market Models: General Setting ˜ = (F˜t )t=0,...,T , describing the flow of information revealed • A filtration F to the investors. • One savings account, named the bond, which is worth 1 at the initial time t = 0 and accumulates compund interest over time, based on a (possibly random, F˜t -measurable) interest rate rt ≥ 0 over the time period (t − 1, t], for all t = 1, . . . , T . The stochastic process S 0 = {St0 }t=0,1,...,T describes the dynamics of the bond value, given by 0 S00 = 1 and St0 = St−1 (1 + rt ), t = 1, 2, . . . , T.

• d risky securities, where d ∈ N, named the stocks, available for trading at each time t ∈ {0, 1, . . . , T } at a price which is observed on the market. Their price dynamics is denoted respectively by S 1 , . . . , S d where, for all i = 1, . . . , d, S i = (Sti )t=0,1,...,T is a non-negative, real˜ valued, F-adapted stochastic process representing the price dynamics of one share of the ith stock. Moreover the initial price of all stocks is known, which means that S0 is a constant in Rd+ . We denote by S the value process of all securities available for trading, that ˜ is a Rd+1 -valued F-adapted stochastic process S = {St }t=0,...,T , where St = (St0 , St1 , . . . , Std ) for all t = 0, 1, . . . , T . In this course, we restrict our study to the case of a constant and riskfree interest rate r over all unit time intervals, so that the bond is a riskless security whose value process S 0 is in fact deterministic. Assumption 3.2.1. For all t = 0, 1, . . . , T , St0 = (1 + r)t , where r ≥ 0 is the constant risk-free interest rate (on a unit of time). Since the only source of randomness that affects us is given by the value of the risky securities, without loss of generality, we can directly work on the path space of the security price process instead of the original probability space, which is an abstract unknown entity. Definition 3.20. The path space (Ω, F, P) relative to the security price process S is defined as follows:

61

62

3. Finite Market Models

• the sample space is the set of all possible trajectories of the security price process, that is: ˜ Ω = {(S0 (ω), S1 (ω), . . . , ST (ω)), ω ∈ Ω} (remember that each St (ω), t = 0, . . . , T , is a vector in Rd+1 + ); • the σ-algebra is the power set of the sample space of paths, i.e. F = P(Ω); • P is the law of S, that is the probability measure on (Ω, F) defined by ˜ −1 (A)) for all A ∈ F; P(A) = P(S • F = (Ft )t=0,...,T is the natural filtration of the coordinate process on (Ω, F), defined by (ω, t) 7→ ω(t) = (ω 0 (t), ω 1 (t), . . . , ω d (t)),

(3.26)

for all ω ∈ Ω and t ∈ {0, 1, . . . , T }. Note that, by (3.26) and the definition of the path space, the price process S coincides with the coordinate space on (Ω, F). Thus Ft = σ ({S0 , . . . , St }) = σ {(S01 , . . . , S0d ), . . . , (St1 , . . . , Std )} ,

t = 0, 1, . . . , T.

Note that, in particular, F0 = {∅, Ω} is the trivial σ-algebra, and FT = F is the power set of the sample space.

3.2.2

Behind filtrations: the information structure

Let us now look in more detail at the role of the filtration in a market model and the reason behind the choice of the natural filtration of the price process. We said that the filtration describes the flow of information revealed to the investors as time goes on, but we want to understand how this flow of information is modeled as a filtration and how the security price process is related to it. Reasonable assumptions on the flow of information revealed to the investors are:

3.2 Finite Market Models: General Setting

1. at time t = 0 no information is given on the true state of the world, all ω ∈ Ω are possibile, with probabilities assigned by P; 2. at time t = T , all investors learn which particular state of the world is realized, hence they come to know the value taken by any random variable on (Ω, F, P); 3. as time evolves from time 0 to time T , more and more information is gradually revealed, in such a way that there is a one-to-one correspondence between the sample space Ω of all possible states of the world and the set of all possible information sequences. The possible information sequences of this kind can be modeled in different equivalent ways: (I) A random sequence {At }t=0,...,T of subsets of Ω such that At : ω 7→ At (ω) 3 ω,

t = 0, . . . , T,

which models at each time the set of states of the world that are considered possible: at time t ∈ {0, 1, . . . , T } the investors observe the outcome of At , i.e. At (ˆ ω ) where ω ˆ is the (unknown) true state of the world, and know that it contains the true state of the world. In order to satisfy the three assumptions of the information flow, the sequence {At }t=0,...,T must satisfy certain properties. Assumption 1. implies that A0 = Ω, because at time 0 all states in Ω must be viewd as possible. Assumption 2. instead implies that AT (ω) = {ω} for all ω ∈ Ω, because full information means that we know exactly what the true state of the world is and hence the set of possible states is a singleton. From assumption 3., we can deduce the following conditions. First, since investors have a more and more detailed information as time goes on, the set of possible scenarios must be smaller and smaller, which means that {At }t=0,...,T must be a decreasing sequence, i.e. At ⊇ At+1 for all t = 0, 1, . . . , T − 1. Moreover, it must satisfy [ At+1 (ω) = At (ˆ ω ). ω∈At (ˆ ω)

63

64

3. Finite Market Models

Indeed: the inclusion ⊆ follows from the fact that the sequence {At }t=0,...,T is decreasing, and if there was some ω ∈ At (ˆ ω ) that was not contained in any possible consequent realization of At+1 , then it would have been ruled out already at time t hence it couldn’t be in At (ˆ ω ) in the first place. Finally, the collection of possible subsets taken by At+1 following At (ˆ ω ) must be mutually exclusive, because otherwise there would be two information sequences associated with a single state of the world, which contradicts the second part of assumption 3. Summarizing, the sequence {At }t=0,...,T is such that: A0 = Ω, AT (ω) = {ω} for all ω ∈ Ω, ω ∈ At (ω) and At ⊇ At+1 for all t = 0, 1, . . . , T − 1, and the collection of realizations of At+1 consequent to a realization At (ˆ ω ) is a partition of At (ˆ ω ). This leads to the second equivalent representation for the information flow.

(II) A sequence of partitions {Pt }t=0,...,T of Ω in a one-to-one correspondence with the random sequences described in (I), in the following way: P0 = {Ω} = {A0 }, and for each time t = 1, . . . , T the partition Pt is the collection of all possible realizations of At consequent to any realization of At−1 . With this one-to-one association, the properties of the sequence {At }t=0,...,T in (I) also imply that PT = {{ω}, ω ∈ Ω} and that for each time t = 1, . . . , T Pt is a refinement of Pt−1 .

(III) A filtration F = (Ft )t=0,...,T , in a one-to-one correspondence with the sequences of partitions in (II): for each σ-algebra Ft , t = 0, . . . , T , there exists a unique partition Pt = {Fn } of Ω such that Fn ∈ Ft for

3.2 Finite Market Models: General Setting all Fn ∈ Pt 1 ; conversely, given a partition Pt , there is a unique algebra Ft obtained performing all possibile finite sequences of elementary set operations (complement, intersection, union) from the elements of Pt ; in particular, Ft is the σ-algebra2 generated by Pt . With this oneto-one association, the properties of the sequences in (II) imply that the associated σ-algebras are such that F0 = {∅, Ω}, FT = P(Ω) and Ft−1 ⊆ Ft for all t ∈ {1, . . . , T }. Once understood the role of the filtration as a model for the information flow, we look at what kind of filtrations makes sense to use. First, we want that the available information for the investor at any time includes the full knowledge of past and present prices of the traded securities. This motivates the specification of price processes which are adapted to the given filtration. Indeed, the fact that S is adapted to F, that is that St is Ft -measurable for all t = 0, . . . , T , equivalently means that St is constant on each element of the partition Pt associated to Ft . Thus, since at time t the investors know which element of Pt contains the true state of the world, they can deduce the current price of the securities; moreover, since the partitions are successive refinements, the investors can also identify which elements of the previous partitions P1 , . . . , Pt−1 contain the true state of the world, and hence deduce the past security prices S1 , . . . , St−1 . On the other hand, it is reasonable to assume that the investors cannot look into the future, that is they cannot have full knowledge of the future prices of securities. This means that the proce process S is not predictable 1

Given an algebra F on Ω (i.e. a collection of subsets of Ω such that Ω ∈ F, if F ∈ F

then F c ∈ F, and if F, G ∈ F then F ∪ G ∈ F) there exists a unique collection {Fn }n∈N of subsets of Ω such that: (a) Fn ∈ F for all n ∈ N ; (b) Fn ∩ Fm = ∅ for all n, m ∈ N , n 6= m; S (c) n∈N Fn = Ω. 2

On a finite space Ω, an algebra is also a σ-algebra.

65

66

3. Finite Market Models

with respect to the filtration F, i.e. St+1 is in general not Ft -measurable. There are two approaches to the specification of the model. One is to equip the probability space with a filtration, perhaps based on a series of information reports released by financial authorities, and then specify a feasible model for the security prices, that is a stocahstic process adapted to the given filtration. The other is to specify the stochastic process modeling the prices and only then introduce an appropriate filtration. In this second case, one has the possibility to choose the unique filtration that corresponds to learning about the security prices as time goes on and nothing more: the natural filtration of the security prices process, that is the coarsest filtration (i.e. the one containing the fewest subsets) that makes the security prices an adapted process. Now we have a better understanding of the reasoning behind the modeling choices.

3.2.3

Trading strategies

The definitions given in Section 3.1 within the framework of the binomial model, are easily extended to the more general framework of finite market models, and the same explanations still hold. Definition 3.21. A trading strategy is a couple (V0 , α), where V0 ∈ R denotes the value of the initial investment and α = (αt )t=1,...,T is an Fpredictable Rd+1 -valued stochastic process where, for all t = 1, . . . , T , αt = (αt0 , αt1 , . . . , αtd ) is a vector whose first component αt0 denotes the units of the bond and for each i = 1, . . . , d αti denotes the number of shares of the ith stock, both to be held in the portfolio over the time interval (t − 1, t]. The portfolio value process of a trading strategy (V0 , α) is the F-adapted real-valued stochastic process V α = (Vtα )t=0,...,T defined by V0α = V0 , Vtα

= αt · St =

d X i=0

αti Sti

t = 1, . . . , T.

(3.27)

3.2 Finite Market Models: General Setting

67

Definition 3.22. The gain process of a trading strategy (V0 , α) is the Fadapted real-valued stochastic process Gα = (Gαt )t=0,...,T defined by Gα0 = 0, t t X d X X i Gαt = αs · (Ss − Ss−1 ) = αsi (Ssi − Ss−1 ), s=1

t = 1, . . . , T.

(3.28)

s=1 i=0

The random value Vtα defined in (3.27) represents the value of the trading portfolio at time t just before any transaction, that is just before any change in the asset positions which is decided at time t and applied immediately after to be held, constant, over the time period (t, t + 1]. The random value Gαt defined in (3.28) represents the cumulative gain over the time period i (0, t] of the portfolio, since, for all i = 0, . . . , d and s = 1, . . . , t, αsi (Ssi − Ss−1 )

represents the gain or loss due to the ownership of αsi units of the security S i over the period (s − 1, s]. Moreover, the stochastic process defined in (3.28) is the discrete-time version of a stochastic integral, as it is the weighted sum of the values of a predictable process, with the weights being the one-period backward changes of another (the integrator) stochastic process. As usual, we require that there is no addition or withdrawal of money between the initial date t = 0 and and the final date t = T . Namely, only self-financing trading strategies are reasonable. Definition 3.23. A trading strategy (V0 , α) is said to be self-financing if V0 = α1 · S0 =

d X

α1i S0i ,

(3.29)

i=0

and Vtα = αt+1 · St ,

t = 1, . . . , T − 1,

i.e. d X i=0

αti Sti

=

d X

i αt+1 Sti ,

t = 1, . . . , T − 1.

(3.30)

i=0

Note that the left-hand expression in (3.30) is the portfolio value at time t, while the right-hand expression represents the value of the trading portfolio

68

3. Finite Market Models

at time t just after all transactions, that is just after any change in the asset positions which is decided at time t, applied immediately after, and carried forward to the next trading time t + 1. So, the self-financing condition means that at any trading time, the portfolio is rebalanced without changing its value. Any change in the portfolio value must be due to a gain or loss in the investments. Remark 3.24. A trading strategy (V0 , α) is self-financing if and only if Vtα = V0 + Gαt

∀t = 0, . . . , T.

(3.31)

The equivalence between the conditions in (3.31) and (3.30) is simply proven by algebraic rearrangement of those equations. As we remarked in the particular case of the binomial model (Remark 3.3), a self-financing trading strategy is univocally specified by the securities holding process α, therefore we will denote self-financing trading strategies (V0 , α) only by α, being the initial value univocally determined by (3.29). Analogously, the following holds true. Remark 3.25. Given V0 ∈ R and d F-predictable R-valued stochastic processes α1 , . . . , αd , there exists a unique F-predictable R-valued stochastic process α0 such that (V0 , α), where α = (α0 , α1 , . . . , αd ), is a self-financing strategy, namely αt0

= V0 −

d X

α1i S0i

i=1

t−1 X d X Ssi i − (αs+1 − αsi ), 0 Ss s=1 i=1

t = 1, . . . , T.

(3.32)

We analogously define the normalized market with respect to the bond: S ∗ = {St∗ }t=0,...,T ,

where St∗ :=

St , t = 0, . . . , T, St0

denotes the discounted security price process, V ∗ = {Vt∗ }t=0,...,T ,

where V0∗ = V0 , Vt∗ := αt · St∗ =

Vtα , t = 1, . . . , T, St0

denotes the discounted value process of a trading strategy (V0 , α), and ∗

G =

{G∗t }t=0,...,T ,

where

G∗t

:=

t X s=1

∗ αs · (Ss∗ − Ss−1 ), t = 0, . . . , T,

3.2 Finite Market Models: General Setting

69

denotes the discounted gain process of a trading strategy (V0 , α). The self-financing conditions (3.29)-(3.30) and (3.31) have an equivalent for the discounted market. Remark 3.26. A trading strategy (V0 , α) is self-financing if and only if the correspondent discounted value process V ∗ satisfies Vt∗ = αt+1 · St∗ ,

t = 0, . . . , T − 1,

(3.33)

t = 0, . . . , T,

(3.34)

or, equivalently, Vt∗ = V0 + G∗t ,

where G∗ is the correspondent discounted gain process. Note that the discounted gain process of a trading strategy (V0 , α) only depends on the stock positions α1 , . . . , αd , because the discounted bond price is constant, S 0 ≡ 1. Thus, if the strategy is self-financing, also the discounted value process only depends on the stock positions α1 , . . . , αd , and V0 , by (3.34). Note that, since the discounted gain process G∗ of a trading strategy (V0 , α) is a function of the stock holdings (and the stock prices) only, we can denote it by G∗ (ˆ α), where α ˆ = (α1 , . . . , αd ). As usual, the presence of trading strategies that turn a null initial investment into a non-negative gain which is stricly positive in at least one scenario is not consistent with economic equilibrium. So, we look for models that rule out arbitrage opportunities, which are defined analogously as in the binomial model: Definition 3.27. An arbitrage opportunity (in the primary market S) is a self-financing trading strategy α such that: (i) V0α = 0, (ii) VTα ≥ 0, (iii) P(VTα > 0) > 0, or, equivalently, EP [VTα ] > 0. Note that the equivalence in (iii) only holds in finite probability spaces.

70

3. Finite Market Models

3.2.4

The First Fundamental Theorem of Asset Pricing

We start by defining a special class of probability measures that make the discounted price process in the market become martingales. Definition 3.28. An equivalent martingale measure (EMM) for S ∗ (also called a risk-neutral probability in Finance) is a probability measure Q on (Ω, F) such that Q ∼ P and S ∗ is a (Q, F)-martingale, that is ∗ EQ [St∗ |Ft−1 ] = St−1

∀t = 1, . . . , T.

(3.35)

We remark that St∗ is a d + 1-dimensional vector, thus the equality in (3.35) is intended component-wise. Remark 3.29. Since, by assumption, P({ω}) > 0 for all ω ∈ Ω, in a finite market model a probability measure Q is equivalent to P if and only if Q({ω}) > 0 for all ω ∈ Ω. The following property of EMMs is a direct consequence of Proposition A.37 in the appendix and the self-financing condition (3.34). It will be useful to prove the fundamental result presented in this section, which is also a milestone in the modern theory of Mathematical Finance, that is the first fundamental theorem of asset pricing (see [2] for a thorough presentation of this theorem in all its different versions and a collection of related papers by the same authors). Lemma 3.30. Let Q be an EMM for S ∗ . The discounted value process V ∗ of any self-financing trading stretegy is a (Q, F)-martingale, that is ∗ EQ [Vt∗ |Ft−1 ] = Vt−1

∀t = 1, . . . , T.

In particular: V0∗ = EQ [Vt∗ ] ∀t = 1, . . . , T.

(3.36)

The following notation is valid for all financial market models, not only the finite market model addressed in this chapter.

3.2 Finite Market Models: General Setting

71

Definition 3.31. A financial market model satisfies the no-arbitrage (NA) condition, or equivalently it is said to be arbitrage-free), if ¯ V ∩ RN + = {0},

(3.37)

where N ∈ N is the cardinality of the sample space Ω, RN + is the positive (closed) orthant of RN , i.e. N RN + := {x = (x1 , . . . , xN ) ∈ R | x1 ≥ 0, x2 ≥ 0, . . . , xN ≥ 0},

and V the set of discounted attainable contingent claims with price 0, i.e. V := {G∗T (ˆ α) | α ˆ Rd -valued F-predictable processes}. Condition (3.37) means that there do not exist arbitrage opportunities in the market. Note that, since the sample space Ω is finite, #Ω = N , we can identify a random variable X on (Ω, F) by the vector of RN composed of the values that it takes on each elementary event. Namely, enumerating the elementary events as Ω = {ω1 , . . . , ωN }, we denote xi = X(ωi ) for all i = 1, . . . , N , and we can then identify the random variable X on (Ω, F) with the vector (x1 , . . . , xN ) of RN . Therefore, the interpretation of the NA condition (3.37) is that, by trading in the primary market, the only non-negative final discounted gain we can obtain is the null random variable, or equivalently the null vector ¯0 ∈ RN . Restricting the set of trading strategies to self-financing ones, this is equivalent to say that the only non-negative final portfolio value that can be obtained with a null initial investment is zero. This means that there do not exist arbitrage opportunities, because if it was possible to obtain a final value which is both non-negative and strictly positive in at least one scenario, then the associated self-financing strategy would be an arbitrage opportunity. Theorem 3.32 (First Fundamental Theorem of Asset Pricing (FFTAP)). A finite market model satisfies NA if and only if there exists an equivalent martingale measure.

72

3. Finite Market Models

The proof of Theorem 3.32 is split in two parts; one implication is easy to prove and we see it here, the other is longer and technical and is postponed to the Appendix B.2. Proof. (∃ EMM ⇒ NA). Assume there exists at least an EMM, and choose one, say Q. Suppose that an arbitrage opportunity α exists, then its discounted value process V ∗ must satisfy V0∗ = 0, VT∗ ≥ 0 and EP [VT∗ ] > 0. Since Q ∼ P, the latter is equivalent to EQ [VT∗ ] > 0. But we also know, by Lemma 3.30, that EQ [VT∗ ] = V0∗ = 0, hence a contradiction. So, we proved that if an equivalent martingale measure for S ∗ exists, then the market is arbitrage-free. Now we see the implications of Theorem 3.32 in terms of asset pricing. Definition 3.33. A European contingent claim (ECC) in the finite market model is represented by a random variable on (Ω, F). A ECC X is said to be simple, or non-path-dependent, if the payoff only depends on the final security prices, that is X = f (ST ) for some measurable function f : Rd+1 → R; in all other cases it is said path-dependent. A ECC X is said attainable if there exists a self-financing trading strategy α such that VTα = X. Such a trading strategy, if it exists, is called a replicating/hedging strategy for X. Intuitively, the random variable X represents the payoff, equivalently the value, of the ECC itself. The simplest examples of derivatives, the call and put options, are simple contingent claims, while the lookback, barrier and Asian options are classical examples of path-dependent options, for they depend on the whole history of the security price. We denote the discounted value of an ECC X by X ∗ :=

X . ST0

Theorem 3.34. Assume that the NA condition holds true and that X is an attainable ECC. Then, for any replicating strategies α for X, the portfolio value process V α is the same, and the the discounted value process V ∗ = V ∗ (α1 , . . . , αd ) is given by Vt∗ = EQ [X ∗ |Ft ],

t = 0, . . . , T,

(3.38)

3.2 Finite Market Models: General Setting for all equivalent martingale measures Q for S ∗ . Note that the equality in (3.38) holds surely, that is the two random variables take the same value on all elementary events. This is thanks to the model assumption that the only set with zero probability under P is the empty set, and the fact that Q ∼ P, hence the two probability measures have the same null sets. In fact, without the first assumption, the equality in (3.38) would only hold Q-almost surely, as we have seen in the Appendix that the conditional expectation is defined up to a modification. Proof. By the FFTAP, there exists at least an EMM, say Q. Given a hedging strategy α, by definition VTα = X and by Lemma 3.30 the discounted portfolio value of α is a Q-martingale, hence the equation (3.41) holds, where the right-hand member does not depend on α, and so niether do V ∗ and V α . Moreover, the left-hand member in (3.41) does not depend on the particular EMM chosen, therefore the claim holds for all EMMs. To be more precise, when we talk about ’equalities’ between stochastic processes, we can have two different types of identification. Definition 3.35. Let X = (Xt )t∈T , Y = (Yt )t∈T be two stochastic processes on a probability space (Ω, F, P). Then, X and Y are modifications of one another if P(Xt = Yt ) = 1

∀t ∈ T.

Instead, X and Y are indistinguishable if P(Xt = Yt ∀t ∈ T) = 1. Note that in Definition 3.35 there is no condition on the set T of time indexes, which can be either discrete (discrete-time stochastic processes) or dense (continuous-time stochastic processes), as well as either finite (finitehorizon stochastic processes) or not (infinite-horizon stochastic processes). In general, it is clear that two indistinguishable stochastic processes are also

73

74

3. Finite Market Models

modifications of one another, but the converse is not true. However, in continuous time, we can easily prove that almost-surely right- or left-continuity is enough to make the converse hold true. In a finite market model, we deal with finite-horizon discrete-time stochastic processes, where T = {0, 1, . . . , T }. In this framework, the two notions of modifications and indistinguishable processes are equivalent. Indeed: assume X and Y are modifications of one another and denote Nt := {ω ∈ Ω : Xt (ω) 6= Yt (ω)},

t = 0, 1, . . . , T ;

then, by Definition 3.35, all Nt , t = 0, . . . , T , are null sets, and so is their union, i.e. P(N ) = 0,

N :=

[

Nt ;

t=0,1,...,T

but the null set N is exactly the complement of {ω ∈ Ω : Xt (ω) = Yt (ω) ∀t = 0, 1, . . . , T } =

\

Ntc ,

t=0,1,...,T

which has thus probability one under P. Therefore, X and Y are modifications one of another if and only if they are indistinguishable. Then, if we consider another probability measure Q ∼ P, it is clear that indistinguishable processes under P are also indistinguishable under Q and viceversa. Moreover, in a finite market model, since P({ω}) > 0 for all ω ∈ Ω, then all almost-sure equalities are in fact sure equalities, which means that X and Y are indistinguishable processes on (Ω, F, P), if and only if they are indistinguishable on (Ω, F, Q), if and only if Xt (ω) = Yt (ω) ∀t = 0, 1, . . . , T, ∀ω ∈ Ω. Now, we improve Theorem 3.34 by showing that not only the portfolio value process of any hedging strategy for X is unique, but it also is the unique arbitrage-free price process for the ECC X. This result generalizes Theorem 3.8, already proven for the Binomial model, and Corollary 3.10. In order to do that, we have to introduce the notion of trading strategies in the extended

3.2 Finite Market Models: General Setting

75

market where, besides the stock and the bond, the (attainable) ECC X can also be traded at any time t = 0, 1, . . . , T with price process C = (Ct )t=0,...,T , that is a real-valued F-adapted stochastic process such that CT = X. Definition 3.36. Given an attainable ECC X with price process C = (Ct )t=0,...,T , a trading strategy in the secondary market (S, X) is a collection (V0 , α, γ), where V0 ∈ R denotes the value of the initial investment, α = (α0 , α1 , . . . , αd ) = {(αt1 , αt2 )}t=1,...,T is a Rd+1 -valued F-predictable stochastic process representing the holdings in the bond and stock as usual (see Definition 3.21), and γ = (γt )t=1,...,T is a real-valued F-predictable stochastic process where γt represents the number of contingent claims held in the portfolio over the time period (t − 1, t], t = 1, . . . , T . The portfolio value of (V0 , α, γ) at any time t ∈ {1, . . . , T } is given by Vtα,γ = αt · St + γt Ct .

(3.39)

The trading strategy (V0 , α, γ) in the secondary market (X, S) is self-financing if it satisfies ( P V0 = α1 · S0 + γ1 C0 = di=0 α1i S0i + γ1 C0 , αt · St + γt Ct = αt+1 · St + γt+1 Ct ,

t = 1, . . . , T.

(3.40)

A self-financing strategy3 (α, γ) in the secondary market (X, S) is an arbitrage if (i) V0α,γ = 0, (ii) VTα,γ ≥ 0, (iii) P(VTα,γ > 0) > 0, or equivalently EP [VTα,γ ] > 0. Note that, if we consider the ECC X as a primary security and we add it to the other risky securities S 1 , . . . , S d , we get a finite market model based on d + 2 securities, with price porcess S = (S 0 , S 1 , . . . , S d , C), and Definition 3.36 gets covered by the definitions given in Section 3.2.3. 3

As usual, we identify a self-financing trading strategy by its holding processes only,

since the initial investment is univocally determined by (3.40).

76

3. Finite Market Models

Now, using the above definitions and Theorem 3.34, we are able to prove the following theorem, which generalizes the results obtained in the binomial model. Theorem 3.37. Assume that the NA condition holds true and that X is an attainable ECC. Then, the stochastic process 0 Q ∗ St E [X |Ft ],

t = 0, . . . , T

(3.41)

is the unique arbitrage-free price process for X, where Q is an equivalent martingale measure for S ∗ . Proof. (Proof of Theorem 3.37) First, by Theorem 3.34, the process in (3.41) coincides with the portfolio value of any hedging strategy α for X, that is Vtα = St0 EQ [X ∗ |Ft ],

t = 0, . . . , T.

Now, let C = (Ct )t=0,...,T be the price process for X, where CT = X, we have to prove two things: that if C is different from (3.41) then the extended market allows for arbitrage opportunities, and that if C coincides with (3.41), no arbitrage opportunities can be realized. (Uniqueness). Suppose that, for some s ∈ {0, . . . , T − 1}, P(Cs > Vsα ) > 0, and denote A := {ω ∈ Ω : Cs (ω) > Vsα (ω)} ∈ Fs . Consider the following trading strategy: invest nothing until at time s, then, if in a scenario ω ∈ A, at time s sell one unit of the ECC, invest an amount Vsα (ω) of the proceeds in the primary market according to αs+1 and invest the remainder Cs (ω)−Vsα (ω) in the bond (to be held fixed over (s, T ]), then continue to trade according to α over the following time periods. Formally, this strategy is defined by: V0 = 0 and 0, (˜ αt , γ˜t )(ω) = α0 (ω) + t

Cs (ω)−Vsα (ω) Ss0

t ≤ s; , αt1 (ω), . . . , αtd (ω), −1 1A (ω), t > s.

The strategy (V0 , α ˜ , γ˜ ) is self-financing: until time t = s − 1 the conditions in (3.40) are trivially satisfied, as they are from time t = s+1 to maturity, since

3.2 Finite Market Models: General Setting

77

the portfolio evolves as the self-financing portfolio α, being the additional holdings in the ECC and bond fixed over the time period (s, T ]; at time ˜γ t = s we have Vsα,˜ = 0 and Cs − Vsα 0 α ˜ s+1 · Ss − Cs = αs+1 · Ss + Ss − Cs 1A = 0. Ss0

Moreover, the final portfolio value is Cs − Vsα 0 α,˜ ˜γ VT = αT · ST + ST − CT 1A Ss0 0 α ST = (Cs − Vs ) 0 1A Ss ≥ 0, ˜γ and P(VTα,˜ > 0) = P(A) > 0. Thus, (˜ α, γ˜ ) is an arbitrage.

Analogously, supposed that, for some s ∈ {0, . . . , T −1}, P(Cs < Vsα ) > 0, and denoted B := {ω ∈ Ω : Cs (ω) < Vsα (ω)} ∈ Fs , we could prove that the strategy (ˆ α, γˆ ), defined by 0, (ˆ αt , γˆt )(ω) = −α0 (ω) + t

Vsα (ω)−Cs (ω) Ss0

t ≤ s; , −αt1 (ω), . . . , −αtd (ω), 1 1B (ω), t > s.

is an arbitrage. Therefore, if P(∃s ∈ {0, . . . , T − 1} : Cs 6= Vsα ) > 0, then there are arbitrage opportunities. (Arbitrage-free). The second part of the proof is about showing that if the price process for X is defined as in (3.41), then it does not allow for arbitrage opportunities. Following a reductio ad absurdum, suppose that C = V α and that (¯ α, γ¯ ) is an arbitrage. Denote by V ∗ and V¯ ∗ the discounted ¯γ value process of α and (¯ α, γ¯ ) respectively, Vt∗ := Vtα /St0 and V¯t∗ := Vtα,¯ /St0

for all t = 0, . . . , T , and compute EQ [V¯t∗ |Ft−1 ] = EQ [¯ αt · St∗ + γ¯t Vt∗ |Ft−1 ] =α ¯ t · EQ [St∗ |Ft−1 ] + γ¯t EQ [Vt∗ |Ft−1 ] ∗ ∗ =α ¯ t St−1 + γ¯t Vt−1 ∗ = V¯t−1

78

3. Finite Market Models

by linearity of the conditional expectation, by Lemma 3.30 and by selffinancing of (¯ α, γ¯ ). Hence, V¯ ∗ is also a (Q, F)-martingale, and its expectation under Q is constant: EQ [V¯T∗ ] = EQ [V¯0∗ ] = 0, since an arbitrage has zero initial value. On the other hand, an arbitrage has non-negative final value V¯ ∗ ≥ 0. A non-negative random value with null T

expectation is necessarily almost-surely null, i.e. Q(V¯T∗ = 0) = 1, but Q ∼ P and so it would follow that P(V¯ ∗ = 0) = 1, a contradiction. T

3.2.5

The Second Fundamental Theorem of Asset Pricing

Once solved the hedging and pricing problems for attainable ECCs, it remains to identify the attainable claims. A more convenient situation would be a model where all ECCs can be replicated, that is the subject of the present section. Definition 3.38. A financial market model is complete if all European contingent claims are attainable. In a complete finite market model, Theorem 3.37 solves the arbitragefree pricing problem for all ECCs. A necessary and sufficient condition for completeness of the model is given by the following fundamental theorem. Theorem 3.39 (Second Fundamental Theorem of Asset Pricing (SFTAP)). A finite market model satisfying NA is complete if and only if there is a unique equivalent martingale measure for S ∗ . Proof. (⇒) Assume that the market is arbitrage-free, i.e. satisfies NA, and complete. By contradiction, suppose that there are two different equivalent martingale measures for S ∗ , Q1 , Q2 . Now, consider any event A ∈ F and take the European contingent claim which takes value 1 on A and 0 otherwise, X =

3.2 Finite Market Models: General Setting

79

1A . Since the market is complete, X is attainable, thus by Theorem 3.37 its arbitrage-free price at time 0 is unique and equal to Q2 1A Q1 1A =E . E ST0 ST0 Since A was arbitrarily chosen in F, the equation above translates into Q1 (A) = Q2 (A)

∀A ∈ F,

therefore Q1 = Q2 , that is there cannot be two distinct EMMs for S ∗ . (⇐) Assume that the equivalent martingale measure for S ∗ , Q, is unique. By contradiction, suppose that the (arbitrage-free) market is not complete. As in the proof of the FFTAP, we identify random variables with vectors in RN . We denote by W := {VTα,∗ | α self-financing strategy} the set of all discounted attainable claims, which is a vector space, and remark that the market is not complete if and only if W ( RN . Let us define a scalar product h·,·iQ in RN as follows: hX, Y iQ = E [XY ] = Q

N X

xi yi Q({ωi }),

for all X, Y, ∈ RN ,

i=1

X = (x1 , . . . , xN ), Y = (y1 , . . . , yN ). Then, there exists a vector ξ ∈ RN \ {0} orthogonal to W, i.e. such that hξ, XiQ = EQ [ξX] = 0 ∀X ∈ W. In particular, as every constant belongs to W (it can be reached by taking both the initial investment and α0 equal to the constant discounted by ST0 and α1 = . . . = αd = 0), by taking X = 1 we get EQ [ξ] = 0. Now, chosen arbitrarily δ > 1, we define another probability measure Qδ as

Qδ ({ωi }) = 1 +

ξi δkξk∞

Q({ωi }),

i = 1, . . . , N,

80

3. Finite Market Models where kξk∞ := max |ξi |. If we can prove that Qδ is another EMM for S ∗ , 1≤i≤N

then we have a contradiction, that ends of the proof. First, Qδ is a probability measure, since Qδ (Ω) = =

N X i=1 N X

Qδ ({ωi }) Q({ωi }) +

i=1

=1+

N 1 X ξi Q({ωi }) δkξk∞ i=1

1 EQ [ξ] = 1. δkξk∞

Moreover, Qδ ∼ P because, by definition, it assigns strictly positive probability to each elementary event. It remains to be proved that S ∗ is a (F, Qδ )martingale. Again, by Proposition A.37, this is equivalent to prove that, for all bounded R-valued predictable processes γ and all k ∈ {1, . . . , d}, EQδ [Gt (γ, S ∗ k )] = 0,

t = 1, . . . , T.

For all k = 1, . . . , d, we have EQδ [GT (γ, S ∗ k )] =

N X

GT (γ, S ∗ k )(ωi )Q({ωi }) +

i=1

= EQ [GT (γ, S ∗ k )] +

N 1 X ξi GT (γ, S ∗ k )(ωi )Q({ωi }) δkξk∞ i=1

1 EQ [ξGT (γ, S ∗ k )] δkξk∞

= 0, since Q is an EMM for S ∗ and GT (γ, S ∗ k ) ∈ W. Then, since this holds for any arbitrary bounded R-valued predictable processes γ, for any time t = 1, . . . , T − 1 we can set γt+1 = . . . = γT = 0 and we also get EQδ [Gt (γ, S ∗ k )] = 0, for all bounded R-valued predictable processes γ, since Gt (γ, S ∗ k ) =

t X

γs (S ∗ ks − S ∗ ks−1 )

s=1

only depends on the first t values taken by γ.

3.2 Finite Market Models: General Setting

81

Another equivalent characterization of completeness is provided by the following version of the martingale representation theorem. Theorem 3.40. Assume that the finite market model satisfies NA and let Q be an equivalent martingale measure for S ∗ . Then, the model is complete if and only if every real-valued (F, Q)-martingale M = (Mt )t=0,...,T has a representation of the form Mt = M0 +

t X

∗ ), α ˆ s · (Sˆs∗ − Sˆs−1

t = 1, . . . , T,

(3.42)

s=1

for some Rd -valued F-predictable stochastic process α ˆ = (α1 , . . . , αd ), where we denoted Sˆ∗ = (S ∗,1 , . . . , S ∗,d ). Proof.

(⇒). Suppose the model is complete and take an arbitrary (F, Q)-

martingale M = (Mt )t=0,...,T . Then, X = ST0 MT is an ECC and, by completeness, there exists a hedging strategy α for X. Denoted by V ∗ the discounted portfolio value process of α, it holds VT∗ = X ∗ = MT , and Vt∗ = EQ [X ∗ |Ft ] = EQ [MT |Ft ],

t = 0, . . . , T,

by Theorem 3.34. But also, since M is a martingale, Mt = EQ [MT |Ft ],

t = 0, . . . , T.

Thus, by (3.31), we obtain Mt = Vt∗ = V0 + G∗t = V0∗ + = M0 +

t X s=1 t X

∗ ) αs · (Ss∗ − Ss−1

∗ α ˆ s · (Sˆs∗ − Sˆs−1 ),

s=1

where α ˆ = (α1 , . . . , αd ) and we denoted by G∗ the discounted gain process of α. This gives a representation in the form of (3.42).

82

3. Finite Market Models

(⇐). Suppose the martingale representation holds and take an arbitrary ECC X. We can define a (F, Q)-martingale M = (Mt )t=0,...,T by setting Mt = EQ [X ∗ |Ft ],

t = 0, . . . , T.

By Lemma 3.25, we know that, given α ˆ as in the statement of the theorem, there exists a unique real-valued predictable process α0 such that α = (α0 , α ˆ ) = (α0 , α1 , . . . , αd ) is a self-financing trading strategy with initial value M0 . The discounted value V ∗ = V α /S 0 of such strategy at any time t ∈ {0, . . . , T } is Vt∗ = V0∗ + G∗t = M0 + = M0 +

t X s=1 t X

∗ αs · (Ss∗ − Ss−1 )

∗ α ˆ s · (Sˆs∗ − Sˆs−1 )

s=1

= Mt , from which VTα = MT ST0 = X, hence α is a hedging strategy for X and X is attainable.

3.2.6

Incomplete markets

To sum up the financial results of the last two sections, we have seen how to replicate and price any attainable contingent claim in a finite market model which is free of arbitrage opportunities. If the model is also complete, then we already know all we need to sell an European contingent claim on the market. On the contrary, if the market is not complete, that is not all contingent claims are attainable, then we still know how to deal with attainable claims, but not with the other ones. We end the chapter providing some hint on how dealing with pricing unattainable European contingent claims: we cannot find a unique arbitrage-free price for the derivative, but we are able to identify an interval within which the fair price should fall.

3.2 Finite Market Models: General Setting

83

Definition 3.41. Let X be an ECC, a super-hedging strategy for X is a self-financing trading strategy α such that VTα ≥ X. For simplicity, let T = 1, i.e. we consider here a single-period finite market model. We also assume that the no-arbitrage condition is satisfied, but the model is not complete, hence there are more than one EMMs and not all European contingent claims are attainable. Let M denote the set of all equivalent martingale measures for S ∗ . Consider an unattainable ECC X, and define the quantity V+ (X) := inf{V0α : α super-hedging strategy for X}.

(3.43)

Note that V+ (X) is well defined, because the set in (3.43) is non-empty: any strategy α1 = (λ, 0, 0, . . . , 0), with λ ∈ R large enough to have λ(1 + r) ≥ X, is a super-hedging strategy for X. Moreover, it can be rewritten as V+ (X) = inf{EQ [Y ∗ ] : Y attainable ECC, Y ≥ X} ∀Q ∈ M

(3.44)

This equality comes from Theorem 3.34, which implies that the initial value of any super-hedging strategy α for X is given by EQ [Y ∗ ], where Y := VTα ≥ X, independently of the chosen Q ∈ M. Remark 3.42. V+ (X) is an upper bound on the fair price for X, that is a fair price C0 for X should satisfy C0 ≤ V+ (X). Indeed, if the derivative was priced at a higher value C0 > V+ (X), then one could construct an arbitrage opportunity, as follows: - at time 0, sell X and invest an amount EQ [Y ∗ ] of the proceeds, where Y is an attainable ECC such that Y ≥ X and V+ (X) ≤ EQ [Y ∗ ] < C0 , in the replicating portfolio for Y , and the remaining amount C0 − EQ [Y ∗ ] in the bond; - at time 1, the portfolio value will be Y − X + (C0 − EQ [Y ∗ ])(1 + r) > 0. Remark 3.43. A lower bound on V+ (X) is given by sup{EQ [X ∗ ], Q ∈ M}.

84

3. Finite Market Models

This is a consequence of the equation (3.44) and monotonicity of the expectation. Remark 3.44. The infimum in the definition (3.43) is attained. Indeed, we could rewrite (3.43) as the linear program minimize

α · S0

subject to α · S1 ≥ X, α ∈ Rd+1 . The feasible region for this problem can be interpreted as the set of all attainable contingent claims Y ≥ X, whose initial price is given by the objective function. If an optimal solution to this linear program exists, then it must be equal to V+ (X), which would then attain the infimum in (3.43). As observed previously, the feasible region for the linear program is non-empty and, by Remark 3.43, the objective function is bounded from below, thus an optimal solution always exists. Let α+ denote the super-hedging strategy for X whose initial value is equal to V+ (X), called a minimal super-hedging strategy for X. V+ (X) is the price that allows the seller of the derivative to invest in a minimal superhedging strategy in order to be protected from the risks coming from the sale. On the other hand, we can analogously obtain a limit from below of the fair price for X, following a symmetric reasoning to the previous one from the point of view of the buyer instead of the seller. Let us define the quantity V− (X) := sup{V0α : α self-financing strategy, VTα ≤ X},

(3.45)

which is well-defined, and we can rewrite it as V− (X) = sup{EQ [Y ∗ ] : Y attainable ECC, Y ≤ X} ∀Q ∈ M.

(3.46)

Remark 3.45. V− (X) is a lower bound on the fair price for X, that is a fair price C0 for X should satisfy C0 ≥ V− (X). Indeed, if the derivative was priced at a lower value C0 < V− (X), then one could construct an arbitrage opportunity, as follows:

3.2 Finite Market Models: General Setting

- at time 0, short sell a portfolio with final value Y ≤ X and initial investment EQ [Y ∗ ] such that C0 < EQ [Y ∗ ] ≤ V− (X), buy X with the proceeds from the sale, and invest the remaining amount EQ [Y ∗ ] − C0 in the bond; - at time 1, the portfolio value will be X − Y + (EQ [Y ∗ ] − C0 )(1 + r) > 0. Remark 3.46. V− (X) is bounded from above by inf{EQ [X ∗ ], Q ∈ M}. This is a consequence of the equation (3.46) and monotonicity of the expectation. Remark 3.47. The suprimum in the definition (3.45) is attained. Indeed, we can equivalently look for a solution to the linear program maximize

α · S0

subject to α · S1 ≤ X, α ∈ Rd+1 . Analogously as for V+ (X), this problem also has an optimal solution, and this must be equal to V− (X), which then attains the suprimum in (3.45). If X is unattainable, as we assumed, then it is possible to prove (using functional analysis) that V− (X) < V+ (X). In this case, according to the arguments presented above, it is clear that any arbitrage-free initial price for X must lie in the open interval (V− (X), V+ (X)). The ends are not included because otherwise X would be attainable, a contradiction. On the other hand, if we consider an attainable ECC X, then, by Theorem 3.37, V+ (X) = V− (X) would be the unique arbitrage-free initial price for X. From these last considerations, it directly follows that A ECC X is attainable if and only if EQ [X ∗ ] has the same value for all Q ∈ M. By linear programming theory, we can furthermore prove another important representation of the two bounds on the fair price for X:

85

86

3. Finite Market Models

For any ECC X, V+ (X) = sup{EQ [X ∗ ], Q ∈ M}, V− (X) = inf{EQ [X ∗ ], Q ∈ M}.

Concluding, for any ECC X, the arbitrage-free initial price C0 must satisfy: inf EQ [X ∗ ] = V− (X) ≤ C0 ≤ V+ (X) = sup EQ [X ∗ ].

Q∈M

3.3

Q∈M

Calibration and Convergence of the Binomial Model

We come back to the specific case of a binomial model to solve two issues of a financial market model: one of practical interest, the calibration to the real market, and one of theoretical interest, the stability of the model when converging towards continuous-time trading (i.e. when the trading frequency goes to infinity). In the present section, we will introduce a complexity in the parametrization of the binomial model, in order to make the dependance on the time steps between consecutive trading dates explicit. Let us the time period between any two consecutive trading dates is not unitary any more, but is measured as δ := T /N , where T is still the time horizon of the model (measured in years) and N + 1 is the number of equally spaced N N N N trading dates in [0, T ], i.e. 0 = tN 0 < t2 < . . . < tN = T with δ = ti − ti−1

for all i = 1, . . . , N . By this change, the price processes and the sequence of random variables driving the stock price will be indexed over {0, . . . , N } (N )

instead of {0, . . . , T }, e.g. S (N ) = (Si stock at time

tN i .

(N )

)i=0,...,N where Si

is the price of the

Despite the unconvenient notation, which motivated the

use of unitary intervals in the previous sections, there is no other change in the model than indexes.

3.3 Calibration and Convergence of the Binomial Model

3.3.1

87

Calibration

The calibration of a model consists in deriving the values for the parameters specifying the model from the observed market data. A binomial model is identified by the values attributed to the following parameters: • the constant risk-free interest rate r, such that Bi = Bi−1 (1 + r) for all i = 1, . . . , N ; • the relative magnitude of the ‘up’ and ‘down’ moves of the stock price, that is u and d; • the probability p of an ‘up’ move of the stock price, which determines univocally the objective/real world/historical probability P. The data that we can either directly get or infer from the observation of the market are: • the market annual interest rate rˆ, that accrues continuously in time according to the compound rule, i.e. an investment of value x at time s ∈ [0, T ] in a savings account grows to a value xerˆ(t−s) at time t ∈ [s, T ], provided that time is measured in years; • statistical estimates (from historical time series) m, ˆ σ ˆ of the average rate and of the volatility of the stock price log-returns, that is N 1 X S¯i mδ ˆ = log ¯ , N i=1 Si−1 2 N 1 X S¯i 2 , σ ˆ δ = log ¯ − mδ ˆ N − 1 i=1 Si−1

where S¯0 , S¯1 , . . . , S¯N are the stock prices observed on the market at times 0, tN 1 , . . . , T respectively. Let us see how to calibrate the model parameters to the market data. The relation between the interest rates is 1 + r = erˆδ .

(3.47)

88

3. Finite Market Models

Then, we define the annual rate of return of the stock log-price as µ ∈ R such that ST = S0 eµT , whose expectation gives the average rate of return m := EP [µ]. This can be computed thanks to the following remarks: ! N N Y X µT = log ξi = log ξi , i=1

i=1

from which the independence of the sequence of random variables (ξi )i=1,...,N gives " N # 1 P X N m= E log ξi = EP [log ξ1 ] = δ −1 (p log u + (1 − p) log d). (3.48) T T i=1 Analogously, the volatility σ of the stock log-price is defined as the standard p deviation of the log-price return per unit of time, i.e. σ := T Var(µ), which satisfies N X 1 σ 2 = Var log ξi T i=1

! =

N u 2 Var (log ξ1 ) = δ −1 p(1 − p) log . T d

(3.49)

By replacing in (3.48) and (3.49) the variables m, σ with their respective historical estimates m, ˆ σ ˆ , we obtain a non-linear system of two equations in three unknowns, the three model parameters u, d, p. In order to find a solution, we have to impose another a-priori condition. The most common modeling choices for this aim are p = 1/2 and ud = 1. Using p = 1/2, the system becomes ud = e2δm , √

u = e2σ δ , d with solution u = eσ

√

δ+mδ

,

d = e−σ

√

δ+mδ

while using ud = 1, the system becomes 2p = 1 + mδ , log u σ 2 δ = 4p(1 − p)(log u)2 ,

,

3.3 Calibration and Convergence of the Binomial Model

89

with solution √ q 2 σ δ 1+δ ( m σ )

u=e

,

d=e

√ q 2 −σ δ 1+δ ( m σ )

.

In both cases, the solution is of the form √ √ = 1 + σ δ + o( δ), √ √ √ √ d = e−σ δ+o( δ) = 1 − σ δ + o( δ), u = eσ

√

√ δ+o( δ)

(3.50)

for δ going to 0.

3.3.2

Convergence and the Black-Scholes formula

We want to assess the consistency of the binomial model, by verifying whether it is stable or not, i.e. whether the discrete-time model converges to some continuous-time limit. We do this by studying the behavior of the price of European contingent claims when the trading frequency increases to infinity. Indeed, once fixed the time horizon T , for each number of trading periods N ∈ N we have a different binomial model, providing different prices for the traded securities; we expect prices to be more and more precise as N , hence the trading frequency, increases. In order for the model to make sense, prices cannot diverge or oscillates for N going to inifinity. What we’ll find out is that not only the binomial model is stable, but it also is, under some a priori assumption, a discrete-time approximation of the well known Black-Scholes model in continuous time. Since we want to make the trading frequency vary, we make the dependence of the parameters on the number of trading dates explicit: we denote by r(N ) , u(N ) , d(N ) , p(N ) the parameters of the binomial model on [0, T ] with (N )

time step δ (N ) = T /N , by (ξi

)i=1,...,N the random variables driving the

stock price S (N ) , and by q (N ) the value defining the risk-neutral probability Q(N ) . As observed before, r(N ) satisfies 1 + r(N ) = erˆδ

(N )

, where rˆ denotes the

annual risk-free rate (observable on the market). Moreover, coherently with the results of Section 3.3.1, we assume that u(N ) , d(N ) are of the following

90

3. Finite Market Models

form: u(N ) = eσ

√ δ (N ) +αδ (N )

,

d(N ) = e−σ

√

δ (N ) +βδ (N )

,

(3.51)

where α, β ∈ R. First, we observe that the asymptotic behavior of the risk-neutral measure does not depend on α, β. Lemma 3.48. If u(N ) , d(N ) are of the form (3.51), then 1 lim q (N ) = . N →∞ 2 Proof. Under the assmptions, the definition of the risk-neutral measure gives us (N )

√

(N )

(N )

1 + r(N ) − d(N ) erˆδ − e−σ δ +βδ √ √ q (N ) = = . u(N ) − d(N ) eσ δ(N ) +αδ(N ) − e−σ δ(N ) +βδ(N ) Hence, by a Taylor expansion of the exponentials, as N → ∞ we get 2 √ √ 1 (N ) (N ) (N ) (N ) + o δ (N ) rˆδ + σ δ − βδ − 2 βδ − σ δ √ √ σ δ (N ) + αδ (N ) + σ δ (N ) − βδ (N ) + o (δ (N ) ) √ 2 rˆδ (N ) + σ δ (N ) − βδ (N ) − σ2 δ (N ) + o δ (N ) √ 2σ δ (N ) + (α − β)δ (N ) + o (δ (N ) ) √ 2 σ δ (N ) + rˆ − σ2 − β δ (N ) + o δ (N ) √ 2σ δ (N ) + (α − β)δ (N ) + o (δ (N ) ) √ √ 2 σ + rˆ − σ2 − β δ (N ) + o δ (N ) 1 √ −−−→ , √ N →∞ 2 2σ + (α − β) δ (N ) + o δ (N ) (N )

q (N ) = =

=

=

(3.52)

(3.53)

where in the both the numerator and the denominator of (3.52) we used the second-order Taylor expansions and we directly included the terms of second order in the small o of δ (N ) . For any simple ECC φ(ST ), the initial price in the Binomial model with N + 1 trading dates on [0, T ] is given by h i 1 Q(N ) −ˆ r T Q(N ) X (N ) E [φ(S )] = e E φ(S e ) , T 0 (1 + r(N ) )N

3.3 Calibration and Convergence of the Binomial Model

where X

(N )

:= log

N Y

(N ) ξi

i=1

=

N X

(N )

Yi

(N )

(N )

(N )

Q(N ) (Y1

(N )

Yi

(N )

:= log ξi

.

i=1

Note that the random variables {Yi Q(N ) (Y1

,

}i=1,...,N are i.i.d., with

√ = σ δ (N ) + αδ (N ) ) = q (N ) , √ = −σ δ (N ) + βδ (N ) ) = 1 − q (N ) .

We now prove, through the following two lemmas, the convergence in distribution of the sequence of random variables {X (N ) , N ∈ N}, which implies the convergence of contingent claim prices (see Theorem 3.51 below).

Lemma 3.49. We have: Q(N )

lim E

N →∞

(N )

lim VarQ

N →∞

(N ) X = X (N )

σ2 rˆ − T, 2

= σ 2 T.

Proof. First, using (3.53), we compute 1 − q (N ) for N small:

1 − q (N )

√ √ −β δ (N ) + o δ (N ) √ =1− √ (N ) (N ) 2σ + (α − β) δ + o δ √ √ √ 2 2σ + (α − β) δ (N ) − σ − rˆ − σ2 − β δ (N ) + o δ (N ) √ = √ 2σ + (α − β) δ (N ) + o δ (N ) √ √ σ2 (N ) (N ) σ + α − rˆ + 2 δ +o δ √ . = (3.54) √ 2σ + (α − β) δ (N ) + o δ (N ) σ + rˆ −

σ2 2

91

92

3. Finite Market Models

Then, using (3.53) and (3.54), for N small we compute h i √ √ (N ) (N ) Y1 = q (N ) σ δ (N ) + αδ (N ) + (1 − q (N ) ) −σ δ (N ) + βδ (N ) EQ √ √ 2 √ σ + rˆ − σ2 − β δ (N ) + o( δ (N ) ) √ = σ δ (N ) + αδ (N ) √ (N ) (N ) 2σ + (α − β) δ + o δ √ √ 2 √ σ + α − rˆ + σ2 δ (N ) + o( δ (N ) ) √ + −σ δ (N ) + βδ (N ) √ 2σ + (α − β) δ (N ) + o δ (N ) √ σ2 2 (N ) (N ) (N ) σ δ + ασδ + σδ rˆ − 2 − β + o(δ (N ) ) √ = √ (N ) (N ) 2σ + (α − β) δ + o δ √ 2 σ 2 δ (N ) − βσδ (N ) + σδ (N ) α + σ2 − rˆ + o(δ (N ) ) √ − √ 2σ + (α − β) δ (N ) + o δ (N ) α+β σ2 (N ) (N ) σδ (α + β) + 2σδ rˆ − 2 − 2 + o(δ (N ) ) √ = √ 2σ + (α − β) δ (N ) + o δ (N ) 2 2σδ (N ) rˆ − σ2 + o(δ (N ) ) √ = √ (N ) (N ) 2σ + (α − β) δ + o δ 2 2σδ (N ) rˆ − σ2 + o(δ (N ) ) = 2σ (1 + o(1)) σ2 (N ) =δ rˆ − + o(δ (N ) ). (3.55) 2 Then, for N small we get Q(N )

E

h i (N ) (N ) Q(N ) X = NE Y1 =

σ2 rˆ − T + o(1), 2

which proves the first statement. Next, using (3.55), for N small we get 4

4

You can prove that, for a random variable ξ on (Ω, F, P) with distribution Pξ =

pδu + (1 − p)δd , it holds EP [ξ 2 ] = (u + d)EP [ξ] − ud. Indeed: EP [ξ 2 ] = u2 p + d2 (1 − p) ± ud(1 − p) = (u + d)(up + d(1 − p)) − ud.

3.3 Calibration and Convergence of the Binomial Model

(N )

EQ

93

h i (N ) h (N ) i (N ) Y1 − log u(N ) log d(N ) (Y1 )2 = log u(N ) + log d(N ) EQ h i √ √ (N ) (N ) Q(N ) (N ) (N ) (N ) (N ) = δ (α + β)E Y1 − σ δ + αδ −σ δ + βδ = σ 2 δ (N ) + o(δ (N ) ),

(3.56)

hence (N )

VarQ

(N ) (N ) Y1 X (N ) = N VarQ h i h i2 (N ) 2 (N ) Q(N ) Q(N ) =N E (Y1 ) − E Y1 = σ 2 T + o(1)

ends the proof. Lemma 3.50. The sequence of random variables {X (N ) , N ∈ N} converges in distribution (under Q(N ) ) to a random variable X

5

with normal distribu-

tion, precisely X ∼ Nr− σ2 T,σ2 T . 2

Proof. By the L´evy’s theorem, if the characteristic function ϕ(N ) of X (N ) converges pointwise to a function ϕ which is continuous at the origin, then ϕ is the characteristic function of a random variable X and X (N ) converges to X in distribution. Thus, in order to prove the lemma, it is enough to prove the pointwise convergence of the characteristic function ϕ(N ) , (N )

ϕ 5

(η) = E

Q(N )

h

iηX (N )

e

i

η ∈ R,

,

Recall that a sequence of Rd -valued random variables (Xn )n∈N , Xn defined on a

probabiliy space (Ωn , Fn , Pn ) for all n ∈ N, converges in distribution (or in law ) to an Rd -valued random variable X on (Ω, F, P) if and only if lim EPn [ϕ(Xn )] = EP [ϕ(X)]

n→∞

∀ϕ ∈ Cb (Rd ),

where Cb (Rd ) is the space of continuous and bounded functions Rd → R.

94

3. Finite Market Models

to the caracteristic function of a Gaussian variable with mean

r−

σ2 2

T

2

and variance σ T . (N )

We denote Y := Y1

(N )

and recall that the random variables {Yi

}i=1,...,N are

independent and identically distributed. By using the properties of characteristic functions and the expressions found in (3.55) and (3.56), we have (N ) iηY N ϕ(N ) (η) = EQ e √ N iη δ (N ) √ Y Q(N ) (N ) δ =E e 2 N 2 (N ) √ Y Y η δ (N ) (N ) Q (N ) Q √ E + o(δ ) = 1 + iη δ (N ) E − 2 δ (N ) δ (N ) N σ2 η2 2 1 = 1+ , for N small. iηT rˆ − − T σ + o(1) N 2 2 Thus: lim ϕ

N →∞

(N )

(η) = e

2 2 iηT rˆ− σ2 − η2 T σ 2

∀η ∈ R,

which is the characteristic function of a Gaussian variable with mean rˆ −

σ2 2

2

and variance σ T . Theorem 3.51 (Convergence of prices in the binomial model). Let φ be a bounded continuous function. The initial arbitrage-free price of the simple (N )

European contingent claim φ(ST ) in the N -period binomial model converges as N goes to infinity to e−ˆrT EQ [ϕ(X)],

(3.57)

(N )

where ϕ is a real map such that φ(ST ) = ϕ(X (N ) ) and Q is aprobability 2 measure under which X has a Gaussian distribution with mean rˆ − σ2 T and variance σ 2 T . This theorem follows immediately from Lemma 3.50 and the footnote on page 97. Note that the expectation in (3.57) can be explicitly computed, thanks to the underlying Gaussian density. In particular, in the case of European Put and Call options, we obtain closed-formulae, the so-called Black-Scholes formulae, as shown in Corollary 3.52 below.

T

3.3 Calibration and Convergence of the Binomial Model (N )

Corollary 3.52 (Black-Scholes formulae). Let P0

(N )

, C0

95

denote the initial

price of an European Put option and of an European Call option, respectively, with the same strike price K and maturity T , in an N -period binomial model with parameters r(N ) , u(N ) , d(N ) given by 1 + r(N ) = erˆδ

(N )

√

u(N ) = eσ

,

δ (N ) +αδ (N )

,

d(N ) = e−σ

√

δ (N ) +βδ (N )

,

where α, β ∈ R. Then, as the number N of time intervals goes to infinity, the price converges to the Black-Scholes price: (N )

lim P0

N →∞

= P0BS = Ke−ˆrT Φ(−d2 ) − S0 Φ(−d1 )

(3.58)

for the European Put option and (N )

lim C0

N →∞

= C0BS = S0 Φ(d1 ) − Ke−ˆrT Φ(d2 )

(3.59)

for the European Call option, where Φ is the standard Normal distribution function, i.e. 1 Φ(x) = √ 2π and d1 = d2 =

Z

x

y2

e− 2 dy,

x ∈ R,

−∞

2 S log( K0 )+ rˆ+ σ2 T √ , σ T 2 S √ log( K0 )+ rˆ− σ2 T √ d1 − σ T = . σ T

(3.60)

Proof. By Theorem 3.51, in order to obtain the formula (3.58) we only have to prove that e−ˆrT EQ

h

K − S0 eX

+ i

= Ke−ˆrT Φ(−d2 ) − S0 Φ(−d1 ),

then the formula (3.59) is obtained by using the Put-Call parity formula illustrated in Corollary 1.5. Note that we couldn’t apply Theorem 3.51 to the Call option directly, for the relative payoff function is not bounded. So, we rewrite X in terms of a standard Gaussian variable Z ∼ N0,1 : √ σ2 T + σ T Z, X = rˆ − 2

96

3. Finite Market Models

and EQ

h

K − S0 eX

+ i

= EQ

K − S0 eX 1{ST

By a simple computation, we see that ST = S0 eX < K if and only if Z < −d2 . Thus: Q

E

h

X +

K − S0 e

i

= I1 − I2 ,

where I1 = KQ(Z < −d2 ) = KΦ(−d2 ) and 2 √ rˆ− σ2 T +σ T Z I2 = S0 E e 1{Z<−d2 } Z −d2 √ σ2 x2 1 rˆT = e S0 √ e− 2 T +σ T x− 2 dx 2π −∞ Z −d2 −σ√T y2 1 = erˆT S0 √ e− 2 dy 2π −∞ Q

√ (by c.o.v. y = x − σ T )

= erˆT S0 Φ(−d1 ). Finally, by the put-call parity formula, we have C0BS = P0BS + S0 − Ke−ˆrT = S0 (1 − Φ(−d1 )) − Ke−ˆrT (1 − Φ(−d2 )) = S0 Φ(d1 ) − Ke−ˆrT Φ(d2 ).

Appendix A Elements of Probability Definition A.1. A probability space is a triple (Ω, F, P), where Ω is a nonempty set, F is a σ-algebra, i.e. F ⊆ P(Ω) such that (i) ∅ ∈ F, (ii) A ∈ F

⇒

Ac ≡ Ω \ A ∈ F,

(iii) An ∈ F ∀n ∈ N

⇒

∪ An ∈ F,

n∈N

and P is a probability measure on (Ω, F), i.e. a map P : F → [0, 1] such that (i) P(∅) = 0, P(Ω) = 1 (ii) An ∈ F∀n ∈ N, An ∩ Am = ∅ ∀n 6= m X ⇒ P ∪ An = P(An ). n∈N

n∈N

Given a probability space (Ω, F, P), Ω is called the sample space, any element of the σ-algebra, A ∈ F, is called an event, and P(A) is the probability of the event A. The intuition behind the concept of σ-algebra in Probability is that it contains the information on a specific random phenomenon or experiment: Ω is the set of all possible outcomes, and F is the set of all the events of which we want to compute the probability. 97

98

APPENDIX Definition A.2. Let M ⊆ P(Ω), the smallest σ-algebra containing M is called the σ-algebra generated by M and is denoted by \ σ(M) := F. F σ-algebra M⊆F

In the case Ω = RN , there is a special σ-algebra, the one generated by the Euclidean topology on RN : B(RN ) := σ {open sets in RN } . The collection of all Lebesgue-measurable sets is another σ-algebra and it strictly contains the Borel σ-algebra. Unless otherwise specified, we always consider RN equipped with the Borel σ-algebra B(RN ), denoted briefly by B when there is no ambiguity. Definition A.3. A RN -valued random variable (briefly denoted r.v.) X on a probability space (Ω, F, P) is a measurable function from (Ω, F) to (RN , B), i.e. X : Ω → RN ,

such that X −1 (H) ∈ F ∀H ∈ B.

The probability measure PX on (RN , B) defined by PX (H) = P(X −1 (H)),

H ∈ B,

is called the distribution (or law ) of X, and we write X ∼ PX . We will use the more intuitive notation P(X ∈ H) := P(X −1 (H)) = P{ω ∈ Ω : X(ω) ∈ H}. Given a random variable X, the σ-algebra generated by all inverse images of Borel sets under X is denoted by σ(X) := σ {X −1 (H), H ∈ B} ⊆ F and is called the σ-algebra generated by X. To better explain the meaning of information associated with σ-algebras, let us consider the following example.

Elements of Probability

99

Example A.4. Consider the rolling of a die: the sample space is the set of all possible outcomes Ω = {1, 2, 3, 4, 5, 6} and the set of all possible events is F = P(Ω). Suppose that we are only interested in knowing whether the result of rolling will be an even or an odd number. Then, we can define a random variable X as X(n) =

1,

if n is even

,

n ∈ Ω,

−1, if n is odd, which generates the σ-algebra σ(X) = {∅, Ω, {2, 4, 6}, {1, 3, 5}} ⊂ F. The information that is needed to study the phenomenon we are interested in is contained in σ(X): we only need to compute the probabilities of the events in σ(X). These probabilities are given by the distribution of X: P X ({1}) = P(X = 1) = P({2, 4, 6}) and P X ({−1}) = P(X = −1) = P({1, 3, 5}). A further example is given by considering two random variables X, Y such that “X depends on Y ”. This situation is mathematically described in the following lemma. Lemma A.5. Let X, Y be R-valued random variables on (Ω, F). Then: X is σ(Y )-measurable if and only if there exists a Borel-measurable function f : R → R such that X = f (Y ). Thus the property ‘X is σ(Y )-measurable’ is translated into intuitive terms as “X depends on the information generated by Y ”. More generally, if G is a σ-algebra, saying that “X depends on the information contained in G” means that X is G-measurable. Definition A.6. A probability measure on (RN , B) is called a distribution (on RN ).

100

APPENDIX

Remark A.7. To specify the distribution of a certain random variable on some probability space, we do not need to know the underlying probability space. Indeed, two random variables X, Y defined on different probability spaces, d

can have the same distribution, in which case we write X = Y . For instance, a real-valued random variable X on (Ω, F, P) has the same distribution of the constant random variable Id on (R, B, PX ) defined as the identity function Id(x) = x for all x ∈ R. Thanks to the Lebesgue integration theory, it is very simple to construct a distribution. Proposition A.8. Let f : RN → R+ be a non-negative Borel-measurable function such that Z f (x)dx = 1, RN

then the measure P on (RN , B) defined by Z f (x)dx, P (H) =

H∈B

(A.1)

H

is a distribution on RN . The function f is called the density of P with respect to the Lebesgue measure. For simplicity, consider the case N = 1. The simplest example of a distribution is given by the so-called Dirac delta function (or Dirac distribution): given x0 ∈ R, the Dirac distribution concentrated at x0 is defined by 1, if x0 ∈ H, δx0 (H) = 0, if x ∈ / H. 0

A.0.1

Finite Probability Spaces

Assumption A.0.1. Let (Ω, F, P) be a finite probability space where the sample space is a finite set Ω = {ω1 , ω2 , . . . , ωn },

n ∈ N.

Elements of Probability

101

When the sample space is finite, any random variable can only take finitely many values. Let us denote x1 , x2 , . . . , xm ∈ R the possible values taken by a real-valued random variable X on (Ω, F, P), that is X(Ω) = {x1 , . . . , xm },

m ∈ N, m ≤ n.

In this case, the distribution of X takes the form of a linear combination of Dirac delta functions: X

P (H) =

m X

δxi (H)P(X = xi ),

H ∈ B.

i=1

Definition A.9. The mean, or expectation, of X on (Ω, F, P) is defined by P

E [X] =

m X

xi P(X = xi ).

i=1

The variance of X on (Ω, F, P) is defined by h 2 i P P = EP X 2 − EP [X]2 . Var(X) = E X − E [X] Let Y be another real-valued random variable on (Ω, F, P), the covariance of X, Y is defined by Cov(X, Y ) = EP

X − EP [X]

Y − EP [Y ]

.

When there is no ambiguity about the probability measure considered, we simply denote by E[X] the expectation of X. Proposition A.10. Let X be a real-valued random variable on (Ω, F, P) and f : R → R a measurable function, then f ◦ X is a random variable on (Ω, F, P) and X

EP [f (X)] = EP [f ]. Proof. Since X is a random variable and f : (R, B) → (R, B) is measurable, for every Borel set H ∈ B, we have f −1 (H) ∈ B, hence (f ◦ X)−1 (H) = X −1 f −1 (H) ∈ F.

102

APPENDIX

Then, assuming f ({x1 , . . . , xm }) = {y1 , . . . , yk }, k ≤ m, we have: P

E [f (X)] = =

m X i=1 k X

f (xi )P(f (X) = f (xi )) yj P(f (X) = yj)

j=1

=

k X

yj P(X ∈ {xi : i ∈ {1, . . . , m}, f (xi ) = yj })

j=1

=

k X

yj PX (f = yj )

j=1 X

= EP [f ].

Example A.11. The distribution of the random variable ξ characterizing the Binomial model is P ξ = pδu + (1 − p)δd ,

0 < d < u, p ∈ (0, 1).

The mean of ξ is EP [ξ] = up + d(1 − p). The variance of ξ can be computed by applying Proposition A.10 to f : x 7→ x2 and X = ξ − up − d(1 − p), and it is equal to Var(ξ) = EP (ξ − up − d(1 − p))2 = p(1 − p)(u − d)2 . Now we see various versions of the concept of independence in Probability theory. Definition A.12. Two events A, B ∈ F are independent if P(A ∩ B) = P(A)P(B). Definition A.13. Given a non-negligible event B, i.e. B ∈ F such that P(B) > 0, the conditional probability P(·|B) given B is a probability measure on (Ω, F, P) defined as P(A|B) =

P(A ∩ B) , P(B)

A ∈ F.

Elements of Probability

103

The conditional probability P(A|B) represents the probability that the event A occurs given that the event B occurred. Remark A.14. If B is a non-negligible event, A, B ∈ F are independent if and only if P(A|B) = P(A). Remark A.15. If A, B ∈ F are independent events, then also Ac , B c , Ac , B and A, B c are independent events. Indeed: P(Ac ∩ B c ) = P ((A ∪ B)c ) = 1 − P(A ∪ B) = 1 − (P(A) + P(B) − P(A ∩ B)) = (1 − P(A))(1 − P(B)). The other cases are proven analogously. Definition A.16. Two sub-σ-algebras G, H of F, i.e. G, H ⊆ F σ-algebras on Ω, are independent if P(A ∩ B) = P(A)P(B),

∀A ∈ G, B ∈ H.

Two random variables X, Y on (Ω, F, P) are independent if the correspondent σ-algebras σ(X), σ(Y ) are independent. The definition and properties of independence extend to the case of families of random variables. Definition A.17. Let {Xi }i∈I be a finite family of real-valued random variables on (Ω, F, P), to any subset of indexes F ⊆ I, #F = k ∈ N, we associate a random variable XF : Ω → Rk ,

XF : ω 7→ (Xi (ω))i∈F .

The family {Xi }i∈I is said independent if, for all disjoint subsets of indexes L, M ⊂ I, L ∩ M = ∅, the random variables XL , XM are independent.

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Note that the independence of a family of random variables is stronger than the pairwise independence. In particular, we have the following equivalent condition. Lemma A.18. The family of random variables {Xi }i∈I is independent if and only if: for any finite subset of indexes F ⊂ I and any family of Borel sets (Hi )i∈F , Hi ∈ B, ! P

\

{Xi ∈ Hi }

i∈F

=

Y

P (Xi ∈ Hi ) .

i∈F

The proof is based on the Dynkin’s Lemma, which is outside the scope of this course. Proposition A.19 (To be proven in the exercise sheet 3). If X, Y are independent real-valued random variables on (Ω, F, P), then: (i) E[XY ] = E[X]E[Y ]; (ii) Cov(X, Y ) = 0, in particular Var(X + Y ) = Var(X) + Var(Y ); (iii) if Z is a σ(Y )-measurable random variable, then X, Z are independent; (iv) if f, g : R → R are measurable functions, then f (X), g(Y ) are independent random variables; (v) if X is σ(Y )-measurable, then X is constant P-almost surely (i.e. there exists an x ∈ R such that for some ω ∈ Ω, X(ω) = x ∈ R, and denoting Ωx = X −1 ({x}) 6= ∅ we have P(A) = 0 for any A ∈ F, A ⊆ (Ωx )c ). Note that the converse of the implication (ii) is not true: there are couples of random variables with zero covariance but that are not independent.

A.0.2

Conditional Expectation

Assumption A.0.1 is equivalent to having a generic (possibly not finite) sample space and only consider simple random variables, that is random

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variables X taking only a finite number of values: X(Ω) = {x1 , . . . , xn } ∈ Rn . Such a random variable can be written as X=

n X

x i 1A i ,

where Ai = X −1 (xi ) ∈ F.

i=1

In the present section, we do not require Assumption A.0.1, and so (Ω, F, P) is a generic probability space, but we mostly consider simple random variables. On a generic (non-finite) sample space, the expectation of a random variable is defined as the integrable of the random variable with respect to the P probability measure. In the case of a simple random variable X = ni=1 xi 1Ai on (Ω, F, P), we have Z

Z XdP ≡

P

E [X] = Ω

X(ω)P(dω) = Ω

n X

xi P(Ai ).

i=1

Note that for any event A ∈ F, we have EP [X1A ] =

R A

XdP.

Assumption A.0.2. The probability space (Ω, F, P) is complete, that is: for any event E ∈ F such that P(E) = 0, all its subsets A ⊆ E are also measurable events, i.e. A ∈ F, and consequently P(A) = 0. All events with null probability are called negligible events. Note that, in case (Ω, F, P) was not complete, we could always extend it to a probability ¯ that includes the negligible sets in the following way: denote ¯ P) space (Ω, F, N := {A ⊂ E | E ∈ F, P(E) = 0} and let F¯ be the smallest σ-algebra containing both F and N , i.e. F¯ := σ(F ∪ N ) = σ ({B ⊆ Ω | B = E ∪ A, E ∈ F, A ∈ N }) , ¯ on F¯ by defining, for all and finally extend P to a probability measure P ¯ ¯ P(B) B ∈ F, = P(E), where B = E ∪ A and E ∈ F, A ∈ N . In financial applications, the available information plays an important role and needs to be considered when computing the expectation of a random variable, e.g. the future price of a security. From this, the introduction of the concept of ‘conditional expectation’ is natural. In particular, given a random

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variable X and a σ-algebra G, we would like to compute an estimate of X which takes into account the set of information contained in G. To define this object, we go through gradual steps. Let X be a simple random variable on (Ω, F, P). (I) Given an event B ∈ F, the conditional expectation of X given B is defined as

Z 1 E[X|B] := XdP, (A.2) P(B) B which is equal to the expectation of X under the conditional probability P(·|B), i.e. E[X|B] = EP(·|B) [X]. Indeed: Z P(·|B) X(ω)P(dω|B) E [X] = Ω Z 1 = X(ω)1B (ω)P(dω) P(B) Ω Z 1 X(ω)P(dω). = P(B) B

Note that in this case the conditional expectation is a constant. (II) Given an event B ∈ F such that 0 < P(B) < 1, the conditional expectation of X given the σ-algebra σ({B}) generated by the singleton {B} is defined as E[X|σ(B)](ω) :=

E[X|B]

if ω ∈ B,

∀ω ∈ Ω

(A.3)

E[X|B c ] if ω ∈ B c ,

Note that E[X|σ(B)] is a random variable. (III) Given another simple random variable Z on (Ω, F, P), taking values Z(Ω) = {z1 , . . . , zm } ∈ Rm , the conditional expectation of X given the σ-algebra generated by Z is defined as E[X|σ(Z)](ω) :=

m X

E[X|{Z = zj }]1{Z=zj } (ω),

ω ∈ Ω.

j=1

(A.4)

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We will use the notation E[X|Z] := E[X|σ(Z)]. Note that the definition in (A.4) is only possible in the discrete setting (i.e. finite probability space or simple random variables).

Before introducing the general definition of conditional expectation, let us look at the properties of the objects defined in (A.3)-(A.4). Lemma A.20. Let G be the σ-algebra generated by either the event B or the random variable Z. Then: a random variable Y is a version of the conditional expectation with respect to G, i.e. Y = E[X|G] P-almost surely, if and only if it satisfies (i) Y is G-measurable; R

(ii)

XdP = A

R A

Y dP for all A ∈ G.

Proof. (⇒) The measurability with respect to G is a direct consequence of the definitions in (A.3)-(A.4). Regarding the property (ii), we distinguish the two cases. If G = σ(B) = {∅, B, B c , Ω}, then Z

Z E[X|G]dP =

(E[X|B]1B (ω) + E[X|B c ]1B c (ω)) P(dω)

A

A

=

0 R

B

if A = ∅, if A = B,

E[X|B]P(dω)

R E[X|B c ]P(dω) if A = B c , Bc R R E[X|B]P(dω) + E[X|B c ]P(dω) if A = Ω B Bc and Z

Z E[X|B]P(dω) = E[X|B]P(B) =

B

(analogously for B c ).

X(ω)P(dω) B

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If G = σ(Z) = σ({Z = zj }, j = 1, . . . , m), then, for all j = 1, . . . , m, Z

Z E[X|{Z = zj }]P(dω)

E[X|G]dP = {Z=zj }

{Z=zj }

= E[X|{Z = zj }]P({Z = zj }) Z = XdP {Z=zj }

(⇐) Since both Y and E[X|G] are G-measurable, also Y − E[X|G] is a Gmeasurable random variable. Thus the event {Y − E[X|G] > 0} = {(Y − E[X|G])−1 ((0, +∞))} ∈ G is contained in G, so that we can apply the property (ii) on it: Z

Z (Y − E[X|G])dP =

Y −E[X|G]>0

(X − X)dP = 0. Y −E[X|G]>0

This imply that P(Y −E[X|G] > 0) = 0. Through an analogous procedure we prove P(Y −E[X|G] < 0) = 0. So eventually we have P(Y 6= E[X|G]) = 0. Another property of the conditional expectation of a simple random variable X given the σ-algebra generated by another simple random variable as defined in (A.4) is that it can be rewritten as a sum of the possible values of X weighted with the respective conditional probabilities, analogously as the expectation of X is the sum of its values weighted with the respective probabilities. Lemma A.21. Let X, Z be two simple random variables on (Ω, F, P), then E[X|Z] =

n X

xi P({X = xi }|Z),

i=1

where X(Ω) = {x1 , . . . , xn } and we denote P(F |Z) := E[1F |Z] for any F ∈ F. Proof. Suppose Z(Ω) = {z1 , . . . , zm }. For all j = 1, . . . , m, for any ω ∈ {Z =

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zj }, we have E[X|Z](ω) = E[X|{Z = zj }] Z 1 XdP = P({Z = zj }) {Z=zj } n X 1 = xi P({X = xi } ∩ {Z = zj }) P({Z = zj }) i=1 =

n X

xi P({X = xi }|{Z = zj }).

i=1

Note that, given a simple random variable Z on(Ω, F, P), the conditional expectation of X with respect to Z can be written as: E[X|Z](ω) = E[X|Z = Z(ω)],

ω ∈ Ω.

(A.5)

The following is another equivalent condition for independence of simple random variables using the conditional expectation. Lemma A.22. Let X, Z be two simple random variables on (Ω, F, P), X(Ω) = {x1 , . . . , xn }, then X, Z are independent if and only if the random variable P({X = xi }|Z) is a constant on (Ω, F, P), for every i = 1, . . . n. Proof. By (A.5) and(A.2), for any ω ∈ Ω we have P({X = xi }|Z)(ω) = E[1{X=xi } |Z = Z(ω)] Z 1 1{X=xi } dP = P(Z = Z(ω)) {Z=Z(ω)} P({X = xi } ∩ {Z = Z(ω)} = P(Z = Z(ω)) = P({X = xi }|{Z = Z(ω)}. If X, Z are independent, this is clearly a constant, by Remark A.14. Conversely, suppose that Z(Ω) = {z1 , . . . , zm }. If, for all j = 1, . . . , m and for all ω ∈ Z −1 ({zj }), P({X = xi }|Z)(ω) = P ({X = xi }|{Z = zj }) = pi ,

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then P ({X = xi } ∩ {Z = zj }) = pi P(Z = zj ), and summing over j = 1, . . . , m we obtain P(X = xi ) = pi . By substitution in the previous equation, we proved that the events {X = xi } and {Z = zj } are independent. Since this holds for all i = 1, . . . , n and j = 1, . . . , m, that means that X, Z are independent.

A.0.3

Conditional Expectation: General Definition and Properties

The properties (i)-(ii) shown in Lemma A.20 for the conditional expectation with respect to a particular kind of σ-algebra, namely the one generated by either a non-negligible event or another simple random variable, can be extended to the definition of the conditional expectation with respect to a generic σ-algebra. Theorem A.23. Let X be an integrable random variable on (Ω, F, P) and G be any σ-algebra contained in F, G ⊆ F. There exists a random variable Y on (Ω, F, P) satisfying (i) Y is integrable, i.e. (ii)

R

XdP = A

R A

R Ω

|Y |dP < ∞, and G-measurable;

Y dP , or equivalently E[X1A ] = E[Y 1A ], for all A ∈ G.

Moreover, Y is unique up to a negligible event, that is: if there exists another random variable Z on (Ω, F, P) satisfying (i)-(ii), then it must be a version of Y , i.e. Y = Z P-almost surely. Definition A.24. The conditional expectation of X with respect to G is any member of the equivalence class of random variables on (Ω, F, P) satisfying (i)-(ii) in Theorem A.23. It is denoted E[X|G]. Definition A.24 is given under the most general assumptions on the probability space and random variables considered. In the case of simple random

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variables or finite sample spaces, we do not require the integrability of Y in (i), because it is always satisfied. Note that the conditional expectation E[X|G] is G-measurable even when X is not. It represents the best estimate of X based on the information contained in G. Proof (Theorem A.23 - Almost sure uniqueness). We proceed in a similar way to the proof of the implication (⇐) in Lemma A.20. Assume that there exist two random variables Y, Z on (Ω, F, P) satisfying (i)-(ii). Then, {Y > Z} = {(Y − Z)−1 ((0, ∞))} ∈ G by (i), and by (ii) we have Z Z (Y − Z)dP = (X − X)dP = 0. {Y >Z}

{Y >Z}

Thus P(Y > Z) = 0. In a symmetric way we obtain P(Y < Z) = 0, and so Y = Z P-almost surely. In order to prove the existence, we have to resort to a classical result in Probability. First, given two measures P, Q on (Ω, F), we say that Q is P-absolutely continuous if, for all A ∈ F such that P(A) = 0 , we also have Q(A) = 0. In this case we write Q F P, or simply Q P when there is no ambiguity. If Q is P-absolutely continuous and P is Q-absolutely continuous, we say that P and Q are equivalent and we write Q ∼ P. Note that the notion of absolute continuity is related to the σ-algebra considered. Lemma A.25 (Radon-Nikodym Theorem). Let P, Q be finite measures on (Ω, F) such that Q P. Then, there exists a map L : Ω → [0, ∞) such that: 1. L is F-measurable (i.e. L is a random variable), 2. L is P-integrable, 3. Q(A) =

R A

LdP for all A ∈ F.

Moreover, L is unique up to a negligible event. L is called the density, or the Radon-Nikodym derivative, of Q with respect to P on F, and the notation L=

dQ dQ ≡ |F dP dP

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is used. For instance, any distribution P defined as in Proposition A.8 is absolutely continuous with respect to the Lebesgue measure m, that is P B m. Actually the inverse also holds: all measures which are m-absolutely continuous, can be written in the form (A.1). We are now able to prove the existence of the conditional expectation. Proof (Theorem A.23: Existence). Assume that X ≥ 0. Define a measure Q R on (Ω, G) by Q(A) = A XdP for all A ∈ G. Then Q is finite, because X is Pintegrable. Moreover, we have Q G P, thus, by Lemma A.25, there exists a R G-measurable and P-integrable random variable Y such that Q(A) = A Y dP for all A ∈ G. Such random variable Y satisfy the properties (i)-(ii). Remark A.26. Property (ii) in Definition A.24 is equivalent to the following: (ii bis) E[XV ] = E[Y V ] for all bounded and G-measurable random variables V on (Ω, F, P). Proof. We only prove it in the case where both X and V are simple random variables. (ii)⇒(ii bis). It is thanks to the fact that V is G-measurable and to the linearity of the expectation. Indeed, by assumption, V =

k X

where Bj ∈ G ⊆ F, ∀j = 1, . . . , k.

vj 1Bj ,

j=1

Then, by (ii), " E[XV ] = E X

k X i=1

# vi 1Bi =

k X i=1

vi E [X1Bi ] =

k X

vi E [Y 1Bi ] = E[Y V ].

i=1

(ii bis)⇒(ii). It is enough to consider all random variables of the form V = 1A , A ∈ G, to get (ii). Proposition A.27 (Properties of the conditional expectation). Let X, Y two integrable random variables on (Ω, F, P) and H, G ⊆ F two sub-σ-algebras of F, then:

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1. If X is G-measurable, then X = E[X|G]. 2. If X is independent of G, i.e. σ(X), G are independent, then E[X|G] = E[X]. 3. E [E[X|G]] = E[X]. 4. [Linearity on the argument] If a, b ∈ R, then E[aX+bY |G] = aE[X|G]+ bE[Y |G]. 5. [Linearity w.r.t. convex combinations of measures] Let λ ∈ [0, 1] and P, Q two probability measures on (Ω, F), then EλP+(1−λ)Q [X|G] = λEP [X|G] + (1 − λ)EQ [X|G]. 6. [Monotonicity] If X ≤ Y , then E[X|G] ≤ E[Y |G]. 7. If Y is G-measurable and bounded, then E[Y X|G] = Y E[X|G]. 8. If Y is independent of σ(X, G), then E[Y X|G] = E[Y ]E[X|G]. 9. If H ⊆ G, then E[E[X|G]|H] = E[E[X|H]|G] = E[X|H]. 10. [Jensen inequality] Let ϕ : R → R be a convex function such that ϕ(X) is P-integrable, then ϕ(E[X|G]) ≤ E[ϕ(X)|G]. Proof.

1. Trivial, by Definition A.24.

2. E[X] is a constant, thus σ(E[X]) = {∅, Ω} ⊆ G, i.e. E[X] is Gmeasurable. Then, for every bounded and G-measurable random variable V , X, V are independent and E[XV ] = E[X]E[V ] = E [E[X]V ] . 3. By (ii bis) with V = 1. 4. By Definition A.24, since the set of G-measurable random variables is a vector space, and the integral is linear.

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5. Again by linearity of the integral, but with respect to the measure. 6. By monotonicity of the integral. 7. Consider the random variable Z := Y E[X|G]. We want to prove that Z satisfies the properties (i)-(ii bis). Since both Y and E[X|G] are G-measurable, so is Z; for every bounded and G-measurable random variable V , E[ZV ] = E [Y E[X|G]V ] = E[Y V X], by (ii bis) for E[X|G]. Thus Z = E[XY |G]. 8. As before, let us consider the random variable Z := E[Y ]E[X|G] and prove (i)-(ii bis). Z is G-measurable as the product of a constant and a G-measurable variable. Moreover, for every bounded and G-measurable random variable V , E[ZV ] = E [E[Y ]E[X|G]V ] = E [E[Y ]XV ] = E[Y ]E[XV ] = E[Y XV ], by (ii bis) for E[X|G] and the independence of Y, XV . Thus Z = E[XY |G]. 9. Consider Z := E [E[X|G]|H]. It is H-measurable by definition; for every bounded and H-measurable random variable V , E[ZV ] = E [E [E[X|G]|H] V ] = E [E [V E[X|G]|H]]

(by 7.)

= E [V E[X|G]]

(by 3.)

= E [E[V X|G]]

(by 7.)

= E [V X] ,

since V is H-measurable and H ⊆ G. Thus Z = E[X|H].

(by (ii bis))

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115

10. We recall a property of convexity, that is: any convex function ϕ is the supremum of all linear functions dominated by it, i.e. for all x ∈ R ϕ(x) = sup l(x), l∈Lϕ

Lϕ := {l : R → R| l(x) = ax + b, a, b ∈ R, l ≤ ϕ}.

Then,

E[ϕ(X)|G] = E sup l(X)|G l∈Lϕ

≥ sup E[l(X)|G] l∈Lϕ

(by 6.) = sup l (E[X|G])

(by 4.)

l∈Lϕ

= ϕ (E[X|G]) .

A.1

Stochastic Processes in Discrete Time

Random variables are variables with an unknown value that depends on the ‘state of the world’, and describe outcomes at fixed points in time, e.g. the static position of a particle or the payoff of a security. When we are interested in studying random phenomena that change with time, we need to introduce a dynamics. This is done with stochastic processes. Definition A.28. Given a probability space (Ω, F, P), a discrete-time stochastic process on (Ω, F, P) with values in RN is a family X = (Xt )t=0,1,... of RN -valued random variables Xt on (Ω, F, P), t ∈ N0 = N ∪ {0}. In the more general terms, a stochastic process is defined as a family X = (Xt )t∈I of random variables on the same probability space, where I is the set of times. Then, it is called discrete when the set of times I is discrete and continuous when I is dense. In particular, in the present section, we restrict to the discrete-time setting, that is to the processes defined in

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Definition A.28. For simplicity, we avoid repeating the specification “discretetime” each time. In the literature, the first index t = 0 is often dropped and the process starts at t = 1. In many applications, including finance, the phenomena of interest have a finite lifetime, therefore it is useful to work in a finite-horizon probabilistic model, where stochastic processes are finite families X = (Xt )t=0,1,...,T and the time horizon T is finite. We can see a stochastic process X = (Xt )t=0,1,... as a function X : (N0 , Ω) → RN ,

(t, ω) 7→ Xt (ω)

of two variables, the time t ∈ N0 and the scenario ω ∈ Ω. For a fixed time t, we have the random variable Xt ; for a fixed scenario ω, we have a function of time X(ω) : N0 → RN , t 7→ Xt (ω), called the trajectory of X in the state ω. The sequence of images of such map, (X0 (ω), X1 (ω), X2 (ω), . . .), may be equivalently referred to as the trajectory of X in the state ω. For all t ∈ N0 , we denote by FtX := σ ({X0 , X1 , . . . , Xt }) = σ {Xn−1 (H), n = 0, . . . , t, H ∈ B}

the σ-algebra generated by the first t + 1 random variables, i.e. by the stochastic process X up to time t. Note that, for every t ∈ N0 , FtX ⊆ F, since X by definition. all elements X0 , . . . , Xt are random variables, and FtX ⊆ Ft+1

Definition A.29. A filtration on a probability space (Ω, F, P) is an increasing family (Ft )t∈N0 of sub-σ-algebras of F, that is Ft ⊆ F and Ft ⊆ Ft+1 , for every t ∈ N0 . The collection FX := (FtX )t∈N0 is thus a filtration, in particular it is called the natural filtration of X. Intuitively, a filtration represents an increasing flow of information. For instance, FtX represents the information available at time t on the phenomenon described by X. This information can only increase with time.

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Definition A.30. Let X and F = (Ft )t∈N0 be respectively a stochastic process and a filtration on (Ω, F, P). We say that X is adapted to F if Xt is Ft -measurable for every t ∈ N0 , or equivalently FtX ⊆ Ft for every t ∈ N0 . When not otherwise specified and only one filtration F is given, we say that a process is adapted when it is adapted to F. Note that X is adapted to its natural filtration FX , in particular FX is the smallest filtration which X is adapted to. Intuitively, if Xt represents the value of a risky security at time t, the σ-algebra FtX represents the set of information available on the security at time t, given by all its past (and the current) market values. Accordingly, the natural filtration FX represents the increasing flow of information contained in the dynamics of X. We say that X is an integrable process on (Ω, F, P) if Xt is P-integrable for every t ∈ N0 . When a probability space (Ω, F, P) is equipped with a filtration F = (Ft )t∈N0 , we call (Ω, F, (Ft )t∈N0 , P) a filtered probability space. Definition A.31. A real-valued integrable stochastic process M = (Mt )t∈N0 on a filtered probability space (Ω, F, (Ft )t∈N0 , P) is called a • martingale if Mt = EP [Mt+1 |Ft ] ,

∀t ∈ N0 ;

• sub-martingale if it is adapted and Mt ≤ EP [Mt+1 |Ft ] ,

∀t ∈ N0 ;

• super-martingale if it is adapted and Mt ≥ EP [Mt+1 |Ft ] ,

∀t ∈ N0 .

An integrable process with values in RN is called a martingale, or submartingale, or super-martingale if each component is a martingale, or submartingale, or super-martingale respectively. Without loss of generality, henceforth we only consider real-valued processes.

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We stress the dependence of Definition A.31 on the probability measure P and the filtration F = (Ft )t∈N0 . For this reason, when more probability measures or filtrations are involved, we will specify that M is a (F, P)-martingale (respectively (F, P)-super-martingale, and (F, P)-sub-martingale). Note that, by definition, a martingale is an adapted (to the reference filtration) stochastic process. Let us see a few important properties of martingales. Remark A.32. If M is a martingale (respectively a sub- or a super-martingale), then, for all t, s ∈ N0 , 0 ≤ s ≤ t: Ms = E [Mt |Fs ]

(A.6)

(respectively Ms ≤ E [Mt |Fs ] and Ms ≥ E [Mt |Fs ]), and E[Mt ] = E[Ms ] = E[M0 ]

(A.7)

(respectively E[Ms ] ≤ E[Mt ] and E[Ms ] ≥ E[Mt ]). Indeed, by the property 9 in Proposition A.27, E [Mt |Fs ] = E [E[Mt |Ft−1 ] |Fs ] = E [Mt−1 |Fs ] = . . . = Ms . Moreover, by the property 3 in Proposition A.27, E[Mt ] = E[E[Ms |Ft ]] = E[Ms ]. Remark A.33.

• If M is a martingale and ϕ a convex function on R such

that ϕ(M ) is integrable, then ϕ(M ) is a sub-martingale. • If M is a sub-martingale and ϕ a convex and increasing function on R such that ϕ(M ) is integrable, then ϕ(M ) is a sub-martingale. Indeed, by the Jensen inequality, ϕ(Mt ) = ϕ EP [Mt+1 |Ft ] ≤ EP [ϕ(Mt+1 )|Ft ] ,

∀t ∈ N0 .

For instance, if M is a martingale, then |M | and M 2 are sub-martingales, which is not the case if M is just a sub-martingale.

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Remark A.34. Let X be an integrable random variable on (Ω, F, (Ft )t∈N0 , P), then the process M defined by Mt = EP [X|Ft ] ,

t ∈ N0

is a martingale. Indeed, since Ft ⊆ Ft+1 , then EP [Mt+1 |Ft ] = EP [E[X|Ft+1 ]|Ft ] = EP [X|Ft ] = Mt . Definition A.35. A stochastic process A = (At )t∈N0 on (Ω, F, (Ft )t∈N0 , P) is called a predictable process if At+1 is Ft -measurable for all t ∈ N0 . Note that a predictable process is an adapted process. Theorem A.36 (Doob’s decomposition). Every adapted and integrable stochastic process X can be uniquely (up to a negligible event) decomposed as a sum X = M + A of a martingale M such that M0 = X0 and a predictable process A such that A0 = 0. Moreover, X is a sub-martingale (respectively a super-martingale) if and only if A is increasing (respectively decreasing). Proof. (Existence). We define recursively the processes M, A as follows: M0 = X0 , M = M + X − E[X |F ], t ∈ N , t+1 t t+1 t+1 t 0 A0 = 0, A = A − (X − E[X |F ]) , t ∈ N . t+1 t t t+1 t 0 By recurrence for M , we have Mt+1 = Mt + Xt+1 − E[Xt+1 |Ft ] = (Mt−1 + Xt − E[Xt |Ft−1 ]) + Xt+1 − E[Xt+1 |Ft ] = ... = X0 +

t X n=0

(Xn+1 − E[Xn+1 |Fn ]) .

120

APPENDIX

If we take on both sides the conditional expectation with respect to Ft , we obtain E[Mt+1 |Ft ] = X0 +

t−1 X

(Xn+1 − E[Xn+1 |Fn ]) = Mt ,

n=0

thus M is a martingale, being also integrable. By recurrence for A, we have At+1 = At − (Xt − E[Xt+1 |Ft ]) = (At−1 − (Xt−1 − E[Xt |Ft−1 ])) − (Xt − E[Xt+1 |Ft ]) = ... = −

t X

(Xn − E[Xn+1 |Fn ]) ,

n=0

which is Ft -measurable. Moreover, by definition, we have Mt + At = X0 +

t−1 X

(Xn+1 − Xn ) = Xt ,

n=0

hence the existence is proved. (Uniqueness). By assumption, we have Xt+1 − Xt = Mt+1 − Mt + At+1 − At . Taking on both sides the conditional expectation with respect to Ft , we obtain E[Xt+1 |Ft ] − Xt = At+1 − At , which gives exactly the definitions of A and M above. Finally, by the definition of A, it follows that At+1 ≥ At (respectively At+1 ≤ At ) if and only if X is a sub-martingale (respectively super-martingale). Proposition A.37. If M is a martingale and α a bounded predictable process, then the stochastic process G(α, M ), defined by G0 (α, M ) := 0 and Gt (α, M ) :=

t X n=1

αn (Mn − Mn−1 ),

t ∈ N,

(A.8)

A.2 Stopping times

121

is a martingale with null expectation. Moreover, if α ≥ 0 and M is just a sub-martingale (respectively a super-martingale), then G(α, M ) is a submartingale (respectively a super-martingale). Conversely: given an adapted process M , if the process G(α, M ) defined in (A.8) has constant null expectation, that is E[Gt (α, M )] = 0

∀t ∈ N,

(A.9)

for all bounded predictable processes α, then M is also a martingale. Proof. First, by definition (A.8) and the assumptions on α, M , G(α, M ) is an adapted and integrable process. Then, for all t ∈ N0 we have Gt+1 (α, M ) − Gt (α, M ) = αt+1 (Mt+1 − Mt ), and taking on both sides the conditional expectation with respect to Ft , we obtain E[Gt+1 (α, M )|Ft ] − Gt (α, M ) = αt+1 (E[Mt+1 |Ft ] − Mt ). From the equation above, it follows that G(α, M ) is a martingale if M is a martingale, or a sub-martingale (respectively a super-martingale) if α ≥ 0 and M is a sub-martingale (respectively a super-martingale). Conversely, assume that G(α, M ) has constant null expectation for all bounded R-valued predictable processes α. It is enough to prove that E[Mt 1A ] = E[Mt−1 1A ] for all A ∈ Ft−1 , for all t ∈ N. For any A ∈ Ft−1 , we consider the predictable process α defined by αs =

1A

if s = t,

0

if s 6= t,

which, applying (A.9), proves the claim.

A.2

Stopping times

Definition A.38. Given a probability space (Ω, F, P), a random variable τ : Ω → N0 ∪{∞} is a stopping time with respect to the filtration F = (Ft )t∈N0 if, for all t ∈ N0 , {τ ≤ t} ∈ Ft .

122

APPENDIX

Remark A.39. Given a random variable τ : Ω → N0 ∪ {∞}, the following conditions are equivalent: (i) τ is a stopping time with respect to F; (ii) for all t ∈ N0 , {τ = t} ∈ Ft ; (iii) for all t ∈ N0 , {τ > t} ∈ Ft ; (iv) for all t ∈ N0 , {τ ≥ t} ∈ Ft−1 . Proof. [(i) ⇒ (ii)] If τ is a stopping time, then, for all t ∈ N0 , {ω ∈ Ω : τ (ω) = t} = {w ∈ Ω : τ (ω) ≤ t} \ {ω ∈ Ω : τ (ω) ≤ t − 1} ∈ Ft . [(ii) ⇒ (iii)] For all t ∈ N0 , {ω ∈ Ω : τ (ω) > t} = =

t \

{ω ∈ Ω : τ (ω) 6= n}

n=0 t \

{ω ∈ Ω : τ (ω) = n}c ∈ Ft .

n=0

[(iii) ⇒ (iv)] For all t ∈ N0 , {ω ∈ Ω : τ (ω) ≥ t} = {ω ∈ Ω : τ (ω) > t − 1} ∈ Ft−1 ⊆ Ft . [(iv) ⇒ (i)] For all t ∈ N0 , {ω ∈ Ω : τ (ω) ≤ t} = {ω ∈ Ω : τ (ω) > t}c = {ω ∈ Ω : τ (ω) ≥ t + 1}c ∈ Ft , thus τ is a stopping time. If the reference filtration is the natural filtration of some stochastic process X modeling some random phenomenon, we can interpret a stopping time as a moment in which we take a decision based on the information provided by the outcomes of that phenomenon observed up to that time.

A.2 Stopping times

123

Example A.40 (Exit times). Let X be an adapted process and H ∈ B a Borel set of R. The exit time of X from H is defined as ν = min{t ∈ N0 : Xt ∈ / H}, where min ∅ := ∞. We can check that ν is a stopping time: {ν = t} =

t−1 \

{Xn ∈ H} ∩ {Xt ∈ / H} ∈ Ft .

n=0

Given a stopping time τ and a stochastic process X, we call Xτ ∧· = (Xt∧τ )t∈N0 , where s ∧ t := min{s, t}, the corresponding stopped process at time τ . Note that, if τ is almost surely finite, i.e. τ (ω) < ∞ ∀ω ∈ Ω \ N, P(N ) = 0, then Xτ is a well defined random variable outside of a negligible event: Xτ (ω) = Xτ (ω) (ω) ∀ω ∈ Ω \ N. Definition A.41. The σ-algebra associated with the stopping time τ is defined as Fτ := {A ∈ F : A ∩ {τ ≤ t} ∈ Ft ∀t ∈ N0 } = {A ∈ F : A ∩ {τ = t} ∈ Ft ∀t ∈ N0 } The definition is well-posed, that is Fτ is a σ-algebra and if τ = t ∈ N0 then Fτ = Ft . Moreover, the following lemma holds true. Lemma A.42. Let X be an adapted process and τ be an almost-surely finite stopping time, then Xτ is Fτ -measurable. Proof. For any Borel set H ∈ B, we have {Xτ ∈ H} ∩ {τ = t} = {Xt ∈ H} ∩ {τ = t} ∈ Ft .

Remark A.43. Let X be a stochastic process and τ be a stopping time. Then:

124

APPENDIX

1. If X is adapted, then Xτ ∧· is adapted. 2. If X is a martingale, then Xτ ∧· is a martingale. 3. If X is a sub-martingale (respectively a super-martingale), then Xτ ∧· is a sub-martingale (respectively a super-martingale). Indeed, we can rewrite the stopped process as Xτ ∧t

t X = X0 + (Xn − Xn−1 )1{n≤τ } ,

t ∈ N.

n=1

Thus, by definition (A.8), we have Xτ ∧t = X0 + Gt (α, X), where αt = 1{τ ≥t} for all t ∈ N0 . Since α is bounded, predictable and non-negative, all claims follow from Proposition A.37. Now we are ready to introduce and deal with American contingent claims in the framework of a multi-period binomial model.

Appendix B The First Fundamental Theorem of Asset Pricing Proof B.1

The finite market model is arbitrage-free “if ” there exists an EMM

Assume there exists at least an EMM, and choose one, say Q. Suppose that an arbitrage opportunity α exists, then its discounted value process V ∗ must satisfy V0∗ = 0, VT∗ ≥ 0 and EP [VT∗ ] > 0. Since Q ∼ P, the latter is equivalent to EQ [VT∗ ] > 0. But we also know, by Lemma 3.30, that EQ [VT∗ ] = V0∗ = 0, hence a contradiction. So, we proved that if an equivalent martingale measure for S ∗ exists, then the market is arbitrage-free.

B.2

The finite market model is arbitrage-free “only if ” there exists an EMM

Conversely, we assume that the market is arbitrage-free and we prove the existence of an EMM for S ∗ , that is some probability measure Q ∼ P, under 125

126

B. The First Fundamental Theorem of Asset Pricing - Proof

which S ∗ is a martingale. By Proposition A.37, it is enough to prove that, for all bounded R-valued F-predictable processes γ, EQ [GT (γ, S ∗ i )] = 0 ∀i = 1, . . . , d

(B.1)

where we recall that ∗i

GT (γ, S ) =

T X

γt (S ∗ it − S ∗ it−1 ).

t=1

Indeed, if this holds, by choosing γ defined by ( 1A , t = s γs = 0, t 6= s, for t ∈ {1, . . . , T } and A ∈ Ft−1 , the expectation in (B.1) would give ∗ EQ [γt (St∗ − St−1 )] = 0

⇒

∗ EQ [1A St∗ ] = EQ [1A St−1 ].

Since t is arbitrarily chosen in {1, . . . , T } and A arbitrarily chosen in Ft−1 , this proves that S ∗ is a martingale. But, by Lemma 3.25, that is equivalent to show that the discounted gain process of any self-financing trading strategy α = (α0 , α1 , . . . , αd ) at the terminal trading date has null expectation under Q, i.e.

" EQ [G∗T (ˆ α)] = EQ

T X

# ∗ α ˆ t · (Sˆt∗ − Sˆt−1 ) = 0,

t=1

where we denote α ˆ := (α , . . . , α ) and Sˆ∗ := (S ∗,1 , . . . , S ∗,d ). 1

d

Note that in a finite space of dimension N , we can identify both random variables and probability measures with vectors of RN . Precisely, denoted Ω = {ω1 , . . . , ωN }, we identify a R-valued random variable X on (Ω, F), X(ωi ) = xi for i = 1, . . . , N , with the vector (x1 , . . . , xN ) of RN , and a probability measure Q on (Ω, F), Q({ωi }) = qi for i = 1, . . . , N , with the vector (q1 , . . . , qN ) of RN . We would like to (strictly) separate the disjoint convex sets V and RN + \{0} by a hyperplane induced by a linear functional on RN , from which we can then construct an EMM. Unfortunately, the standard separation theorems

B.2 The finite market model is arbitrage-free “only if ” there exists an EMM 127 do not directly apply. However, in finite dimension, we can overcome this difficulty by introducing a convex compact subset of RN + \ {0} and using a separation result obtained from the following lemmas. Lemma B.1. Let C be a convex closed subset of RN that does not contain the origin. Then, there exists an element ξ ∈ C such that |ξ|2 ≤ x · ξ

∀x ∈ C.

Proof. Since C is closed, it contains an element ξ such that |ξ| = dist(C, 0). Indeed, dist(C, 0) = inf{|x|, x ∈ C}, which means there exists a bounded sequence {xn }n∈N of elements of C such that |xn | −−−→ dist(C, 0), with a subn→∞

sequence {xnk }k∈N converging to a vector ξ of RN such that |ξ| = dist(C, 0), i.e. |ξ| ≤ |x| for all x ∈ C, and ξ ∈ C because C is closed. Moreover, since C is convex, for all x ∈ C and all t ∈ [0, 1], ξ + t(x − ξ) ∈ C, and so |ξ|2 ≤ |ξ + t(x − ξ)|2 = |ξ|2 + 2tξ · (x − ξ) + t2 |x − ξ|2 . Thus, for all t ∈]0, 1], 0 ≤ 2ξ · (x − ξ) + t|x − ξ|2 , and going to the limit for t converging to 0 from the right, we get ξ · (x − ξ) ≥ 0, which ends the proof. Lemma B.2. Let K be a convex compact subset of RN and V a vector subspace of RN such that K ∩ V = ∅. Then, there exists an element ξ ∈ RN such that x · ξ = 0 ∀x ∈ V,

y · ξ > 0 ∀y ∈ K.

Proof. We consider the set C = K − V := {y − x|x ∈ V, y ∈ K}, which does not contain the origin, because K and V are disjoint. We would like to apply Lemma B.1, so we have to check if C is convex and closed. We can easily show that C is convex: for all x1 , x2 ∈ V, y1 , y2 ∈ K, for all t ∈ [0, 1], t(y1 − x1 ) + (1 − t)(y2 − x2 ) = (ty1 + (1 − t)y2 ) − (tx1 + (1 − t)x2 ) , which is in C because both K and V (as any vector space) are convex. Moreover, we can prove that C is also closed: let {zn }n∈N be a sequence of elements of C converging to z ∈ RN , for all n ∈ N there must be two elements

128

B. The First Fundamental Theorem of Asset Pricing - Proof

xn ∈ V, yn ∈ K such that zn = yn − xn ; since K is compact, there is a subsequence {ynk }k∈N converging to y ∈ K; since a vector space is always closed, we also have xnk −−−→ x ∈ V, hence znk = ynk − xnk −−−→ y − x ∈ C, n→∞

n→∞

as well as zn −−−→ y − x = z ∈ C. n→∞

Finally, by Lemma B.1, there exists ξ ∈ C such that |ξ|2 ≤ (y − γx) · ξ = y · ξ − γ(x · ξ) ∀x ∈ V, γ ∈ R, y ∈ K. Since this holds for all γ ∈ R, we necessarily have x · ξ = 0, hence also y · ξ ≥ |ξ|2 > 0.

Now, we denote by K the convex hull of the set of random variables 1{ωn } n∈N , that is ) ( N N X X µn = 1 K := µn 1{ωn } µn ≥ 0 for n = 1, . . . , N, n=1 n=1 ) ( N X = x = (x1 , . . . , xN ) ∈ RN xn = 1 , + n=1

which is a convex, compact subset of RN + . Since K ∩ V = ∅ and V is a vector space, we can apply Lemma B.2 and find an element ξ = (ξ1 , . . . , ξN ) ∈ RN such that (i) ξ · x = 0 for all x ∈ V; (ii) ξ · y > 0 for all y ∈ K. Since the canonical base of RN is contained in K, we get that ξn = ξ · n > 0 for all n = 1, . . . , N , i.e. ξ has strictly positive components. Thus, we can normalize ξ to obtain a probability measure equivalent to P. Precisely, we define a probability measure Q on (Ω, F) as associated to a vector (q1 , . . . , qN ) of RN , in the following way: Q({ωn }) = qn , where

PN

n=1 qn

qn :=

ξn ξn > 0, = PN |ξ| n=1 ξn

n = 1, . . . , N,

= 1 for all n = 1, . . . , N . Then, Q is a probability measure

equivalent to P; moreover, (i) translates into EQ [G∗ (ˆ α)] = 0 for all Rd -valued predictable processes α ˆ = (α1 , . . . , αd ), which ends the proof of the FFTAP.

Bibliography [1] Tomas Bjork. Arbitrage Theory in Continuous Time, chapter 20, pages 302–305. Oxford University press, Great Clarendon Street, Oxford OX2 6DP, second edition, 2004. [2] Freddy Delbaen and Walter Schachermayer. The mathematics of arbitrage. Springer Finance. Springer-Verlag, Berlin, 2006. [3] Andrea Pascucci. PDE and martingale methods in option pricing, volume 2 of Bocconi & Springer Series. Springer, Milan; Bocconi University Press, Milan, 2011. [4] Stanley R. Pliska. Introduction to Mathematical Finance (Discrete time models), chapter 3,6, pages 106–111,200–221. Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK, first edition, 2001. [5] Ruth J Williams. Introduction to the Mathematics of Finance, volume 72. American Mathematical Society, 2006.

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