1. Estimate the values of derivatives with the following characteristics using Binomial Trees with 20 steps (time steps). a. European and American call options with underlying asset IBM stock with today price So=$60, strike price X=59, risk-free rate r=6%, dividend yield d=4% and time to maturity T=1. Using the actual stock prices to calculate the stock volatility. What do you notice? What would we observe if d=0? [25%] b. Suppose we have one more asset, that of Microsoft with today's value Vo=70, dividend yield d=5% and standard deviation =20%. The risk-free rate remains the same. To estimate the price of a derivative with the highest payoff (maximum) from the two underlying assets, IBM and Microsoft stocks (Margrabe 1978).

2. The cash-or-nothing call option pays zero if the stock price at expiration of the derivative is below the strike price, and pays a fixed amount Q if the stock price at expiration of the derivative is above the strike price. Using Binomial Trees with 10 steps (time steps) estimate the price of a cash-or-nothing call option with underlying asset IBM stock and Q equal to $55.

3. A compound option is an option with an underlying asset and another option with a different underlying asset (option on option). The exercise price of a compound option is a function of the price of the other option. Thus there are 2 strike prices (X1 and X2) and 2 maturities (T1 and T2), one for each option. If X1 and X2 are 2 and 8 respectively, and T1 and T2 are 1 and 2 years respectively, calculate the value of the compound option with underlying asset of the second option the IBM stock with the data of 1a. Both options are European.