Consider a European put option on a non-dividend-paying stock where the stock price is $40, the strike price is $45, the continuously compounded risk- free rate is 4% per annum, the volatility of log return is o = 30% per annum and the time to maturity is 6 months

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Consider a European put option on a non-dividend-paying stock where the stock price is $40, the strike price is $45, the continuously compounded risk- free rate is 4% per annum, the volatility of log return is o = 30% per annum and the time to maturity is 6 months. (a) (20 marks] Calculate u, d and p for a two-step risk-neutral tree with At = 0.25, where u and d are gross returns for an 'up' and 'down' move on the tree, respectively, and p is the risk-neutral probability of the stock price going 'up'. (b) [20 marks] Plot the two-step risk-neutral tree (At = 0.25) for stock prices. (c) [20 marks] Value the option using a two step risk-neutral tree. (d) [20 marks] Value the option using the Black and Scholes option pricing formula. Show intermediate calculations. (e) [20 marks] Assume that the Black and Scholes assumptions hold. Sup- pose you hold the put option described in this question in your portfolio. You want to hedge it dynamically by taking a position in the underlying stock S. How many shares of the underlying stocks should you pur- chase or short to construct a risk-free portfolio.