Show that the formula for Delta (N(d1)) given in the text is correct. Hint: Euler, homogeneous.

The replication method implicit in the "tree" argument for the Black-Scholes formula shows that the option is a combination of the underlying stock and a riskless bond. Both these instruments have positive expected returns. How is it possible that an option can have negative theta?

Use put-call parity to show that calls and puts have the same implied volatility. Out-of-the money puts on almost any equity instrument will have high implied volatility. (Look up "Volatility Smile".) How do you reconcile these two facts?

Exercises 3.1, 3.2, 3.9, 3.13, 4.1, 4.3, 4.6

The downloable table shows option prices for MWD as of August 24, 2005. At the time of these quotes, the underlying stock was at $52.29 per share. The stock is expected to pay two dividends of $0.27, at the end of September and at the end of December. Assume that the dividends are certain and that bid-asked spreads are zero. Find as many arbitrage opportunities as you can.