FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A

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1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHE7013A Module Contact: Dr Davide Proment, MTH Copyright of the University of East Anglia Version: 1

2 (i) Address the following points related to risk free assets using clear explanation and precise mathematical notation. (a) Define maturity-independent yields. Show that if maturity-independent yields are deterministic but not constant an arbitrage opportunity would arise. (b) Define the duration D of a coupon bond. (ii) Two types of investment are available in the money market. The first investment, investment A, consists of a 5-year coupon bond paying annual coupons of 100 and face value of 500. The second investment, investment B, is a 2-year coupon bond paying coupons of 20 every 6 months and face value of 50. Today s maturity-independent yield value is y(0) = 1.5% per year. (a) Compute today s values and durations of the two investments A and B. (b) An investor is interested to invest 10, 000 in the money market for 3 years. Build a portfolio (y A,y B ) where y A and y B are the positions on the bonds A and B respectively, such that the total investment would be insensitive to small variations of the yields.

3 (i) Address the following points using clear explanation and precise mathematical notation. (a) State the No-Arbitrage Principle used as the main axiom in financial mathematics and explain its meaning using your own words. (b) Give the definition of European call and put options and express mathematically their payoffs. (ii) A financial portfolio whose value is V 1 is made today by buying one European call option with strike price X 1 = 100 and taking a short position in one European call option with strike price X 2 = 160, both having exercise time T and the same underlying risky asset S. (a) Calculate the value V 1 of the portfolio at time T. (b) How this type of portfolio is called in the financial jargon and why? (c) Draw carefully the profit plot of the portfolio assuming that the premiums of the two European call options are C (1) E and C(2) E respectively. Consider a second portfolio whose value is V 2 built today by taking a position in the underlying asset S and by buying one European put option with strike price X 3 = 60, exercise time T and underlying risky asset S. (d) What value of needs to be taken if we request that V 2 = V 1 at expiry for the underlying asset having value (i) S(T) = 180 and (ii) S(T) = 50? MTHE7013A PLEASE TURN OVER Version: 1

4 From past market data and assuming a binomial model to price stock, a risky asset whose value today is S 0 = $80, has estimated returns per step of U = 0.2 and D = 0.3 if the shares go up with probability p = 0.4 and down with probability 1 p = 0.6, respectively. The return per step on bonds is given by R = (i) Find the value S of the shares using a 2-step binomial model. Compute the expected value and variance of the shares value after two steps. (ii) Find the premium and the values at all steps of a European put option having strike price X = $90 and exercise time after two steps. (iii) Define American call and put options. Find the premium of an American put option having strike price X = $90 and exercise time after two steps. (iv) Show that if the condition D < R < U in the binomial model is broken, then an arbitrage opportunity arises when trading with stock.

5 The Put-Call Parity relation for European put and call options having premiums P E and C E respectively, same strike price X and exercise time T, and whose underlying asset does not pay any dividend is C E P E = S(0) Xe rt, where S(0) is the underlying asset s value today and r is the risk-free asset interest rate compounded continuously. (i) Prove the Put-Call parity relation by assuming the No-Arbitrage Principle. [8 marks] (ii) From past market data, and assuming a binomial model to price stock, a risky asset whose value today is S 0 = 100 has estimated returns per step U = 0.1 and D = 0.05 if the shares go up and down, respectively. The return per step on bonds is given by R = (a) Calculate the values of the shares using a 3-step binomial model. (b) Compute the premium of a European call option with strike price X = 115 and expiry time T = 3h where h is the time interval in the 3-step binomial model. (c) Compute the premium of a European put option with strike price X = 115 and expiry time T = 3h. Verify the Put-Call Parity relation. (d) Calculate the premium and the values at all steps of an American call option with strike price X = 115 and expiry time T = 3h. Comment on your answer. [12 marks] MTHE7013A PLEASE TURN OVER Version: 1

6 (i) Assume that the risky asset s price follows the Black Scholes model for stock S(t) = S(0)e µt+σw(t), where µ and σ are the drift and the volatility of the shares respectively and W(t) is a Wiener process. (a) Compute the infinitesimal variation of the share prices ds up to the order dt using the instruments of stochastic calculus. (b) By explaining clearly what you are doing and the assumptions taken, derive the Black Scholes equation V t σ2 S 2 2 V V +rv S2 S rv = 0, for the evolution of the price V(S,t) of a financial claim whose underlying risky asset S has volatility σ and risk-free asset s interest rate (compounded continuously) is given by r. [12 marks] (ii) The value of a European call option C(S,t) with strike price X and exercise time T satisfies the Black Scholes equation if C(S,t) = SN(d 1 ) Xe r(t t) N(d 2 ) where d 1 = log(s/x)+(r +σ2 /2)(T t) σ T t, d 2 = d 1 σ T t and N(x) = 1 2π x e y2 /2 dy. Calculate the position on stock and on bonds that an option writer has to take in order to hedge the European call option. [8 marks]

7 The values of the London Stock Market Exchange FTSE 250 index recorded at the beginning of each month from January 2016 to December 2016 are: Jan 17, May 16, Sep 17, Feb 16, Jun 17, Oct 18, Mar 16, Jul 16, Nov 17, Apr 16, Aug 17, Dec 17, (i) Compute the FTSE 250 index s monthly logarithmic returns, drift and volatility. [6 marks] (ii) Make use of the log-normal random walk model with N steps to mimic the shares prices where the random walk is described by S(t) = S(0)e µt+σw N(t), W N (t) = h(y 1 + +Y N ) and Y i = { +1 with p = with q = 0.5, i = 1,...,N. (a) By explaining clearly what you are doing, compute the future values in 2 months of the FTSE 250 using (i) a 2-step model and (ii) a 4-step model. (b) Assume that the risk-free asset s interest rate compounded continuously is r = 2% per year. Using the future FTSE 250 index values obtained with the 2-step model, find the premium and the values at each time steps of a European call option having strike price X = 165, expiry time in 1 month and whose underlying asset is a virtual asset whose value is a hundredth of the FTSE 250 index expressed in GBP. (c) Compute the expected value of the shares at time t for the log-normal random walk model as a function of the N steps and in the limit N. [14 marks] END OF PAPER