i) Consider a European option on an underlying stock whose payoff is 51 if ST>K and 0 otherwise. Assume that the initial stock price S0=70 and that the stock price either increases by a factor u=10/7 or decreases by a factor d=7/10, the time to expiry T=1, the strike price K=65, and the interest rate r=0.05. The interest rate compounds continuously. Consider one time step of the binomial option pricing model, where the option price either increases to uS0 or decreases to dS0 at expiry time T. Consider a portfolio in which we short-sell shares of the stock (short position) of the underlying asset and one European option on the stock (long position). a) What is the value of such that the value of the portfolio at expiry time is the same irrespective of whether the stock price increases to uS0 or decreases to dS0 ? b) What is the value of the portfolio at time 0 ? c) What is the value of the European option at time 0 ? ii) Use the inversion method and uniformly distributed UU[0,1] to calculate a stochastic variable X with cumulative distribution F(x)=1ex(xa),xa. iii) The stock price St is assumed to follow a geometric Brownian motion StdSt=dt+dWt,t(0,T], where Wt is a standard Brownian motion, , are positive constants and S0 is given. a) What is the distribution of St ? b) Describe how can you simulate the asset prices St at times 0=t0K and 0 otherwise. Assume that the initial stock price S0=70 and that the stock price either increases by a factor u=10/7 or decreases by a factor d=7/10, the time to expiry T=1, the strike price K=65, and the interest rate r=0.05. The interest rate compounds continuously. Consider one time step of the binomial option pricing model, where the option price either increases to uS0 or decreases to dS0 at expiry time T. Consider a portfolio in which we short-sell shares of the stock (short position) of the underlying asset and one European option on the stock (long position). a) What is the value of such that the value of the portfolio at expiry time is the same irrespective of whether the stock price increases to uS0 or decreases to dS0 ? b) What is the value of the portfolio at time 0 ? c) What is the value of the European option at time 0 ? ii) Use the inversion method and uniformly distributed UU[0,1] to calculate a stochastic variable X with cumulative distribution F(x)=1ex(xa),xa. iii) The stock price St is assumed to follow a geometric Brownian motion StdSt=dt+dWt,t(0,T], where Wt is a standard Brownian motion, , are positive constants and S0 is given. a) What is the distribution of St ? b) Describe how can you simulate the asset prices St at times 0=t0