MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED BOOK examination. The mark for this module is calculated from 50% of the percentage mark for this paper plus 50% of the percentage mark for associated coursework. Answer any FOUR questions. Candidates are permitted to use approved portable electronic calculators in this examination. Marks shown in questions are merely a guideline.
1. (a) Explain what is meant by a European call option on a risky asset. In what way is an American call option different from a European one? Show that, if the underlying asset does not pay dividends, it is never optimal to exercise an American call option early. State carefully any assumptions that you make. (8) (b) Two European call options C 1 and C 2 on the same asset have the same exercise price E but have expiry times T 1 and T 2 respectively, where T 1 < T 2. Give an arbitrage argument to show that, at any time t < T 1, their prices satisfy C 1 (t) C 2 (t). (c) Now consider three European call options C 1, C 2, C 3, all with the same expiry time T, but with exercise prices E 1, E 2, E 3 respectively, for which E 1 < E 2 < E 3 and E 2 = (E 1 + E 3 )/2. Show that, at any time t T, their values satisfy C 1 (t) + C 3 (t) 2C 2 (t). (Hint: What is the payoff at time T of the portfolio obtained by buying one option C 1 and one option C 3, and short-selling two options C 2?) (9) (8) Page 2 of 5 MASM006/continued...
2. (a) The share price of a certain company is now 100p. In one year s time, the share price will either have risen to 130p or dropped to 90p. The riskfree interest rate (continuously compounded) over the next year will be 5%. Calculate the risk-neutral probability p for a rise in the share price. What would be the fair (No Arbitrage) price for a European put option on this share, with exercise price 110p, which expires one year from now? (8) (b) In the (multistep) Binomial Method for option pricing, the price S n of the underlying asset is modelled as { usn with probability p, S n+1 = ds n with probability 1 p, where u, d and p are parameters depending on the underlying asset and on the timestep δt. Outline the Binomial Algorithm, based on this model, for finding the price of a European put option with expiry time T = Nδt and exercise price E. You should indicate the various steps of this algorithm. You may give a listing in pseudocode or in MATLAB (or another programming language) if you wish, but you should also include a description of the purpose of each step. (10) (c) State formulae for the mean and variance of log(s N /S 0 ), in terms of u, d and p, for the model in part (b). How are these quantities related to the expected return µ and the volatility σ of the shares? State (without proof) how suitable values of u and d can be chosen for a given asset. (7) 3. Let (W t ) t 0 be a standard Brownian motion. (a) Explain what it means for a continuous time stochastic process (X t ) t 0 to be a martingale. Using Itô s Lemma, or otherwise, determine whether or not each of the following processes (X t ) t 0 is a martingale: (i) X t = 2W t ; (ii) X t = tw t ; (iii) X t = W 3 t 3tW t ; (iv) X t = e 2Wt t. (15) (b) Let (X t ) t 0 be the continuous time stochastic process defined by the stochastic differential equation dx t = 2e 2Xt dt + 2e Xt dw t ; X 0 = 0. Solve this equation to find an explicit formula for X t in terms of t and W t. Show that the solution becomes singular in a finite, random time. (10) Page 3 of 5 MASM006/continued...
4. The Black-Scholes partial differential equation is V t + 1 2 σ2 S 2 2 V V + rs rv = 0. S2 S The price C(t, S) of a European call option (with exercise price E and expiry time T ) satisfies the Black-Scholes equation, and is given by C(t, S) = SN(d 1 ) Ee r(t t) N(d 2 ). Here r is the riskfree interest rate (assumed constant), N(x) is the cumulative normal distribution function, N(x) = 1 2π x e s2 /2 ds, and d 1, d 2 = ln(s/e) + (r ± 1 2 σ2 )(T t) σ. T t (a) State (without proof) the Put-Call Parity Formula for European options, and use it, together with the formula for C(t, S) above, to deduce the Black-Scholes formula for the price P (t, S) of a European put option. (7) (b) Show that a function of the form V (t, S) = e r(t t) f(t, S) satisfies the Black-Scholes equation if and only if f satisfies the partial differential equation f t + 1 2 σ2 S 2 2 f f + rs S2 S = 0. (9) (c) The function f(t, S) = N(d 2 ) can be shown to satisfy the partial differential equation in part (b). (You do not need to verify this). Deduce from this that each of the functions V 1 (t, S) = Ee r(t t) N(d 2 ) and V 2 (t, S) = SN(d 1 ) is a solution of the Black- Scholes equation. By considering the behaviour of these functions as t T, give a description of the financial derivatives whose prices are given by V 1 and V 2. (9) Page 4 of 5 MASM006/continued...
5. (a) Explain what is meant by a BTCS (backward difference in time, central difference in space) numerical scheme to solve a partial differential equation relating a first derivative with respect to time and a second derivative with respect to space. Write down an BTCS scheme to solve the time-reversed Black- Scholes equation V τ 1 2 σ2 S 2 2 V V rs S2 S + rv = 0 (where τ = T t). Explain how this scheme can be expressed in terms of matrix algebra. (15) (b) Describe briefly the advantages and disadvantages of a BTCS scheme over a FTCS (forward difference in time, central difference in space) scheme. (4) (c) What boundary conditions should be imposed if the BTCS scheme is implemented to price (i) a European call option; (ii) a European put option? (6) Page 5 of 5 MASM006/END OF PAPER