The movement of a share price over the next two months is to be modelled using...

  

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The movement of a share price over the next two months is to be modelled using a two-period recombining binomial model. Over each month, it is assumed that the share price will either increase or decrease by 10%. (1) Over each month, the risk-neutral probability of an up-step is q=0.55. Calculate the monthly risk-free force of interest r that has been used to arrive at this figure. [1] (ii) The current share price is 1. The annualised expected force of return on the share is = 30%. Calculate the state-price deflators in each of the three possible final states of the share price. (iii) Calculate the value of each of the following two-month derivatives: (a) a derivative with payoff profile (1,0,0) (b) a derivative with payoff profile (0.1,0) (c) a derivative with payoff profile (0,0,1) (d) a European call option with a strike price of K = 0.95 (e) a European put option with a strike price of K=1.05 (f) a derivative whose payoff is 2x|5-0.98, where S is the share price at the end of the two months. [5] (1) (ii) (ii) Let X, be a diffusion that satisfies X, (X,,t)dt o (X₁.) dB, where B, is a standard Brownian motion. Let f(x, t) be a function of r and X₁. By considering Taylor's theorem, suggest a partial differential equation that must be satisfied by /(X.) in order that it is a martingale. [3] Verify that your equation holds when f(x,1)=B² -1. Find g (1) such that B, +g (1) B, is a martingale [1] E [2] [Total 6] In the following. B, denotes a standard Brownian motion. (i) Write down the general solution of the stochastic differential equation: dX, =-yX,dt + odB, (ii) Hence determine the solution of the stochastic differential equation: dR, = 0.8 (4-R,)dt +dB, where Ro = 5. Find the distribution of the process R, at times and in the long-term. [1] [1] 3 [3] [Total 5] The movement of a share price over the next two months is to be modelled using a two-period recombining binomial model. Over each month, it is assumed that the share price will either increase or decrease by 10%. (1) Over each month, the risk-neutral probability of an up-step is q=0.55. Calculate the monthly risk-free force of interest r that has been used to arrive at this figure. [1] (ii) The current share price is 1. The annualised expected force of return on the share is = 30%. Calculate the state-price deflators in each of the three possible final states of the share price. (iii) Calculate the value of each of the following two-month derivatives: (a) a derivative with payoff profile (1,0,0) (b) a derivative with payoff profile (0.1,0) (c) a derivative with payoff profile (0,0,1) (d) a European call option with a strike price of K = 0.95 (e) a European put option with a strike price of K=1.05 (f) a derivative whose payoff is 2x|5-0.98, where S is the share price at the end of the two months. [5] (1) (ii) (ii) Let X, be a diffusion that satisfies X, (X,,t)dt o (X₁.) dB, where B, is a standard Brownian motion. Let f(x, t) be a function of r and X₁. By considering Taylor's theorem, suggest a partial differential equation that must be satisfied by /(X.) in order that it is a martingale. [3] Verify that your equation holds when f(x,1)=B² -1. Find g (1) such that B, +g (1) B, is a martingale [1] E [2] [Total 6] In the following. B, denotes a standard Brownian motion. (i) Write down the general solution of the stochastic differential equation: dX, =-yX,dt + odB, (ii) Hence determine the solution of the stochastic differential equation: dR, = 0.8 (4-R,)dt +dB, where Ro = 5. Find the distribution of the process R, at times and in the long-term. [1] [1] 3 [3] [Total 5]

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