Inflation Indexed Swaps and Swaptions - Editorial Express

Inflation Indexed Swaps and Swaptions - Editorial Express

Inflation Indexed Swaps and Swaptions ∗ Mia Hinnerich† Department of Finance, Stockholm School of Economics, Box 6501, SE-113 83 Stockholm, SWEDEN ...

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Inflation Indexed Swaps and Swaptions



Mia Hinnerich† Department of Finance, Stockholm School of Economics, Box 6501, SE-113 83 Stockholm, SWEDEN

September, 2006.

Abstract This article considers the pricing of inflation indexed swaps, inflation indexed swaptions, and options on inflation indexed bonds. To price the inflation indexed swaps, we suggest an extended HJM model. The model allows both the forward rates and the consumer price index to be driven, not only by a standard multidimensional Wiener process but also, by a general marked point process. Our model is an extension of the HJM approach proposed by Jarrow and Yildirim (2003) and later also used by Mercurio (2005) to price inflation indexed swaps. Furthermore we price options on so called TIPS-bonds assuming the model is purely Wiener driven. We then introduce an inflation swap market model to price inflation indexed swaptions. All prices derived have explicit closed form solutions. Furthermore, we formally prove the validity of the so called foreign-currency analogy earlier used by Hughston (1998), Jarrow & Yildirim (2003) and Mercurio (2005). JEL classification: G12; G13 Key words: Inflation, Index-linked securities, Derivatives, Swaps, Swaptions

∗I

gratefully acknowledge financial support from the Louis Fraenckels foundation and the Infina Foundation. I am grateful to Tomas Bj¨ ork and Linus Kaisajuntti at Stockholm School of Economics, Lane Hughston and Andrea Macrina at King’s College and to Tom Hurd from McMaster University for useful ideas and discussions. † Tel: +44 8 736 91 61, E-mail address: [email protected]

1

1

Introduction

Even though some inflation linked products can be traced back to the middle of the 18th century, the modern inflation indexed market did not start until the beginning of the 1980s. Then the United Kingdom issued inflation indexed bonds and shortly thereafter Austria, Sweden and Canada followed. In 1997 the United States began to issue Treasury Inflation Protected Bonds (TIPS) and since then several countries have entered the inflation indexed bond market. About ten years ago, the inflation indexed swap market began to develop in the United Kingdom. Today inflation indexed swaps and other inflation indexed derivatives are traded in for example the United Kingdom, the United States, France, Japan and in the Euro Market. The derivatives market is still young though and most certainly we will experience more products develop. Inflation is defined as the percentage change of a particular reference index. The choice of reference index varies from country to country but usually it is a consumer price index (CPI). The consumer price index measure the price of a representative basket of goods and services. Thus an increase in the consumer price index over a period of time implies that there has been inflation over that period. Inflation indexed products are tied to inflation. The main idea of inflation indexed bonds is that investing in the bond and keeping it until maturity will generate a certain real return. Thus, even though the nominal value of the coupons and principal may change, the real value of these remain the same. A zero coupon inflation indexed bond with principal equal to unity will pay out enough in dollars to buy one unit of the consumer price index basket. A swap is an agreement between two counter parties to exchange cash flows. The agreement specifies the cash flows and the dates when the cash flows are to be paid. The most common types of swaps are interest rate swaps and currency swaps, but inflation indexed swaps are also traded and have gained more and more interest lately. In an inflation indexed swap at least one of the cash flows is tied to inflation. A swaption is an option to enter into a swap at a pre specified date for a pre specified swap rate. An inflation indexed swaption is a swaption where the underlying swap is an inflation indexed swap. Inflation linked products can be used to hedge future cash flow against inflation. That is particularly attractive to investors that seek asset-liability matching such as for example insurance companies. Inflation linked products may also be used for risk diversification and of course for speculation. From the issuers perspective, inflation linked bonds may prove as means of establishing a trustworthy inflation policy. One of the early studies on inflation derivatives was made by Hughston (1998) and introduces a methodology based on the foreign-currency analogy. Here nominal assets are thought of as domestic assets, real assets as foreign assets, and the consumer price index is treated as the exchange rate between the nominal and the real markets. Developing a theory where the dynamics of the consumer price index and the real and nominal discount bonds have the 2

same structure as in a HJM model, Hughston shows how index linked derivatives can be treated in much the same way as foreign-exchange derivatives. However no derivative pricing formulas are calculated explicitly. Also relying on the foreign-currency analogy, Jarrow and Yildirim (2003) have developed a three factor HJM model in order to price TIPS and options written directly on the inflation index. They assume that the volatilities of all asset prices and the consumer price index are deterministic. They use a parameterization of the forward volatility that corresponds to the Hull-White short rate model in order to obtain an explicit formula for the option. Hence they obtain pricing formulas for the case when bond prices are Gaussian. Recently, Mercurio (2005) has studied the pricing of zero coupon inflation indexed swaps, year-on-year inflation indexed swaps, as well as inflation indexed caplets and floorlets. The swaps are priced first using the Jarrow & Yildirim model with the Hull-White parameterization and then using two different market model approaches. The main criticisms, which can be directed to all of the articles above, is that they all assume a priori that the foreign-currency analogy holds. The purpose of this paper is to: • Show how to price indexed derivatives in a HJM framework without assuming the foreign-currency analogy. • In fact, we prove that the foreign-currency analogy holds for a completely arbitrary process, including the case where the process does not even have an economic interpretation. • Extend the HJM model suggested in Jarrow and Yildirim (2003) to allow for more than three factors and also allow for the possibility of jumps in the economy. Thus, in our model the random processes describing the real and nominal market as well as the consumer price index are allowed to be driven by both a standard multidimensional Wiener process and a general marked point process. To ensure closed-form solutions we will assume that the intensity of the point process as well as the volatilities of all asset prices and the consumer price index, with respect to both the Wiener process and the point process, is deterministic. • Price year-on-year inflation indexed swaps, given our multi factor HJM model allowing for jumps. • Price options written on TIPS, given the multi factor HJM model without jumps. • Specify an inflation indexed swap market model. • Price zero coupon inflation indexed swaptions and year-on-year inflation indexed swaptions given our inflation indexed swap market model. Show that the zero coupon inflation indexed swaptions also can be priced using the multi factor HJM model without jumps. 3

The article is organized as follows: In the next Section we will present the multi factor HJM model allowing for jumps. We will also prove the validity of the foreign-currency analogy. Section 3 is devoted to inflation indexed swaps. For preparatory purposes we will in Section 3.1 show how to find the model independent price of the zero coupon inflation indexed swap. Then we will price the year-on-year inflation indexed swap, given the model specified in Section 3.2. In Section 4 we will study inflation indexed swaptions. We will introduce an inflation indexed swap market model and use this to price the year-on-year inflation indexed swaption. This is done in Section 4.1. In Section 4.2 we show that the zero coupon inflation indexed swaption can be priced both by using the inflation indexed swap market model and by the multi factor HJM model without jumps. In Section 5 options on TIPS are priced using the model specified in Section 3 with zero jump-volatilities. Section 6 concludes.

2

The extended HJM model

In this section we specify our HJM model under the objective probability measure. From this specification we derive the dynamics of the nominal bonds, the inflation protected bonds, the inflation process and of what we call the fictive real bonds. Furthermore we will be able to define a real martingale measure. Everything will be done without using any prior assumption about the foreigncurrency analogy. We will instead show that the foreign-currency analogy does indeed hold. Assumption 2.1 We consider a financial market where all objects are defined on a filtered probability space (Ω, F, P, F) where P is the objective probability measure. The probability space carries both an n-dimensional Wiener process W P and a general marked point process µ(dt, dv) on R+ × V with compensator λP (t, dv)dt . The filtration F = {Ft }t≥0 is generated by both W P and µ, i.e. Ft = FtW ∨ Ftµ . Let pn (t, T ) denote the price in dollar at time t of a nominal zero-coupon T-bond that pays out one dollar at the maturity date, T . We will refer to these bonds as T -bonds. Let I(t) be any stochastic process at time t. By pIP (t, T ) we denote the price in dollars at time t of a contract that pays out I(T ) dollar at time T . We will refer to these bonds as T − IP -bonds. If I(T ) denotes CPI at time t, then pIP (t, T ) is the price at time t of a contract that at maturity will pay out the dollar value of one CPI-unit. Hence, in this case pr (t, T ) is the price of an inflation protected zero-coupon T-bond. The inflation protected bonds are often called TIPS-bond, where TIPS stands for treasury inflation protected ˜ · 1 where I(t) ˜ is security. Since I(t) can be any process, we can let I(t) = I(t) the temperature at the top of the Eiffel tower at time t and 1 has the unit of number of dollars over the squared temperature. That is I(t) is measured as number of dollars per temperature degree. Then, pIP (t, T ) is the price at time t of a contract that will pay out an amount in dollars that is equal to the number of degrees on the scale at the top of the Eiffel tower at time T . 4

Define pr (t, T ) as pr (t, T ) =

pIP (t, T ) I(t)

This implies that pr (t, t) = 1. Note that the unit of pr (t, T ) is equal to the number of dollars per one unit of I. Since CPI is expressed as dollars per CPI-basket, then if I(t) is the CPI-index at time t, the unit of pr (t, T ) will be CPI-baskets. Hence pr (t, T ) is the price in CPI-baskets of a (fictive), real bond that pays out one CPI-basket at time T . Suppose for simplicity that the CPI-basket consists of carrots only, then pr (t, T ) will be the price at time t, expressed in carrots, of a contract that pays out one carrot at time T . In the case where I(T ) denotes the temperature the interpretation of pr (t, T ) is a bit awkward, but of course this did not prevent us from defining it. If I(t) is the temperature of the top of the Eiffel tower at time t then pr (t, T ) is the price, expressed as degrees, of a contract that pays out one degree at time T . Assumption 2.2 We assume that there exists a (dollar) market for T -bonds and T − IP -bonds for all maturities T > 0. Furthermore we assume that for every fixt t, pn (t, T ) and pIP (t, T ) are differentiable with respect to the maturity T. Now we can define, for each fixed T , two types of instantaneous forward rates, contracted at time t by fi (t, T ) = −

∂ln pi (t, T ) ∂T

for i = r, n

If I(T ) is the CPI, then these forward rates can be interpreted as the nominal and the real forward rates. Using these forward rates, we now define two types of interest rates by ri (t) = fi (t, t) for i = r, n. Again, if I(T ) is the CPI rn and rr can be interpreted as the nominal and the real short rates. Finally we define Bn and Br by, Rt

Bi (t) = e

0

r i (s)ds

for i = r, n

where, if I(T ) is the CPI, Bn (t) would denotes the nominal money market account at time t and is measured in dollars and Br (t) denotes the real money market account at time t and is thus measured in CPI-baskets. Assumption 2.3 Assume that under the objective probability measure P , the dynamics of fr and fn for every fixed T > 0 and the dynamics of I are given by: Z i i P dfi (t, T ) = α (t, T )dt + σ (t, T )dW + ξ i (t, v, T )µ(dt, dv) for i = r, n V Z I I P dI(t) = I(t)µ (t)dt + I(t)σ (t)dW + I(t−) γ I (t, v)µ(dt, dv) V

where σ r , σ n , σ i are F-adapted and ξ r , ξ n , γ I and λP are F-predictable. 5

In addition to this assumption we need to know that the above processes posses some boundedness and regularity properties, see [2]. Assumption 2.4 We assume that there are no arbitrage possibilities, i.e. the market is arbitrage free. From general theory (see for example [2]) it follows by Assumption (2.4) and (2.2) that there exists an equivalent martingale measure Qn such that, for each fixed T , pn (t, T ) pIP (t, T ) are Qn -martingales Bn (t) Bn (t) This result is the main tool in the proof of the next proposition. Proposition 2.1 If fn (t, T ), fr (t, T ) and I(t) satisfies Assumption (2.3), then I, pn , pIP and pr will under the nominal martingale measure Qn satisfy: Z dI(t) = {rn (t) − rr (t)} dt + σ I (t)dW + γ I (t, v)˜ µ(dt, dv) (1) I(t−) V dpn (t, T ) pn (t−, T ) dpIP (t, T ) pIP (t−, T ) dpr (t, T ) pr (t−, T )

= rn (t)dt + β n (t, T )dW +

Z

δ n (t, v, T )˜ µ(dt, dv)

n

= r (t)dt + β

IP

(t, T )dW +

Z

δ IP (t, v, T )˜ µ(dt, dv)

(3)

δ r (t, v, T )˜ µ(dt, dv)

(4)

V

= a(t, T )dt + β r (t, T )dW +

Z V

where i

β (t, T ) = −

Z

T

σ i (t, s)ds

for i = r, n

t

β IP (t, T ) = σ i (t) + β r (t, T ) δ i (t, v, T ) = e−

RT t

ξ i (t,v,s)ds

−1

for i = r, n

δ IP (t, v, T ) = δ r (t, v, T ) + γ I (t, v) + δ r (t, v, T )γ I (t, v) Z a(t, T ) = rr (t) − σ i (t) · β r (t, T ) − δ r (t, v, T )γ I (t, v)λP t (dv) V

µ ˜(dt, dv) λt (dv) dWtP

(2)

V

= µ(dt, dv) − λt (dv)dt = λP t (dv)(1 + ρ(t, v)) = ht dt + dWt 6

where we have suppressed Qn to shorten notation so W is the Qn -Wiener process, and λ is the intensity of the marked point process under the Qn -measure. Furthermore ht and ρ(t, v) are the Girsanov kernels for the transition from P to Qn with respect to the Wiener process and the marked point process respectively. That is −ht and −ρ(t, v)λ(t, dv)P are the market price of diffusion risk and jump risk respectively. Proof. Given the P -dynamics of fn (t, T ) and fr (t, T ), then [3] show that   dpi (t, T ) 1 2 = ri (t) + Ai (t, T ) + kβ i (t, T )k dt + β i (t, T )dW P pi (t−, T ) 2 Z + δ i (t, v, T )µ(dt, dv) for i = r, n

(5)

V

where i

A (t, T )

=−

Z

=−

Z

T

αi (t, s)ds t

β i (t, T )

T

σ i (t, s)ds t

δ i (t, v, T )

=

e−

RT t

ξ i (t,v,s)ds

− 1 for i = r, n

By using the P -dynamics of pr (t, T ) that we just obtained, together with the P -dynamics of I(t) that is given by Assumption 2.3, Itos lemma gives that   dpIP (t, T ) 1 2 = rr (t) + Ar (t, T ) + kβ r (t, T )k dt pIP (t−, T ) 2  I I r + µ (t) + σ (t) · β (t, T ) dt +



r



I

β (t, T ) + σ (t) dW

P

+

Z

δ IP (t, v, T )µ(dt, dv) V

where δ IP (t, v, T ) = δ r (t, v, T ) + γ I (t, v) + γ I (t, v)δ r (t, v, T ) Next, we would like to change measure from P to the equivalent (nominal) martingale measure Qn . By the Girsanov theorem, we know there exist a P adapted process ht andRa P -predictable process ρ(t, v) ≤ −1∀v ∈ V a such that dLt = ht Lt dW P + V ρ(t, v)˜ µP (dt, dv) where LT = dQ/dP on FT so that P dWt = ht dt + dWt and λt (dv) = λP t (dv)(1 + ρ(t, v)). Here W denotes a Qn -Wiener process and λt (dv)dt is the intensity measure of the marked point process under the Qn -measure. Furthermore we will use µ ˜(dt, dv) to denote the compensated marked point process under Qn that is µ ˜(dt, dv) = µ(dt, dv) − 7

λt (dv)dt. Hence the dynamics of I(t), pn (t, T ) and pIP (t, T ) under Qn are given by: dI(t) I(t−)

=



+

Z

µI (t) + h(t) · σ I (t) dt γ I (t, v) (1 + ρ(t, v)) λP (t, dv)dt

V

+ σ i (t)dW +

Z

γ I (t, v, T )˜ µ(dt, dv)

(6)

V

dpn (t, T ) pn (t−, T )

= µn (t, T )dt + β n (t, T )dW

+

Z

δ n (t, v, T )˜ µ(dt, dv)

(7)

V

 = µIP (t, T )dt + β r (t, T ) + σ I (t) dW Z + δ IP (t, v, T )˜ µ(dt, dv)

dpIP (t, T ) pIP (t−, T )

(8)

V

where 1 2 µn (t, T ) = rn (t) + An (t, T ) + kβ n (t, T )k 2 Z + h(t) · β n (t, T ) + δ n (t, v) (1 + ρ(t, v)) λP (t, dv)

(9)

V

1 µIP (t, T ) = rr (t) + Ar (t, T ) + kβ r (t, T )k2 + µI (t) 2 + σ I (t) · β r (t, T ) + h(t) · β r (t, T ) + h(t) · σ I (t) Z + δ IP (t, v, T ) (1 + ρ(t, v)) λP (t, dv)

(10)

V

As stated above, the assumption of arbitrage free markets implies that pn (t, T ) Bn (t)

pIP (t, T ) Bn (t)

are Qn -martingales

hence the drift of pn (t, T ) and pIP (t, T ) must equal the nominal short rate, that is µn (t, T ) = µIP (t, T ) = rn (t). This condition together with equation (7) and (8) immediately gives that the Qn -dynamics of pn (t, T ) and pIP (t, T ) satisfies equation (2) and (3) respectively. 8

Next we insert the conditions µn (t, T ) = µIP (t, T ) = rn (t) into the drift equations (9) and (10). By noting that this must hold for all T and using that δ IP (t, v, T ) = δ r (t, v, T ) + γ I (t, v) + γ I (t, v)δ r (t, v, T ) we get three drift conditions: An (t, T )

1 = − kβ n (t, T )k2 − h(t) · β n (t, T ) 2 Z − δ n (t, v, T ) (1 + ρ(t, v)) λP t (dv)

(11)

V

Ar (t, T )

1 2 = − kβ r (t, T )k − σ I (t) · β r (t, T ) − h(t) · β r (t, T ) − h(t) · σ I (t) 2 Z  − δ r (t, v, T ) 1 + γ I (t, v) (1 + ρ(t, v)) λP (12) t (dv) V

µI (t)

= rn (t) − rr (t) − h(t) · σ I (t) Z − γ I (t, v) (1 + ρ(t, v)) λP t (dv)

(13)

V

From condition (13) and equation (6) we now see that under Qn the dynamics of I satisfy equation (1). By definition pr (t, T ) = pIP (t, T )/I(t), so by Itos lemma and the equations (1) and (3) we finally see that the Qn -dynamics of pr (t, T ) satisfies equation (4).

Corollary 2.1 The drift conditions that has to be satisfied in order for the market to be free of arbitrage are: ! Z T



αn (t, T ) = σ n (t, T )

σ r (t, s) ds − h(t)

t

+

Z

{δ n (t, v, T ) + 1} ξ n (t, v, T ) (1 + ρ(t, v)) λP t (dv) V

r

r

α (t, T ) = σ (t, T )

Z

T r



I

σ (t, s) ds − σ (t) − h(t)

!

t

+

Z

 1 + γ I (t, v) (1 + ρ(t, v)) (1 + δ r (t, v, T )) ξ r (t, v, T )λP t (dv) V

µI (t)

= rn (t) − rr (t) − h(t) · σ I (t) −

Z

γ I (t, v) (1 + ρ(t, v)) λP t (dv) V

9

The ∗ denotes transpose. The corollary follows from the three drift equation (11), (12) and (13) and by taking the T -derivative of the first two. When comparing Proposition 2.1 with Proposition 2 in the paper by Jarrow & Yildirim (2003), it is clear that one get their result as a special case of Proposition 2.1. That is the special case when all jump sizes are zero and the dimension of the Wiener process equals three. The major difference is what assumptions that are needed for the proof. As commented earlier in this paper, the results in Jarrow & Yildirim rely on the foreign-currency analogy. More specifically Jarrow & Yildirim assume that I(t)Br (t) Bn (t)

is a Qn -martingale

Given this assumption, they show that I(t) has to satisfy equation (1). That is, given their assumption they show that, I(t) will have the same dynamics as the FX-rate in a foreign-currency model. The main objection is that we do not know a priori that I(t)Br (t)/Bn (t) is a Qn -martingale. In other words it is not known if I(t)Br (t) can be considered as a traded asset in the nominal economy and we do not know if Br (t) can be considered as a traded asset in the real economy either. In the proof of Proposition 2.1 we showed, without any a priori information about I(t)Br (t)/Bn (t), that I(t) satisfies equation 1. Furthermore, we show that as a result of Proposition 2.1 I(t)Br (t)/Bn (t) is indeed a Qn -martingale (see the Lemma below). Hence, we have proved that the foreign-currency analogy does hold. It should be noted that this proof does not rely on that I is the CPI, but I can be an arbitrary stochastic process that does not necessarily have an economic interpretation. Lemma 2.1 Define BIP (t) by BIP (t) = I(t)Br (t) Then

BIP (t) Bn (t)

is a Qn -martingale

Proof. By the definition of Br (t) and the Qn -dynamics of I(t) given in equation (1), Itos lemma gives that Z dBIP (t, T ) γ I (t, v)˜ µ(dt, dv) = rn (t)dt + σ I (t)dW + BIP (t−, T ) V

Proposition 2.2 Define QIP by dQIP = LT dQn on FT 10

where Lt =

BIP (t) Bn (0) Bn (t) BIP (0)

then QIP is a martingale measure for the numeraire BIP Proof. Let Π be a stochastic process such that Π(t)/Bn (t) is a Qn -martingale, i.e. so that Π is an arbitrage free price process. We have to show that the process Π(t)/BIP (t) is a QIP -martingale. Let s ≤ t, then Bayes formula gives that h i h i Π(t) Bn (0)  Q Q BIP (t) Π(t)  L(t) E E s s BIP (t) Bn (t) BIP (t) BIP (0) Π(t) IP = = Es BIP (t) L(s) Ls = EsQ



 Π(s) Bn (s) Π(s) Π(t) Bn (s) = = Bn (t) BIP (s) Bn (s) BIP (s) BIP (s)

Corollary 2.2 Define QT −IP by dQT −IP = LT dQn on FT where Lt =

PIP (t, T ) Bn (0) Bn (t) PIP (0, T )

then QT −IP is a martingale measure for the numeraire PIP (t, T ) If one exchange BIP (t) for PIP (t, T ) in the proof of Proposition 2.2 the Corollary follows. Proposition 2.3 Let Πn denote an arbitrage free price process in the nominal economy. Define the process Πr by Πr (t) = Πn (t)/I(t). Define Qr by dQr = LT dQn on FT where Lt =

Br (t)I(t) Bn (0) Bn (t) Br (0)I(0)

Then Qr is a martingale measure for the real numeraire Br (t) and Πr (t) Br (t)

is a Qr -martingale

11

Proof. From Proposition 2.2 it follows that Πn (t)/BIP (t) is a QIP -martingale and that Qr is equal to QIP . Since Πr (t) Πr (t)I(t) Πn (t) = = Br (t) Br (t)I(t) BIP (t) it follows that Πr (t)/Br (t)) is a Qr -martingale. Corollary 2.3 Define QT,r by dQT,r = dQn LT on FT where Lt =

Pr (t, T )I(t) Bn (0) Bn (t) Pr (0, T )I(0)

Then QT,r is a martingale measure for the real numeraire pr (t, T ) and Πr (t) pr (t, T )

is a QT,r -martingale

Corollary 2.4 pr (t, T ) Br (t)

is a Qr -martingale

pr (t, S) pr (t, T )

is a QT,r -martingale

We finish this section by making one additional assumption that will be needed for some of the calculations in the coming sections. Assumption 2.5 We assume that σ r , σ n , σ i , ξ r , ξ n , γ I and λP are deterministic.

3

Inflation indexed swaps

In this section we will study inflation indexed swaps. In particular, we will price the zero coupon inflation indexed swap and the year-on-year inflation indexed swap. We will use martingale methods and the technique of convexity corrections. In what follows Π[t, ·] is used to denote the price, in the dollars, of the payoff (·). A swap is an agreement between two counter parties to exchange cash flows. The agreement specifies the cash flows and the dates when the cash flows are to be paid. In an inflation-indexed swap at least one of the cash flows is dependent on an inflation index or an inflation protected security. If we let T0 , T1 , · · · , TM be a fixed set of increasing times and define αi by αi = Ti − Ti−1

for i = 1, · · · , M 12

Then, typically the swap starts at time T0 and the payments occur at the dates T1 , T2 , · · · , TM . By a receiver swap we refer to a swap where the holder at each payment date receives a fixed amount and pays a floating amount. The fixed payment is known at the start date of the swap while the floating payment is not. By a payer swap we mean a swap where the payments go in the opposite direction to the receiver swap.

3.1

Zero Coupon Inflation Indexed Swap

In this Section we will price Zero Coupon Inflation Indexed Swap (ZCIIS). Mercurio (2005) priced ZCIIS in [6] by using martingale methods and showed that the price is model independent. As a preparation for the next Section on Year-on-Year Inflation Swaps we will also show this result. Furthermore we will provide an alternative proof by a simple replicating argument. In a ZCIIS one party pays a fixed interest rate and receives the inflation rate over the specified time period. The inflation rate is calculated as the percentage return of the consumer price index. The other party of the swap receives the same flows but of opposite signs. As the name indicates, a ZCIIS has only one time interval [T0 , T ] with payments at time T and no intermediary payments. That is, cash flows are exchanged only once. Let Z0 (T, K) denote a payer ZCIIS that starts at time T0 , has payment date at time T and has a swap rate equal to K. Then a fixed amount of (1 + K)T −T0 − 1 is payed out at time T and floating amount of I(T) −1 I(T0 ) is received at time T. Let Z0 (t, T, K) denote the price of a Z0 (T, K) at time t. Then the payoff is I(T ) Z0 (T, T, K) = − (1 + K)T −T0 I(T0 ) and   I(T ) T −T0 Z0 (T0 , T, K) = Π T0 , − (1 + K) I(T0 )     I(T ) = Π T0 , (14) − Π T0 , (1 + K)T −T0 I(T0 ) where the first part is   I(T ) Π T0 , = I(T0 ) =

pn (T0 , T ) T,n ET0 [I(T )] I(T0 )   pn (T0 , T ) T,n I(T )Pr (T, T ) ET0 = Pr (T0 , T ) I(T0 ) Pn (T, T )

13

since I(t)pr (t, T )/pn (t, T ) = pIP (t, T )/pn (t, T ) is a QT , n-martingale. The second part is   Π T0 , (1 + K)T −T0 = pn (T0 , T )(1 + K)T −T0 Hence equation (14) becomes equal to pr (T0 , T ) − pn (T0 , T )(1 + K)T −T0 This result was first stated in [6]. We note that there is also a simple replicating argument that proves the result. Remark 3.1 To replicate the floating leg of the swap: At time T0 buy 1/I(T0 ) TIPS-bonds with maturity date T . Then at time T we will receive the dollar value of 1/I(T0 ) CPI units, that is I(T )/I(T0 ). The price at time T0 of 1/I(T0 ) TIPS-bonds is 1/I(T0 ) times I(T0 )pr (T0 , T ), ie pr (T0 , T ). Remark 3.2 Regardless of whether one uses the martingale method or the replicating argument to price the ZCIIS, it should be noted that no assumptions on the dynamics of the assets are needed. Hence this result is model independent.

3.2

Year-on-Year Inflation Indexed Swaps

In this Section we will price Year-on-Year Inflation Indexed Swaps (YYIIS) using the jump-diffusion model specified in Section 2. Let YmM (K) denote a payer Year-on-Year Inflation Indexed Swap that starts at time Tm with payment dates at Tm+1 , Tm+2 , · · · , TM . For every period [Ti , Ti+1 ] for i = m, · · · , M − 1 a fixed amount of αi+1 K is payed out at time Ti+1 . For the same period a floating amount of αi+1 [Xi+1 − 1] where Xi+1 =

I(Ti+1 ) I(Ti )

is received at time Ti+1 . If we let YmM (t, K) denote the price of a YmM (K) at time t where t ≤ Tm , then YmM (t, K)

=

=

M −1 X

Π [t, αi+1 Xi+1 − 1] −

M −1 X

i=m

i=m

M −1 X

Π [t, αi+1 Xi+1 ] − (K + 1)

i=m

Π [t, αi+1 K] M −1 X i=m

14

αi+1 p(t, Ti+1 )

(15)

Where we have used standard no-arbitrage pricing theory. In order to price PM −1 the YmM we are thus left with the exercise of calculating i=m Π [t, αi+1 Xi+1 ]. However, before we do that, we will define the forward swap rate. We define the forward swap rate of a Year-on-Year Inflation indexed swap to be the value of K for which the price of the swap is zero. We denote the forward swap rate for M M M the swap YmM (K) by Rm (t). Hence YmM (t, Rm (t)) = 0 and so Rm (t) is given by: PM −1 PM −1 M i=m Π [t, αi+1 Xi+1 ] − i=m αi+1 pn (t, Ti+1 ) Rm (t) = (16) PM −1 i=m αj pn (t, Ti+1 ) PM −1 Next we want to calculate i=m Π [t, αi+1 Xi+1 ]. As we will see this expression is model-dependent and in order to calculate it we will use the model that was setup in Section 2. This will enable us to obtain explicit formulas for both the swap price and the forward swap rate. To calculate Π [t, Xi+1 ] when t < Ti we simplify and look at the case when i = 1. That is we calculate Π [t, X2 )] when t < T1 . The general case Π [t, Xi+1 )] is obtained by the obvious extention. Π [t, T2 ] is the value at time t of the payoff X2 = α2 [I(T2 )/I(T1 ) − 1] that is payed out at time T2 and in order to calculate it we will use the T2 -forward measure and iterated expectation along with the fact that I(t)pr (t, T2 )/pn (t, T2 ) = pIP (t, T2 )/pn (t, T2 ) is a QT2 ,n -martingale. We find that   I(T2 ) Π [t, X2 ] = pn (t, T2 )EtT2 ,n α2 I(T1 ) 

 1 ETT12 ,n [I(T2 )] I(T1 )    1 I(T2 )pr (T2 , T2 ) ETT12 ,n = pn (t, T2 )α2 EtT2 ,n I(T1 ) pn (T2 , T2 ) = pn (t, T2 )α2 EtT2 ,n

= pn (t, T2 )α2 EtT2 ,n



pr (T1 , T2 ) pn (T1 , T2 )



(17)

To calculate the expected value in equation (17) we change the numeraire to the QT1 ,n -forward measure by using Bayes formula and the likelihood ratio T ,n/T1 ,n Lt 2 where dQT2 ,n pn (t, T2 ) pn (0, T1 ) T2 ,n/T1 ,n = = Lt dQT1 ,n t pn (t, T1 ) pn (0, T2 )

15

Hence EtT2 ,n



pr (T1 , T2 ) pn (T1 , T2 )



=

h

EtT1 ,n

pr (T1 ,T2 ) T2 ,n/T1 ,n pn (T1 ,T2 ) LT1

i

T ,n/T1 ,n

Lt 2 

 pr (T1 , T2 ) pn (T1 , T2 ) pn (t, T1 ) pn (T1 , T2 ) pn (T1 , T1 ) pn (t, T2 )

=

EtT1 ,n

=

pn (t, T1 ) T1 ,n E [pr (T1 , T2 )] pn (t, T2 ) t

(18)

Combining equation (17) and (18) we find that Π [t, X2 ] = α2 pn (t, T1 )EtT1 ,n [pr (T1 , T2 )]

(19)

This result is also stated in [6]. Note that we have not yet made use of any model assumption. However, the expected value in equation (19) is model dependent. It is calculated in [6] using a diffusion model. We will calculate it using the jump-diffusion model specified in Section 2. We change measure from the nominal QT1 ,n -forward measure to the real T1 ,r Q -forward measure. Again we use Bayes formula and the expected value in equation (19) can thus be rewritten as h i T1 ,n/T1 ,r 1 ,T2 ) L EtT1 ,r pprr (T (T1 ,T1 ) T1 EtT1 ,n [pr (T1 , T2 )] = T ,n/T1 ,r Lt 1 (20) Since T ,n/T1 ,r Lt 1

dQT1 ,n pn (t, T1 ) pr (0, T1 )I(0) = = T ,r 1 dQ pr (t, T1 )I(t) pn (0, T1 ) t

T ,n/T1 ,r

the dynamics of Lt 1

=

n

+

Z

T ,n/T1 ,r

dLt 1

T ,n/T1 ,r

Lt−1

under QT1 ,r is given by: o βtn,1 − βtr,1 − σti dW T1 ,r

V

δtn,1 − δtr,1 + γtI + δtr,1 γtI T1 ,r µ ˜ (dt, dv) 1 + δtr,1 + γtI + δtr,1 γtI

where we have used the simplifying notation βtk,j = β k (t, Tj ), δtk,j = δ k (t, v, Tj ) T ,n/T1 ,r and γtI,j = γ I (t, v, Tj ). Since both pr (t, T2 )/pr (t, T1 ) and Lt 1 are QT1 ,r martingales Ito gives that EtT1 ,n [pr (T1 , T2 )] =

pr (t, T2 )C(t, T1 , T2 ) pr (t, T1 )

16

where

R T1

C(t, T1 , T2 ) = e

t

R

{(βsn,1 −βsr,1 −σsi )·(βsr,2 −βsr,1 )+

and = ∆1,2 t

δtr,2 − δtr,1

!

V

T1 ,r ∆1,2 (ds,dv)}ds s λ

(21)

δtn,1 − δtr,1 + γtI + δtr,1 γtI

1 + δtr,1

1 + δtr,1 + γtI + δtr,1 γtI

Inserting this into equation (19) gives that Π [t, XT2 ] = α2

pn (t, T1 )pr (t, T2 )C(t, T1 , T2 ) pr (t, T1 )

Changing back to the general case with 1 = i and 2 = i + 1 we have that   pn (t, Ti )pr (t, Ti+1 )C(t, Ti , Ti+1 ) Π t, XTi+1 = αi+1 pr (t, Ti ) Hence the pricing equation (15) is found to be YmM (t, K)

=

M −1 X

αi+1

i=m

− (K + 1)

pn (t, Ti )pr (t, Ti+1 )C(t, Ti , Ti+1 ) pr (t, Ti )

M −1 X

αi+1 pn (t, Ti+1 )

(22)

i=m

and the forward swap rate (16) is found to be PM −1 αi+1 pn (t,Ti )pr (t,Ti+1 )C(t,Ti ,Ti+1 ) M (t) Rm

=

i=m

pr (t,Ti ) PM −1 i=m



PM −1 i=m

αi+1 pn (t, Ti+1 )

(23)

αi+1 pn (t, Ti+1 )

If we choose all volatilities to be zero and the volatilities of the real and nominal forward rates to be ξ n (t, T ) = ae−b(T −t) and ξ r (t, T ) = ce−c(T −t) for some positive constants a, b, c, d as in the model by Jarrow and Yildirim (2003) then the pricing formula (22) reduces to that in [6]. The pricing formulas can be rewritten so that they do not depend on any real bond prices or any real volatilities. Instead they will be functions of the inflation protected TIPS-bonds and the volatilities of these inflation protected bonds. This is good news since that saves us the work of trying to strip real zero coupon bond prices from some inflation linked products. The trick is just to use that pIP (t, Tk ) = I(t)pr (t, Tk ). Hence if we extend the right hand side of equation (22) by I(t)/I(t). The time t price of a YmM (K) can be rewritten as YmM (t, K) =

M −1 X

αi+1

i=m

− (K + 1)

pn (t, Ti )pIP (t, Ti+1 )C(t, Ti , Ti+1 ) pIP (t, Ti )

M −1 X

αi+1 pn (t, Ti+1 )

i=m

17

Since pIP (t, Tk ) = I(t)pr (t, Tk ), the Ito formula gives that β IP = β r + β I and δ IP = δ r +δ I +δ I δ r . This is what we will use to rewrite the correction term C(t, Ti , Ti+1 ) to a function of the volatilities of the TIPS-bonds rather than of the volatilities of real bonds. Recall that R Ti

C(t, Ti , Ti+1 ) = e

t

R

{(βsn,i −βsr,i −σsi )·(βsr,i+1−βsr,1 )+

and ∆i,i+1 t

=

δtr,i+1 − δtr,i 1+

!

V

∆si,i+1 λTi ,r (ds,dv)}ds

δtn,i − δtr,i + γtI + δtr,i γtI

δtr,i

1 + δtr,i + γtI + δtr,i γtI

In the nominator of C(t, Ti , Ti+1 ), we extend the second parenthesis in the exponent by +σ I − σ I and use the relation β IP = β r + β I . In the denominator of C(t, Ti , Ti+1 ) we extend ∆i,i+1 by (1 + γ I )/(1 + γ I ) and use the relation δ IP = δ r + δ I + δ I δ r . Finally we change intensity from λTi ,r to λ by the relation i λTi ,r = (1 + βIP )λ which follows from Radon-Nikodym derivative between the Ti ,r Q -measure and the Qn -measure. We find that R Ti

C(t, Ti , Ti+1 ) =

e

t

(βsn,i −βsIP,i )·(βsIP,i+1 −βsIP,i )ds

e−

R Ti R t

{

V

∆i,i+1 λ(ds,dv)}ds s

where ∆i,i+1 t

(δtIP,i+1 − δtIP,i )(δtn,i − δtIP,i ) 1 + δtIP,i

Similarly the forward swap rate in equation (23) can be rewritten as PM αj pn (t,Ti )pIP (t,Ti+1 )C(t,Ti ,Ti+1 ) PM −1 − i=m αi+1 pn (t, Ti+1 ) i=m pIP (t,Ti ) M Rm (t) = PM −1 i=m αi+1 pn (t, Ti+1 )

4

Inflation Indexed Swaptions

A Swaption is an option to enter into a swap at a pre specified date for a pre specified swap rate. An inflation indexed swaption is an option to enter into an inflation indexed swap. In this Section we will price two types of swaptions, the zero coupon inflation indexed swaption (ZCIISO) and the year-on year inflation indexed swaption (YYIISO). We will start with the latter one.

4.1

YYIISwaption

A YYIISwaption is an option to enter into a YYIIS at a pre specified date for a M pre specified swap rate. More specifically, let Y Om (K) denote an option to enter M M into a payer Ym (K) at time Tm with the fixed swap rate K and let Y Om (t, K) M denote the price of this option at time t. Then the payoff of Y Om (K) is: M Y Om (Tm , K) = max[YmM (Tm , K), 0]

18

(24)

Where according to equation (15) in Section 3.2 YmM (t, K) =

M −1 X

Π [t, αi+1 Xi+1 ] − (K + 1)

i=m

M −1 X

αi+1 p(t, Ti+1 )

(25)

i=m

We will state an alternative formulation of the payoff of a YYIISwaption which involves the forward swap rate. To find this formulation, recall from equation (16) the forward swap rate is given by M Rm (t) =

PM −1 i=m

PM −1 Π [t, αi+1 Xi+1 ] − i=m αi+1 pn (t, Ti+1 ) PM −1 i=m αi+1 pn (t, Ti+1 )

(26)

k For each pair m, k such that m < k, define Sm (t) by

k (t) = Sm

k−1 X

αi+1 pn (t, Ti+1 )

i=m k Lemma 4.1 For m ≤ k, Sm is a self-financing portfolio

Proof. Since k (t) = Sm

k−1 X

αi+1 pn (t, Ti+1 )

i=m k is just a weighted sum of traded assets, hence it is a and αi are constants, Sm self-financing portfolio. k Since Sm is a self-financing portfolio there exist a martingale measure for k the numeraire asset Sm which we will denote by Qkm . k Using Sm (t), the forward swap rate can be rewritten as: M Rm (t)

=

PM −1 i=m

M Π [t, αi+1 Xi+1 ] − Sm (t) M Sm (t)

(27)

k Proposition 4.1 The forward swap rate Rm , is a Qkm -martingale.

k Proof. From Lemma 4.1 we know that Sm is a self-financing portfolio. Also Pk j=m+1 Π [t, αj Xj ] is a self financing-portfolio since it is a sum of self-financing k is a Qkm -martingale. portfolios. Hence it follows from equation (27) that Rm

Using the expression for the forward swap rate given in equation (27), the price of the swap YmM (K) can be expressed as:  M M YmM (t, K) = Rm (t) − K Sm (t) (28)

19

and the payoff of the YYIISwaption in equation (24) can be rewritten as: M M M Y Om (Tm , K) = Sm (Tm )max[Rm (Tm ) − K, 0]

(29)

M (K) can be regarded as a call option on the From which we see that the Y Om M forward swap rate, expressed in units of Sm . The natural choice of measure to use for pricing the YYIISwaption is the Qkm -measure. However even if we choose the jump volatilities to be zero in the model specified in Section 2, so that the model is purely Wiener driven, the swap rate will be a sum of lognormal variables and so the swap rate has a nasty distribution in this model. It seems plausible that there does not exist any explicit formula for the YYIISwaption in this case. The problem is similar to that of pricing interest rate swaptions assuming a HJM model for the forward rates. Black’s model is popular among practitioner for pricing interest rate swaptions and is used without much of logical underlying model. Thus, one perhaps naive approach to price YYIISwaptions is to just use Blacks model for this case as well. In Black’s model, it is implicitly assumed that the forward swap rates are lognormal. The use of Blacks model for pricing interest rate swaptions has found some justification after the introduction of swap market models, since in the swap market models the key assumption is that the swap rates are lognormal distributed. In the next Section we introduce an Inflation Indexed Swap Market Model and we will see that the naive approach is not so bad after all.

4.1.1

Inflation indexed Swap Market Models

In this Section we will define an inflation indexed swap market model. Given this model, we will price the YYIISwaption. Definition 4.1 Given a set of increasing resettlement times T0 , T1 · · · TM we define B to be the set consisting of pairs (m, k) of positive integers m and k such that 0 ≤ m < k < M . For any given pair (m, k) in B we assume that the k forward swap rate Rm has dynamics given by k k k k dRm (t) = Rm (t)σm (t)dWm (t) k where Wm is a multidimensional Wiener process under the Qkm -measure and k σm (t) is a vector of non-stochastic functions of time and. M (t, K) at Proposition 4.2 For any given swap market model, the price Y Om M time t where t ≤ Tm of a payer Ym is given by  M M M Y Om (t, K) = Sm (t) Rm (t)N (d1 ) − KN (d2 )

where d1 d2

=

1

  M  1 2 Rm ln + Σm,M K 2

Σm,M = d1 − Σm,M

20

where Σ2m,M =

Z

Tm M kσm (s)k2 ds t

Proof. By assumption R Tm

M M Rm (TM ) = Rm (t)e

t

M M σm (s)dWm (s)− 12

R Tm t

M ||σm (s)||ds

Hence conditional on time t,   Σ2 2 M M (TM ) ∼ N ln Rm (t) − lnRm ,Σ 2 where Σ2

Z

=

T0

2

M kσm (s)k ds t

So by letting fX (x) denote the density function for a standard normal random variable and using the QM m -measure, no-arbitrage pricing gives that M (t, K) Y Om

QM m

M = Sm (t)Et

M = Sm (t)

Z

  M max{Rm (TM ) − K, 0} n o Σ2 M max Rm (t)e− 2 +xΣ − K, 0 fX (x)dx

∞ −∞

=

M Sm (t)

Z

∞ M Rm (t)e−

Σ2 2

+xΣ

fX (x)dx − KN [−x0 ]

x0

 M M = Sm (t) Rm (t)N [−(xo − Σ)] − KN [−x0 ]

where ln x0 =



K Ψ(t,T )



+

Σ2 2

Σ

which proves the proposition.

4.2

ZCIISwaption

In this Section we will price ZCIISwaptions, given a standard HJM model without jumps. A payer ZCIISwaption is an option to enter into a ZCIIS for a given pre specified swap rate at a pre specified time. Let ZO0 (t, T, K) denote the price at time t, of an option with maturity date T0 to enter into a payer ZCIIS

21

that starts at time T0 , has payment date at time T and a swap rate equal to K. Then the payoff of this option is given by: ZO0 (T0 , T, K) = max{Z0 (T0 , T, K), 0}

(30)

where as in previous sections Z0 (t, T, K) is the price at time t of a payer ZCIIS that starts at time T0 , has payment date at time T and a swap rate equal to K. From Section 3.1 we know that Z0 (T0 , T, K) = pr (T0 , T ) − pn (T0 , T )(1 + K)T −T0 Hence equation (30) is equal to  max pr (T0 , T ) − pn (T0 , T )(1 + K)T −T0 , 0 By letting (1 + K)T −T0 = G and defining Ψ(t, T ) =

pr (t, T ) pn (t, T )

we can rewrite the payoff again so that equation (30) is equal to pn (T0 , T ) (max {Ψ(T0 , T ) − G, 0}) Using pn (t, T ) as the numeraire, the price of the swaption at time t is: ZO0 (t, T, K) = pn (t, T )EtT,n [max {Ψ(T0 , T ) − G, 0}]

(31)

To calculate the expected value in equation (31) we will assume a standard HJM model without jumps so that Ψ(T0 , T ) is lognormally distributed. Since we have restrict the model to exclude jumps, the Qn -dynamics of pn (t, T ) and pr (t, T ) are given by equation (2) and (4) with the assumption that δ n = 0 and δ r = 0. Hence by Itos lemma the dynamics of Ψ(t, T ) under the nominal risk neutral measure Qn is: dΨ(t, T ) Ψ(t, T )

= {a(t, T ) − rn (t) + β n (t, T )β n ∗ (t, T ) − β(t, T )r β n ∗ (t, T )} dt + {β r (t, T ) − β n (t, T )} dWt

To change measure to the nominal T -forward measure QT,n we use that dQT,n pn (t, T ) 1 T,n/n Lt = = dQn t B(t) p(0, T ) hence the dynamics of Radon-Nykodym is given by T,n/n

dLt

T,n/n

= Lt

βn (t, T )dWtT,n

So by the Girsanov Theorem dWt = βn∗ (t, T )dt + dWtT,n 22

Hence the dynamics of Ψ(t, T ) under the nominal T -forward measure QT,n is dΨ(t, T ) = {a(t, T ) − rn (t)} dt + {β r (t, T ) − β n (t, T )} dWtT,n Ψ(t, T ) By Itos lemma we find that   1 r n n 2 d ln Ψ(t, T ) = a(t, T ) − r (t) − kβ (t, T ) − β (t, T )k dt+{β r (t, T ) − β n (t, T )} dWtT,n 2 Hence ln Ψ(T0 , T ) ∼ N where M 2

Σ

=

Z

=

Z

  Σ2 2 ,Σ ln Ψ(t, T ) + M − 2

T0

{a(s, T ) − rn (s)} ds t T0

2

kβ r (s, T ) − β n (s, T )k ds t

So by letting fX (x) denote the density function for a standard normal random variable, we can write the expected value in equation (31) as Z ∞ n o Σ2 max Ψ(t, T )eM − 2 +xΣ − G, 0 fX (x)dx −∞ Z ∞ Σ2 Ψ(t, T )eM − 2 +xΣ fX (x)dx − GN [−x0 ] = x0

where ln x0 =



G Ψ(t,T )



−M +

Σ2 2

Σ After some additional but straight forward calculations we find that, given the standard HJM-model without jumps, the price of the ZCIISwaption is: ZO0 (t, T, K) = pr (t, T )eM N [d1 ] − pn (t, T )GN [d2 ]

(32)

where d1 d2 M Σ2

=

pr (t,T ) )+M + ln( Gp n (t,T )

Σ2 2

Σ = d1 − Σ Z T0 = {a(s, T ) − rn (s)} ds =

Z

t T0

2

kβ r (s, T ) − β n (s, T )k ds t

G = (1 + K)T −T0 Since a ZCIISwaption is a special case of a YYIISwaption, the ZCIISwpation can also be priced using the inflation swap market model. More precisely m+1 ZOm (t, Tm+1 , K) = Y Om (t, K). 23

5

TIPStions

In this section we will price a TIPStion, assuming a standard HJM model without jumps. A TIPStion is an option on a TIPS-bond. A call TIPStion gives the buyer the right to purchase a TIPS-bond for a given pre specified price at maturity date. Let t denote the maturity date of a call option on a TIPS-bond that pays out the dollar value of one CPI unit at time T . Let K be the strike price of the option. Then the payoff of the option at maturity is Ψt = max[pIP (t, T ) − K, 0] The price of the option at time 0 is Π[0, Ψt ] = Π[0, pIP (t, T )IA ] − Π[0, KIA ]

(33)

where A = {pIP (t, T ) > K} Define Υ(s, t, T ) =

pn (s, t) pIP (s, T )

Since Υ is the quotient of two traded assets where the inflation protected Tbond is the numeriare, it is a QT −IP -martingale. Assuming no jumps implies that the Qn -dynamics of pn and pIP are given by equations (2) and (3) with δ n = 0 and δ IP = 0. By Itos lemma we find that the volatility of Υ(s, t, T ) under Qn is β n (s, t) − β IP (s, T ). Since the volatility is preserved under measure changes we have that the QIP -dynamic of Υ is dΥ(s, t, T )  n = β (s, t) − β IP (s, T ) dWsT −IP Υ(s, t, T ) so under QT −IP   1 2 2 ln Υ(t, t, T ) ∼ N ln Υ(0, t, T ) − Σ , Σ 2 where Σ2 =

Z

t

2

kβ n (s, t) − β IP (s, T )k ds 0

Define Γ(s, t, T ) =

pIP (s, T ) 1 = pn (s, t) Υ(s, t, T )

Since Γ is a Qt -martingale, the dynamics under Qt is  dΓ(s, t, T ) = − β n (s, t) − β IP (s, T ) dWsT −IP Γ(s, t, T ) hence

  1 2 2 ln Γ(t, t, T ) ∼ N ln Γ(0, t, T ) − Σ , Σ 2 24

Using the nominal T-IP forward measure, we calculate the first part of equation (33) to Π[0, pIP (t, T )IA ]

= pIP (0, T )E0T −IP [IA ] = pIP (0, T )QT −IP (pIP (t, T ) ≥ K) T −IP

= pIP (0, T )Q

T −IP

= pIP (0, T )Q where

ln d1 =





1 pn (t, t) ≤ pIP (t, T ) K



  1 Υ(t, t, T ) ≤ = pIP (0, T )N [d1 ] K

pIP (0,T ) pn (0,t)K





+ 12 Σ2

Σ2 Using the nominal t-forward measure we calculate the second part of equation (33) to Π[0, KIA ]

= pn (0, t)KE0t [IA ] = pn (0, t)KQt (pIP (t, T ) ≥ K)   pIP (t, T ) t ≥K = pn (0, t)KQ pn (t, t) = pn (0, t)KQt (Γ(t, t, T ) ≥ K) = pn (0, t)KN [d2 ]

where

√ d2 = d1 −

Σ2

Hence the price of the TIPStion is Π[0, Ψt ] = pIP (0, T )N (d1 ) − pn (0, t)KN (d2 )

6

Conclusion

We have priced options on TIPS-bonds and zero coupon inflation indexed swaptions given a finite dimensional HJM model for the real and nominal forward rates. Furthermore we have priced year-on-year inflation indexed swaps given this finite dimensional HJM model but extended to also allow the bond prices and the consumer price index to jump. The jumps have been modeled using a general marked point processes. We have proposed an inflation swap market model and used this to price year-on-year inflation indexed swaptions. We have priced the zero coupon inflation indexed swaptions also under this model. Furthermore, we have formally proved the validity of the so called foreign-currency analogy.

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