Question 4 (a) Suppose a stock price S without dividend payments is governed by a geometric Brownian motion. Interest rate r is constant and continuously compounded. (i) Using Ito formula, derive the Black-Scholes (BS) equation for the price of an European call option, written on S. [4] (ii) Using Ito formula, derive the BS equation for the price of a call option, written on S, with payo (3% e Kr. [4] (iii) Using Ito formula, derive the BS equation for the price of a future, written on S. [4] (iv) Discuss the difference between a future's BS equation and an option's BS equation. [4] 3 of 5 (b) Assume a non-dividend stock price 3 is governed by geometric Brownian motionlnterest rate r is constant and continuously compounded. A nancial institution plans to write a derivative security that pays off at time T a dollar amount equal to a x where a is a large positive constant. Use risk neutral valuation to determine an expression for the price of this derivative security at time t in terms of the current stock price S, its volatility a, the risk-free rate r and timetomaturity T t. [4] [Tot a1: 20 marks]