Black-Scholes-Merton Model (one point each per section of each problem, 12 total) 1. Suppose that a stock price has the expected return of 10% per annum and volatility of 30% per annum. The stock price is currently $50. The stock pays no dividends. Calculate the following: (a) The expected stock price in one year. (b) The standard deviation of the stock price in one year. (c) The 95% confidence interval for the stock price in one year. (d) What is the probability that the stock price will be greater than $60 in one year? (e) What is the probability that the stock price will be less than $40 in one year? 2. With the same data as above and assuming the interest rate is 3% per annum, calculate the prices of the following options using the Black-Scholes model (one option contract is always written on 100 shares of stock): (a) A call option with the strike of $60 and one year to expiration. (b) A binary call option that pays ten dollars per share at expiration in one year if the stock is above $60. (c) A put option with the strike of $40 and one year to expiration. (d) A binary put option that pays ten dollars per share at expiration in one year if the stock is below $40. 3. With the same data as above, consider a bull call spread where you buy a call option with the strike of 60 and one year to expiration and sell a call option with the strike of 70 and one year to expiration. (a) What is the price of the call spread? (b) What is the delta of each of the two call options? (c) Assume you took a long position in this call spread. What position in the underlying stock do you need to take to fully delta-hedge this spread?