1. Black Scholes and Delta Hedging Questions (20 points) 1. Use the Black-Scholes formula for the following exercises. Please answer the following questions. You can assume that interest rates and dividend yields are zero. The stock price is 100.

1) Consider the ATM option of strike 100 of maturities 1 day, 1 week, 1 month and 1 year, respectively. Let the volatility be 20% annualized. It doesnt matter if you use a call or a put. What are the prices of these options? Can you figure out the relation between the ATM option as a function of maturity? (You can plot them)

2) Consider options of strikes 95, 100, and 105 of 1-month maturity. If the stock price changes by 1 (i.e. either from 100 to 99, or from 100 to 101), how much does the option price change? Use these changes to estimate the delta of both the call and put.

3) Consider options of strikes 95, 100, and 105 of 1-month maturity. If the volatility changes by 1% annualized (i.e., from 20 to 21 or 19), then how much does the option price change? Use these changes to estimate the vega for both the call and put.

4) Explain how the delta would change if stock price changes and volatility changes were correlated such that a decline in the stock price by 1% increased the volatility by 1%.