p:[0,3]Rp(x)=cx2 where c>0 is a constant. Let f:[0,3]R be given. a) Find the value of the constant c. b) Write down the integral for the expectation value E[f(X)]. c) Find the inverse cumulative distribution function corresponding to p. d) Let YU([0,1]) be a uniformly distributed random variable. How can you use this random variable to generate samples from the random variable X ? e) Write a Matlab code which generates 10 samples with pdf p given above. ii) Consider a European put option on an underlying stock with initial stock price S0=30. Assume that the stock price either increases by a factor u=6/5 or decreases by a factor d=5/6, the time to expiry T=1/4, the strike price K=30.5, and the interest r=0.05. The interest 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 of shares of the stock (long position) of the underlying asset and one European put 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 put option at time 0? iii) Let ,>0,T>0 and X0 be given constants. Consider the random walk Xn=Xn1++Zn,n=1,2,,N, where Z1,Z2,,ZNN(0,1) are independent, standard normal random variables and =NT. a) Write a Matlab code which generates a random vector (X1,X2,,XN). b) What is the distribution of XN ? c) Find the mean and variance of XN. p:[0,3]Rp(x)=cx2 where c>0 is a constant. Let f:[0,3]R be given. a) Find the value of the constant c. b) Write down the integral for the expectation value E[f(X)]. c) Find the inverse cumulative distribution function corresponding to p. d) Let YU([0,1]) be a uniformly distributed random variable. How can you use this random variable to generate samples from the random variable X ? e) Write a Matlab code which generates 10 samples with pdf p given above. ii) Consider a European put option on an underlying stock with initial stock price S0=30. Assume that the stock price either increases by a factor u=6/5 or decreases by a factor d=5/6, the time to expiry T=1/4, the strike price K=30.5, and the interest r=0.05. The interest 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 of shares of the stock (long position) of the underlying asset and one European put 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 put option at time 0? iii) Let ,>0,T>0 and X0 be given constants. Consider the random walk Xn=Xn1++Zn,n=1,2,,N, where Z1,Z2,,ZNN(0,1) are independent, standard normal random variables and =NT. a) Write a Matlab code which generates a random vector (X1,X2,,XN). b) What is the distribution of XN ? c) Find the mean and variance of XN
