ACT 4000, MIDTERM #1 ADVANCED ACTUARIAL TOPICS FEBRUARY 8, 2007 HAL W. PEDERSEN
You have 70 minutes to complete this exam. When the invigilator instructs you to stop writing you must do so immediately. If you do not abide by this instruction you will be penalised. All invigilators have full authority to disqualify your paper if, in their judgement, you are found to have violated the code of academic honesty. Each question is worth 10 points. Provide sufficient reasoning to back up your answer but do not write more than necessary. This exam consists of 8 questions. Answer each question on a separate page of the exam book. Write your name and student number on each exam book that you use to answer the questions. Good luck!
Question 1. Consider a singleperiod binomial market. There are two assets available for trade. The first asset pays 2 in the upstate and 1 in the downstate. The second asset pays 2 in the upstate and 3 in the downstate. The price of asset 1 is PI = 1.39. The price of asset 1 is P2 = 2.29. The market is arbitragefree. (1) [6 points] Compute the price of a risky asset that pays 10 in the upstate and 0 in the downstate. (2) [4 points] What is the implicit effective interest rate in this model?
Question 2. An equity securities market model follows a multiperiod binomial model. At each node of the binomial tree the current stock price S will branch to uS in the upstate and dS in the downstate. You are given that the initial stock price is 10, u = 1.25, d = 0.85 and the interestrate is 5% effective per period. (i) (5 points) Compute the price of a European put option on the stock which expires in 4 periods and has a strike price of 8.5. (ii) (5 points) Compute the price of an American put option on the stock which expires in 4 periods and has a strike price of 8.5 and describe the optimal exercise policy for this American put option.
2
#1
ACT 4000  MIDTERM
Question 3. In North America, it is common. to issue investment products with return guarantees. For instance, a product may offer the returns of the S&P 500 index except the investor is guaranteed a 0% return over the investment horizon and the maximum total return the investor will be credited over the investment horizon is 30%.
You are the pricing actuary for a large insurance company. Your company has decided to sell an investment product for which the investments will be credited a return equal to that on an index that will experience either a 20% gain or a 10% loss at the end of one period (i.e. the singleperiod binomial model). The continuously compounded interestrate (force of interest) for the period is 0.08. The customer is guaranteed a 5% return and you are to set the maximum total return the customer will receive over the period (denoted a) so that your insurance company will break even on the product. Determine
a.
Suppose that the exchange rate is 0.79 €I$. Let r($) = 3%, r(€) = 5%, u = 1.1850, d = 0.8581, T = 15 months, n = 3, and K = 0.8 €. (a) What is the price of a 15month European call? (b) What is the price of a 15month European put?
The price of a 6month dollardenominated call option on the euro with a $0.90 strike is $0.0404. The price of an otherwise equivalent put option is $0.0141. The annual continuously compounded dollar interest rate is 5%. a. What is the 6month dollareuro
forward price?
b. If the eurodenominated annual continuously is 3.5%, what is the spot exchange rate?
compounded
interest rate
A oneperiod arbitragefree model has two assets with the following price dynamics.
Asset
1
12
Asset
2
10
10.4
8
2.5
A call option on asset 1 with a strike price of 9 has a price of 1.8. (i) Compute the state prices implicit in this model. (ii) Compute the risk neutral probabilities for this model. (iii) Compute the implied short rate for this model. [Compute the implied short rate as a force of interest.] (iv) The price of a call option on asset 2 is 2.4. What is the strike price of this call option?
7
Suppose call and put prices are given by Strike Call premium Put premium
50 18 7
55 14 10.75
60 9.50 14.45
Find the convexity violations. What spread would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
8
10 .
Suppose the S&P 500 futures price is 1000, a and n = 3.
=
If )
30%, r
=
5%, 8
=
5%, T
a. What are the prices of European calls and puts for K you find the prices to be equal?
=
b. What are the prices of American calls and puts for K
= $1000?
c. What are the timeO replicating
=
I,
$1 OOO? Why do
portfolios for the European call and put?
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9. 10
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7.
(a) We can calculate the price of the call currency option in a very similar way to our previous calculations. Please pay attention to the fact that we have a strike price and exchange rate in Euros, therefore the foreign interest rate is the $ interest rate! For the European call option, we have:
Time (yrs)
0.416667
0.79 0.095828
0.936117 0.17665

0.677895 0.03079
0.833333
1.25
1.10926 0.31197
1.314427 0.51443
0.803277 0.06837
0.95185 0.15185
0.581698
0.689288
o
o 0.499152
o The value of the option is 0.096€.
(b) for the European put option, we have:
1.25
Time (yrs)
0.416667
0.833333
0.79 0.086435
0.936117
1.10926
1.314427
0.031
o
o '
0.677895 0.13698
0.803277 0.05858
0.95185 o
0.581698 0.20903
0.689288 0.11071 0.499152 0.30085
The value of the option is 0.086€.
Question 9.6. a)
We can use putcallparity
+ C (K,
'* '*
Fo. T
=
T)  P (K, T)
FO,T = erT
to determine the forward price: PV (forward
=
 PV (strike)
erT FO.T _ KerT
+ KerT]
[+C (K, T) . P (K, T)
= eO.05*O.5[$0.0404 = $0.92697.
price)
 $0.0141 + $0.geO.05*O.5]
b) Given the forward price from above and the pricing formula for the forward price, we can find the current spot rate: FO.T ¢}
xo
=
= xoe(rrf)T =
Fo. Te(rrf)T
$0.92697e(O.050.035)O.5
=
$0.92.
'.¢'quations(9.17) and (9.18) of the textbook are violated. To see this, let us calculate the values. ,b;>'t
!~ve: . (K_1_) __C_( K_z_)
Kz 'ch
=
K)
_18__1_4= 0.8 55  50
C (Kz)  C (K3)
and
14  9.50 60  55
K3  Kz
= 0.9,
violates equation (9. I7) and 10.75  7 55  50
= 0.75
and
P (K3)  P (Kz)
14.45  10.75 60  55
K3  K2
= 0.74,
}1 violates equation (9.18) .
.a!culate lambda in order to know how many options to buy and sell when we construct the rfly spread that exploits this form of mispricing.Because the strike prices are symmetric around ~~mbdais equal to 0.5. I.., Pdore, we use a call and put butterfly spread to profit from these arbitrage opportunities. Transaction "Buy 1 50 strike call .Sell 2 55 strike calls ,Buy 1 60 strike call I'~',"
TOTAL
r~
t=O
ST
18 +28 9.50 +0.50
< 50
50 :::
ST :::
o
ST 
50
55 ::: ST ST  50
o
o
110  2
o o
o 50 2: 0
o 60 
ST 
55
:::
60
ST
> 60
ST X ST
110  2 ST 
ST
2: 0
50
o
60
x
ST

t=O 21.50 7 Transaction50 0ST 02STx60
Please note that we initially receive money and have nonnegative future payoffs. Therefore , have found an arbitrage possibility, independent of the prevailing interest rate.
we
Question 10.18. a) We have to use the formulas of the textbook to calculate the stock tree and the prices of the options. Remember that while it is possible to calculate a delta, the option price is just the value of B, because it does not cost anything to enter into a futures contract. In particula~, this yields the following prices: For the European call and put, we have: premium = 122.9537. The prices must be equal due to putcallparity. b) We can calculate for the American call option: premium put option: premium = 124.3347. c) .' We have the following time For the European call option:
e replicating
portfolios:
Buy 0.5371 futures contracts. Borrow 122.9537 For the European put option: Sell 0.4141 futures contracts. Borrow 122.9537
= 124.3347
and for the American