ECMM706: Assignment 1: FOR CREDIT 1. Shares in a certain company are currently priced at 90 pence each. One year from now, their price will either have risen by 20% (with probability 0.5) or fallen 20% (with probability 0.5). Take the annual riskfree interest rate as r = 0:05 (compounded continuously). 1. Calculate the expected discounted payment from selling one share a year from now. 2. Use the One-Step Binomial Model to find the value now of a European call option on one share, with expiry date one year from now and with exercise price 95 pence. 3. Suppose going from the end of Year 1 to end of Year 2, the share price evolves in the same probabilistic and proportional way as specified within Year 1, with annual riskfree rate r = 0:05. By using the One-Step Binomial Model (twice over), find the value now of a European call option on one share, with expiry date T = 2 years from now and with exercise price 95 pence. 4. Use the Put-Call Parity Formula to find the value now of a European put option on one share, with expiry date T = 2 years from now and with exercise price 95 pence. [14] 2. Let fYng be an independent and identically distributed sequence of RVs, with P (Yn = a) = 1 2 = P (Y = −a); for some a > 0: Let Sn be the stochastic process defined by the sequence: Sn = Sn−1(1 + Yn 3); n ≥ 1; S0 = 1; and let (Fn)n≥0 denote the natural filtration associated to this process. 1. Find (in terms of a) the value of the expectations E(Sn), E(Sn 2). 2. For m < n, find E(Sn j Fm) and E(Sn 2 j Fm). 3. Deduce the range of values of a (if any) for which i) Sn is a martingale; ii) Sn 2 is a martingale. [10] 3. Using the It^o isometry formula, compute E "Z0t g(s; Ws) dWs2# in the following cases: (a) g(t; Wt) = te2Wt; (b) g(t; Wt) = Wt sin t. [6] 14. Modify the program CallBinomial.m (or write a program of your own), to price a cash or nothing option by using the Binomial Algorithm. A cash or nothing call option (with expiry time T, exercise price E) has the payoff function given by A; if S(T) > E, 0; if S(T) < E, where A > 0 is fixed. (a) Use your program to price call options with the following parameters: S(0) = 100; E = 100; A = 120; T = 2; r = 0:03; σ = 0:2: (Take the number of timesteps as 50.) (b) Try out your program by varying each of the six parameters in turn (while keeping the others parameters fixed). Plot a graph of the call option price versus the parameter value, and using your plots describe briefly how the option price behaves with the parameters. i.e. does the price appear to change in a monotone way as the parameters are changed? [In Higham's book, Chapter 17, an explicit formula is given for the price of the option in the continuous time limit.] (c) If C(t; S) denotes the price of the call option (in the continuous time limit), decide if the following partial derivatives are always positive, always negative, or of indeterminate sign: @C @S ; @C @E ; @C @A: You do not need to give any formulae for these partial derivatives: you may refer to graphs in part (b). Give financial explanations for the behaviour of these partial derivatives. [20] 2