2. Your client wants to bet on a certain stock which pays no dividends. In particular, she wants to trade a European derivative on the stock that expires at T = 0.5 and has the payoff shown in the figure below. The current stock price is $52.5, and the interest rate is r = 5% 15 10 5 payon 0 -5 - 10 1 -15 40 45 50 55 60 65 ST (a) How can you replicate this payoff with positions in "vanilla" instruments (European puts and calls)? What is your client's prediction on this stock? (b) You have used a five-step binomial model, to price a digital call option with maturity of six months and strike price 50, and determined that the fair price of the digital option is $0.5. Assuming the implied volatility is flat across all option strikes, what is the the Black-Scholes model price for this instrument? (c) Assuming your client can enter the trade using the price your have calculated in part (b), calculate your client's mark to market for different S, in the range [40,65] in one month's time if the volatility of the stock changes to 30%. (d) Calculate the delta of the option portfolio today, and investigate how the delta changes with respect to S, in one month's time assuming the same volatility change as in (c). Propose to your client a modified strategy that yields a more stable delta