(a) Assume a non-dividend paying asset St, undergoes geometric Brownian motion dS = S+dt+S+dWt where W is a standard Brownian motion and and are constants. We assume also that r, the risk-free interest rate, is constant throughout. Show that the Black-Scholes equation t 1 +rs + as 2 202C t = TC where c(t, St) is the value of a European call option may be reduced to Iv rv + = 41/1/0 + (1 - 1/1/0) (1-1) under the transformation S and is a constant. av 2 == Ae; c(t, S) = = Av(T,x); T = T-t (b) Consider a four-month European put option on a stock index. Suppose that the current value of the index is 300, the strike price is 290, the dividend yield is 3% per annum, the risk-free interest rate is 8% per annum with continuously, and volatility is 20% per annum. Show that an increase of 1 in the index increases of the option by approximately 0.01008.