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# Option Valuation and Strategies

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1. A call option is the right to buy stock at \$50 a share. Currently the option has six months to expiration, the volatility of the stock (standard deviation) is 0.30, and the rate of interest is 10 %
a) What is the value of the option according to the Black-Scholes model if the price of the stock is \$45, \$50, or \$55?
b) What is the value of the option when the price of the stock is \$50 and the option expires in six months, three months, or one month?
c) What is the value of the option when the price of the stock is \$50 and the interest rate is 5 %, 10%, or 15%?
d) What is the value of the option when the price of the stock is \$50 and the volatility of the stock is 0.40, 0.30, or 0.10?
e) What generalizations can be derived from the solution to these problems?

2. Put-call parity in effect states that a combination of a put, a call, and a risk-market is in disequilibrium. The resulting arbitrage alters the securities' prices until the value of the three securities equals the value of the stock. Currently the price of the put option is \$1.50, and the rate of interest is 10%-so that the investor may purchase a \$50 discounted note for \$45.50
Given these prices, an arbitrage opportunity exists. Verify this setting up a risk-less arbitrage. Show the possible profit if the price of the stock is \$45, \$50, or \$55? at the expirations of the options (Since the problem does not specify which market is in disequilibrium, all prices would simultaneously change. To facilitate answering the problem, assume that only the market for the call opportunity, the actual final price of each security is irrelevant to the answer)

3. One useful piece of information derived from the Black-Scholes model for the valuation of a call option is the hedge ratio, which gives the slope of the line relating the change in the price of an option to the change in the price of the stock
a) If the delta is 0.6 and the investor owns 600 shares of stock, how may the investor use call options to hedge the position
b) If the investor buys a call option on 100 shares, what position in the stock and how many shares will offset movement in the price of the options?