Consider the risk-neutral dynamics of the price process of a non-dividend-paying stock, dSt=rStdt+StdWt, where r is the constant instantaneous risk-free rate, is a positive constant, and W is a standard Brownian motion under the risk-neutral probability measure Q. (a) Assume that St=g(t,Wt)=S0exp((r22)t+Wt). Use It's Lemma to show that dSt is the above stochastic differential equation. [7 marks] (b) Let Xt=St2+t. Use It's Lemma to find dXt. [8 marks ] (c) Consider a derivative Yt written on the stock which only pays at a prespecified maturity T and has the following payoff: YT=2,1,0,ifififSTK2,K2>STK1,K1>ST, where 00 is a constant agreed upon at time 0 . Draw the payoff diagram of this derivative. Find Z0, the price of this derivative at time 0 . [10 marks] Consider the risk-neutral dynamics of the price process of a non-dividend-paying stock, dSt=rStdt+StdWt, where r is the constant instantaneous risk-free rate, is a positive constant, and W is a standard Brownian motion under the risk-neutral probability measure Q. (a) Assume that St=g(t,Wt)=S0exp((r22)t+Wt). Use It's Lemma to show that dSt is the above stochastic differential equation. [7 marks] (b) Let Xt=St2+t. Use It's Lemma to find dXt. [8 marks ] (c) Consider a derivative Yt written on the stock which only pays at a prespecified maturity T and has the following payoff: YT=2,1,0,ifififSTK2,K2>STK1,K1>ST, where 00 is a constant agreed upon at time 0 . Draw the payoff diagram of this derivative. Find Z0, the price of this derivative at time 0 . [10 marks]