3. i) Consider a European option on an underlying stock whose payoff is 51 if ST...

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3. 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 So = 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 uSo or decreases to dSo at expiry time T. Consider a portfolio in which we short-sell A 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 A such that the value of the portfolio at expiry time is the same irrespective of whether the stock price increases to uSo or decreases to do? 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 U~ - u[o, culate a stochastic variable X with cumulative distribution F(x)=1 - ex(-a), x ≥ a. iii) The stock price St is assumed to follow a geometric Brownian motion d.St St = μdt + odWt, te (0, T], where W is a standard Brownian motion, u, o are positive constants and So is given. to cal- a) What is the distribution of St? Describe how can you simulate the asset prices S, at times 0 = to < t₁ < t₂ < < tm = T. (The points are not necessarily equally spaced, i.e. tn -tn-1 tm- tm-1 for m‡n). c) Suppose there is a financial derivative with a payoff function { ["* Silt - K,0} . Stdt dt Outline a Matlab code which approximates the value of P. P = max 3. 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 So = 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 uSo or decreases to dSo at expiry time T. Consider a portfolio in which we short-sell A 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 A such that the value of the portfolio at expiry time is the same irrespective of whether the stock price increases to uSo or decreases to do? 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 U~ - u[o, culate a stochastic variable X with cumulative distribution F(x)=1 - ex(-a), x ≥ a. iii) The stock price St is assumed to follow a geometric Brownian motion d.St St = μdt + odWt, te (0, T], where W is a standard Brownian motion, u, o are positive constants and So is given. to cal- a) What is the distribution of St? Describe how can you simulate the asset prices S, at times 0 = to < t₁ < t₂ < < tm = T. (The points are not necessarily equally spaced, i.e. tn -tn-1 tm- tm-1 for m‡n). c) Suppose there is a financial derivative with a payoff function { ["* Silt - K,0} . Stdt dt Outline a Matlab code which approximates the value of P. P = max

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