4. [20 marks] Black-Scholes model. Assume that the stock price S is governed under the risk-neutral proba- bility measure P by the Black-Scholes stochastic differential equation dSt = St(rdt to dW+) where o > 0 is the volatility and r is the short-term interest rate. Con- sider the European contingent claim X with maturity T and the following payoff X = KST min (ST, L) where L= ert S, and K > 0 is an arbitrary constant. (a) Sketch the profile of the payoff X as a function of the stock price St at time T and show that X admits the following representation X = (K 1)St + Ct(L) where Cr(L) = (ST L)+ is the payoff at time T of the call option with the strike L. (b) Using the Black-Scholes call option pricing formula, find an explicit expression for the arbitrage price to(X) at time t = 0. (c) Find the limit of the arbitrage price To(X) when T approaches 0. (d) Find the limit of the arbitrage price no(X) when the volatility o goes to Q. (e) Explain why the price of X is positive when K > 1. 4. [20 marks] Black-Scholes model. Assume that the stock price S is governed under the risk-neutral proba- bility measure P by the Black-Scholes stochastic differential equation dSt = St(rdt to dW+) where o > 0 is the volatility and r is the short-term interest rate. Con- sider the European contingent claim X with maturity T and the following payoff X = KST min (ST, L) where L= ert S, and K > 0 is an arbitrary constant. (a) Sketch the profile of the payoff X as a function of the stock price St at time T and show that X admits the following representation X = (K 1)St + Ct(L) where Cr(L) = (ST L)+ is the payoff at time T of the call option with the strike L. (b) Using the Black-Scholes call option pricing formula, find an explicit expression for the arbitrage price to(X) at time t = 0. (c) Find the limit of the arbitrage price To(X) when T approaches 0. (d) Find the limit of the arbitrage price no(X) when the volatility o goes to Q. (e) Explain why the price of X is positive when K > 1