UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS MTHE6026A 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. MTHE6026A Module Contact: Dr Davide Proment, MTH Copyright of the University of East Anglia Version: 1
- 2-1. (i) Address the following points related to risk-free assets using clear explanation and precise mathematical notation. (a) Explain the three methods of simple interest, periodic compounding and continuous compounding to compute the interests of a risk-free asset. (b) State what is the growing factor corresponding to each of the three methods. (c) Derive the growing factor for the continuous compounding method given that the growing factor for the periodic compounding method. [10 marks] (ii) Two types of investments are available in the money market. The first attracts interest with a continuous compounding method having rate r 1 = 1.8% per year. Inthesecond theinterest iscompounded annually andtheinterest rateis r 2 = 2%. Which of the two investments is preferable if the investment time is 3 years? (iii) Define zero-coupon bonds and coupon bonds. What is today s value of a 4-year coupon bond paying coupons of 30 and having face value of 100 if an interest rate of r = 3% per year is considered? MTHE6026A Version: 1
- 3-2. (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. [10 marks] (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? [10 marks] MTHE6026A PLEASE TURN OVER Version: 1
- 4-3. 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 = 0.05. (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. MTHE6026A Version: 1
- 5-4. 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 = 0.02. (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] MTHE6026A PLEASE TURN OVER Version: 1
- 6-5. (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 + 1 2 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] MTHE6026A Version: 1
- 7-6. 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 17,122.15 May 16,730.09 Sep 17,849.63 Feb 16,489.26 Jun 17,061.47 Oct 18,183.52 Mar 16,789.49 Jul 16,465.49 Nov 17,523.30 Apr 16,843.68 Aug 17,131.21 Dec 17,497.34 (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 = 0.5 1 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 MTHE6026A Version: 1
MTHE6026B Feedback on Main Series Examinations Question 1 Overall very well done. Some of you simply stated that the continuous interest method is the limit for m of the periodic compounding without showing the calculations and marks were taken due to this as you were asked to show the calculation. Also some marks were taken for not saying what that m is the number of payments for the periodic compounding method. Question 2 Overall well done. The main mistake was adding the premiums to the value of the portfolio V 1 and marks were taken for that. Also, marks were taken for not saying what is the profit or not taking into account the fact that the initial value of the portfolio V (t = 0) has to be multiplied by the growing factor before subtracting it to the final value V 1 (T ). Around half of the class realised that the portfolio V 1 is an example of a bull spread. Only a few of you correctly solved Q2(ii)(d), realising that is a non-dimensional number that represent the share position (# of shares in the portfolio V 2 ). Question 3 Overall well done. Marks were taken for not saying clearly how to compute the shares value using the recursive formula and for not saying what is the risk-neutral probability p. Many of you computed the standard deviation σ rather than the variance σ 2 and lost marks. Some of you did not use the risk-neutral probability p to compute the values of the option but used instead p obtaining completely wrong results. Many of you answered Q3(iv) by simply saying that if D < R < U is not satisfied the risk-neutral probability may become negative: this is indeed true but you were requested to prove that an arbitrage opportunity arises, hence you did not obtain the full marks. Question 4 Overall very well done. The main mistake in this question was using the Put-Call parity relation to obtain the premium of the European put in Q4(ii)(c). You were instead asked to compute the European put option values at all times (including the initial time, that is finding the premium) and then verified the Put-Call parity. Some of you used the continuous version of the Put-Call parity reported in the text of Q4 rather than modifying it to the discrete version in order to take into account the fact that the discrete binomial model is used to find the option values. Question 5 Q5(i) was overall well done. Marks were taken for not saying clearly that (dw ) 2 = dt for a Wiener process W (t) and that V rep is the value of the self-replicating strategy that equals the option value at any time. Essentially none of you were able to fully solve Q5(ii) but a few noticed that the hedging share position in the European call option solution C(S, t) is simply given by N(d 1 ) and got some marks. Question 6 Overall not very well done. Several mistakes in the calculations of the drift and/or volatility and conceptual mistakes in the formulas used to compute them. Marks were also taken for not writing the dimensional units in µ, σ and h used. Few attempts in solving Q6(ii)(c) and essentially all the answers were not correct. Dr Davide Proment 1/1 2017