ACTSC 446/846 Winter 2013

Mathematical Models for Finance

Assignment 1

Department of Statistics and Actuarial Science

University of Waterloo, Canada

Due: Thursday Feb 7th, 2013 in class. Hard copy please. No electronic version.

To earn the full credit of the assignment, your answer must be well justified. Simply listing the

final answer leads to zero credit. Only about half of questions will be randomly selected for grading,

and your grade on this assignment will be calculated based on your performance on those selected

questions. However, in order to obtain the full credit for this assignment, you have to do all the

assigned questions.

You must place the present page on the top of your solutions as the cover and combine them to-

gether, and put your last name, your first name, and UW ID number very clearly in the corresponding

blanks below.

Last Name:

First Name:

UW ID #:

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1. Assume that the effective 6-month interest rate is 2%, the S&R 6-month forward price is $1020,

and premiums are charged as illustrated in the following table for S&R options with 6 months

to expiration:

Strike Call Put

$950 $120.405 $51.777

$1000 $93.809 $74.201

$1020 $84.470 $84.470

$1050 $71.802 $101.214

$1107 $51.873 $137.167

Further assume the index pays no dividend, and let ST denote the S&R index in 6 months.

(a) Suppose you buy the S&R index for $1000 and buy a 950-strike put.

i) Derive an expression, as a function of ST , of the payoff and profit respectively for this

position.

ii) Construct payoff and profit diagrams for this position.

iii) Verify that you obtain the same payoff and profit diagram by investing $931.37 in

zero-coupon bond and buying a 950-strike call.

(b) Develop an expression for the profit in each of the following ratio spreads positions. Sim-

plify your answer as much as possible and draw the corresponding profit diagrams.

i) Buy 950-strike call, sell two 1050-strike calls.

ii) Buy two 950-strike calls, sell three 1050-strike calls.

2. The S&R index spot price is $1,100, the risk-free rate is 5%, and the dividend yield on the

index is 9%.

(a) Compute the fair 6-month forward price on the S&R index.

(b) Suppose you observe a 6-month forward price of $1,115. What arbitrage would you un-

dertake?

(c) Suppose you observe a 6-month forward price of $1,050. What arbitrage would you un-

dertake?

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For (b) and (c), build up your portfolio and explain the possible profit.

3. The price of a nondividend-paying stock is $100 and the continuously compounded risk-free

rate is 5%. A one-year European call option with a strike price of $100 × e0.05 = $105.127 has

a premium of $11.924. A 1.5 year European call option with a strike price of $105.127 has a

premium of $11.50. What arbitrage would you undertake? Build up your portfolio and explain

the possible profit.

4. In each of the following cases identify the arbitrage and demonstrate how you would make

money by creating a portfolio and showing your payoff:

(a) Consider two European options on the same stock with the same time to expiration. The

90-strike call costs $10 and the 95-strike call costs $4.

(b) Now suppose these options have 2 years to expiration and the continuously compounded

interest rate is 10%. The 90-strike call costs $10 and the 95-strike call costs $5.25. Show

again that there is an arbitrage opportunity.

(c) Suppose that a 90-strike European call sells for $15, a 100-strike call sells for $10, and a

105-strike call sells for $6. Show how you could use an asymmetric butterfly to profit from

this arbitrage opportunity.

5. Suppose that the current spot $/¥ exchange rate is 0.008, the one-year continuously compound-

ed dollar-denominated rate is 5% and the one-year continuously compounded yen-denominated

rate is 1%. If there is no arbitrage opportunity, calculate one year currency forward price on

1¥.

For the next few questions, assume no arbitrage opportunity.

6. Suppose the European call function C(S,K, t, T ) and European put function P (S,K, t, T ) are

differentiable with respect to K. Show that

−1 ≤ ∂C

∂K≤ 0,

0 ≤ ∂P

∂K≤ 1.

7. (Convexity, put version.) Using arbitrage-free argument, show that For K1, K2 ≥ 0,

λP (S,K1, t, T ) + (1 − λ)P (S,K2, t, T ) ≥ P (S, λK1 + (1 − λ)K2, t, T ),

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λp(S,K1, t, T ) + (1 − λ)p(S,K2, t, T ) ≥ p(S, λK1 + (1 − λ)K2, t, T ),

That is, the European put option price P (S,K, t, T ) and the American put option price

p(S,K, t, T ) are convex functions of K on R+.

8. Using the result of (7.), show that if P is twice differentiable with respect to K, then

∂2P

∂K2≥ 0.

9. (Homogeneity.) Show that for λ > 0,

λC(S,K, t, T ) = C(λS, λK, t, T ).

Here, C(λS, λK, t, T ) means a call option written on λ shares of S, with strike price λK. (In

fact, all four types of options C, c, P, p have this property.)

10. (Change of numeraire.) Let c$(£, K, t, T ) be an American call option on £1 with strike price

K in currency $ and p£($K, 1, t, T ) be an American put option on $K with strike price 1 in

currency £. Show that

c$(£, K, t, T ) = p£($K, 1, t, T ).

The next question is for grad students only; for undergrad it is a bonus question.

11. Put-call parity holds only for European options. Show that for American option prices we have

the following inequalities (the underlying pays no dividends):

S0 −K ≤ c− p ≤ S0 −Ke−r(T−t),

where c and p denote the prices of American style call and put options on S with the same

maturity T and strike price K, r is the continuously compounded interest rate, and S0 denotes

the spot price on the underlying asset S.

Hint: For the first part of the relationship consider: (a) a portfolio consisting of a European

call plus an amount of cash equal to K and (b) a portfolio consisting of an American put option

plus one share of S.

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