If you own a stock, buying a put option on the stock will greatly reduce your risk. This is the idea behind portfolio insurance. To illustrate, consider a stock that currently sells for $56 and has an annual volatility of 30%. Assume the risk-free rate is 8%, and you estimate that the stock’s annual growth rate is 12%.

a. Suppose you own 100 shares of this stock. Use simulation to estimate the probability distribution of the percentage return earned on this stock during a one‑year period.

b. Now suppose you also buy a put option (for $238) on the stock. The option has an exercise price of $50 and an exercise date one year from now. Use simulation to estimate the probability distribution of the percentage return on your portfolio over a one‑year period. Can you see why this strategy is called a portfolio insurance strategy?

c. Use simulation to show that the put option should, indeed, sell for about $238.