# 1. Your customers are interested in the following contract. They are willing to commit to buying a stock at a fixed price of \$85 at a future date if the price of the stock P is between \$80 and \$90, bu

1. Your customers are interested in the following contract. They are willing to commit to buying a stock at a fixed price of \$85 at a future date if the price of the stock P is between \$80 and \$90, but would like to pay the market price at that time if the price of the stock P satisfies is either less than \$80, or higher than \$90.

a. Plot the payoff of the contract requested by the client.

b. Is it possible to replicate this position using only the stocks, puts and calls? If you think that it is portfolio. If you think that it is impossible explain why.

2. A bank has just sold a call option on 500,000 shares of a stock. The strike price is 40; the stock price is 40; the risk-free rate is 5%; the volatility is 30%; and the time to maturity is 3 months.

a. What position should the company take in the stock for delta neutrality?

b. Suppose that the bank does set up a delta neutral position as soon as the option has been sold and the stock price jumps to 42 within the first hour of trading. What trade is necessary to maintain delta neutrality? Explain whether the bank has gained or lost money in this situation.

3. A “log-option” with expiration T pays \$ ln(ST ), where \$ST is the price of the underlying stock at time T. Find the Black-Scholes-Merton price of such an option, if the current price of the stock is \$60, the risk- free interest rate is 4%, the volatility is 20%, and the option expires in 6 months.