END-OF-CHAPTER QUESTIONS AND PROBLEMS
1. (A Nobel Formula) The formula has two terms:
C = S0N(d1),
is the present value of the expected value of the stock price, conditional on the option expiring in-the-money discounted at the risk-free rate. The second term,
is the present value of the expected payout of the exercise price at the expiration. N(d2) is the probability of exercise. All of this is based on the condition of risk neutrality, meaning that the probability distribution is based on a risk-free expected return on the stock.
2. (Variables in the BSM Model) a. The delta is the change in the call price for a given change in the stock price. Strictly speaking the delta applies only when the stock price changes by a very small amount. The delta also gives the hedge ratio, which tells how many shares of stock must be held (or sold short) to hedge a given short (or long) position in calls. The delta is between zero and one and, as expiration approaches, converges to one if the option is in-the-money and zero if the option is out-of-the-money.
b. Gamma is the change in delta for a given (again very small) change in the stock price. The gamma measures the risk involved in not adjusting the hedge ratio to equal the delta. The gamma will be large when the option is at-the-money and nearly zero when the option is deep in- or out-of-the-money.
c. Rho measures the change in the option price when the risk-free rate changes. The relationship is nearly linear and is fairly weak. In other words, the call price is not very sensitive to the risk-free rate.
d. Vega (also called kappa and lambda) measures the change in the option price for a change in the volatility. The relationship is nearly linear when the option is at-the-money. The option price is very sensitive to the volatility.
e. Theta is the relationship between the option price and the time to expiration. For European calls, the theta is negative meaning that the option price will fall as expiration approaches.
3. (Estimating the Volatility) If you accept the "theoretically correct" standard deviation as the true volatility, then the market price of the option is implying a higher volatility of the stock than is reasonable. That is, the implied volatility obtained by setting the Black-Scholes-Merton price equal to the market price is higher than it should be. This means that the market price is too high. You would consider selling the overpriced option, perhaps in conjunction with the purchase of the stock.
4. (Estimating the Volatility) a. Implied volatilities can indeed vary for options on the same stock with the same exercise price and different expirations. The volatility is supposed to be the volatility of the stock over the life of the option so it can indeed vary with a different time to expiration. The difference in volatilities is called the term structure of implied volatility.
b. In theory implied volatilities should not vary for options on the same stock with the same expiration and different exercise prices. The volatility is referring to the volatility of the stock over a given period of time. There cannot be but one such volatility. In practice, however, there are multiple volatilities, which is often referred to as the volatility smile or skew.
c. Often option prices are quoted using the implied volatility. A given option price could be quoted by stating its implied volatility, while another option on the same stock could be quoted by stating a different implied volatility. It is assumed that the actual option price is then found by plugging into a model such as Black-Scholes-Merton. The purpose of quoting option prices this way is because it is much easier to see which are the more expensive options.
5. (Stock Price) The primary reason is the inability to trade at no cost over a very small time interval. Delta