UNIVERSITY OF BRISTOL School of Mathematics FINANCIAL MATHEMATICS MATH35400 (Paper code MATH-35400) MAY/JUNE 2020 2 hours 30 minutes This paper contains 4 questions. All answers will be used for assessment. On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions. Do not turn over until instructed. Page 1 of 5 ... .....
Cont ... 35400-20 1. (25 marks) (a) (3 marks) Explain in one sentence each: i. the difference between an option and a forward contract, ii. the difference between a call option and a put option, iii. the difference between a European option and an American option. (b) (8 marks) State and prove the put-call parity. [Hint: You may use without proof that two combinations of financial derivatives with the same prices VT = WT at some time T must also have the same prices Vt = Wt for any time t < T before.] (c) The current price for AKOCOM shares is £37.50 per share. A put option on 1 share of AKOCOM maturing in exactly half a year with a strike price of £30 is traded at £11. The price of the corresponding call option with same strike price and same maturity is currently £18.50. Assume that the interest rate for lending money as well as depositing money on a bank account is 5% per annum. i. (1 mark) Use the put-call parity to show that there exist arbitrage opportunities in the model above. ii. (4 marks) Describe explicitly what actions an arbitrageur would carry out today and in six months time to attain riskless profit. You can assume that the arbi- trageur does not have to pay any fees and that short selling is possible. [Hint: You might need to distinguish two cases to describe the actions in six months time.] (d) (3 marks) Let St be the time t price of some stock and q1 and C? the time t prices of European call options on that stock with maturity T and strike prices K1 and K2 where K1 < K2. We also consider a capped call option X on the same stock with same maturity T which has value XT = {ST - K1, 0, K2 - K1, if K2 ? ST if K1 ? ST ? K2 if ST ? K1. Find the time t price Xt of the capped call option in terms of Cf and C ?. (e) Consider a single-period model with one stock that has a current price of S0 = 1 and a future price of either S1(1) = 1.3 or S1 (2) = 1.1. The interest rate of the risk-free asset is r. i. (4 marks) Find all values for r such that there exists a risk neutral probability measure, and determine that measure in terms of r. ii. (2 marks) Assume that r = 0.4 and that stock holders are getting paid a dividend of 0.1 at time point 1. Describe explicitly what an