An alternative to the protective put is purchasing a call and a Treasury bill in preference to purchasing the stock and the put. (This strategy is not to be confused with purchas- ing a stock and a put versus purchasing a call and a bond. If the financial markets are in equilibrium, put–call parity implies that these two positions produce the same results.
See Problem 6.) Suppose a stock is selling for $100 and the call to buy the stock after one year is $6. A one-year Treasury bill is selling for $96 for a 4.19 percent yield. The investor could purchase the stock or purchase both the call and the bill. Notice that the costs of the two positions differ: $100 for the stock versus $102 for the call plus the bill.
(You may infer that the price of a one-year put option to sell the stock at $100 is $2, because put–call parity requires that the sum of the cost of the stock and the put equal the sum of the cost of the call and the bill.)
What are the potential profits and losses on the two positions at various prices of the stock? The answer to this question is given in the following profit/loss profile.
Price of the Stock
Profit (Loss) on the Stock
Profit (Loss) on the Call
Profit (Loss) on the Bill
Net on the Call 1 Bill
$110 $10 $4 $4 $8
105 5 (1) 4 3
100 0 (6) 4 (2)
95 (5) (6) 4 (2)
90 (10) (6) 4 (2)
As may be seen in the profit/loss profile, the combination of the call and the bill is similar to the stock on the upside but limits the loss on the downside. If the price of the
stock rises, the bill-call strategy generates almost the same profit. The $2 difference is the result of the difference in the initial cash outflows.
If the above profit/loss profile is compared to Exhibit 17.9 in the previous chapter, the profit/loss pattern is the same. Hence, this strategy is essentially another version of the protective put. Both limit the downside loss but do not limit the upside poten- tial. If the price of the stock does not rise or declines, the worst-case scenario for the call-bill combination is a loss of $2, the initial cash outflow. The worst-case scenario for the stock is that the investor could lose the entire $100. This reduction in potential loss suggests that the strategy reduces the investor’s risk but the reduction in risk only marginally limits the potential gain.
SUMMARY
While the previous chapter presented the basics concerning options, this chapter ex- panded that material by covering the Black-Scholes option valuation model; by ex- plaining how the stock, bond, and option markets are interrelated so that changes in one are transmitted to the others; and by illustrating several strategies using options.
The Black-Scholes option valuation model specifies that the value of a call option is positively related to the price and to the volatility of the underlying stock. As the price of the stock rises, the value of the option rises. The same relationship holds for vari- ability of returns, as options on volatile stocks command higher valuations. Call option values are also positively related to the life of the option. As the term of the option diminishes and the option approaches expiration, the option’s value declines.
Although an increase in interest rates generally depresses the value of a financial asset, this negative relationship does not apply to options to buy stock. An increase in interest rates increases the value of the option, because higher rates reduce the present value of a call’s strike price. The lower strike price then increases the value of the option to buy the stock.
In addition to being used to value publicly traded puts and calls, the Black-Scholes model may be applied to options issued by firms to selected employees, especially se- nior management. These options are part of incentive-based compensation packages.
If the firm is successful and the value of its stock rises, the value of the options also increases. Since incentive options are part of the employee’s compensation, an account- ing question arises: Should the cost of the options be expensed? Expensing requires a valuation, and the Black-Scholes model is often used to value incentive options in order to determine their cost.
The Black-Scholes option valuation model also calculates the hedge ratio, which determines the number of options necessary to completely hedge a stock portfolio. A completely hedged portfolio means that any loss generated on one side (e.g., a long position in stocks) is offset by the gain on the opposite side (e.g., a short position in the options). Such a hedging strategy is executed by portfolio managers to reduce risk. Such risk reduction may not be available to individual investors, since the portfolio has to be rebalanced frequently as the hedge ratio changes.
Put–call parity explains the interrelationships among financial markets. In equilib- rium, the price of the underlying stock, the price of the puts and calls on the stock, and the present value of the strike price (as affected by the rate of interest) must balance or
an opportunity for a risk-free arbitrage would exist. As investors seek to execute the arbitrage, the prices of the various securities are affected. An implication of put–call parity is that any change in one of the markets (e.g., an increased demand for stocks) must be transmitted to the other markets.
Strategies using options include the covered put and the protective call, which are the reverse of the covered call and the protective put. Other possible option strategies are straddles; bull and bear spreads; and collars. Straddles and spreads involve buying or selling more than one option on the same stock. Straddles and spreads permit inves- tors to take long, short, or hedged positions in stocks without actually owning or sell- ing the stocks. Collars permit investors who own stock but cannot sell it to lock in the current price. All these strategies using options alter the individual’s potential returns and risk exposure from investing in financial assets. Options, thus, are both a means to speculate on anticipated price movements in the underlying stocks and to manage the risk from actual price movements in the underlying stocks.
Summary of Option Strategies
Chapter 17 covered several strategies such as buying a call or a put, selling a naked call or a covered call, and the protective put. This chapter added several additional strate- gies. These included the following:
1. The covered put: sell a stock short and sell a put.
2. The protective call: sell a stock short and purchase a call.
3. The long straddle: buy a put and a call with the same strike price and expiration date.
4. The short straddle: sell a put and a call with the same strike price and expiration date.
5. The bull spread: purchase a call with a lower strike price and sell a call with a higher strike. Both options have the same expiration date.
6. The bear spread: sell the call with a lower strike price and buy a call with a higher strike. Both options have the same expiration date.
7. The collar: the investor owns a stock whose price has appreciated; this investor sells a call and buys put.
Straddles and spreads were illustrated using calls, but comparable strategies may be constructed using puts. Straddles and spreads may be expanded; Problem 17 illustrates the strip, in which the investor buys one call and two puts, and the strap, in which the investor buys two calls and one put. (These strategies may also be constructed using put options.) Problem 18 illustrates the butterfly spread, in which the investor buys (or sells) one of each of two options with different strike prices and sells (buys) two options with a strike price between the two options that were purchased. Various option strat- egies are also illustrated in the case “Not for the Faint of Heart: Analyzing Different Option Strategies.”
QUESTIONS
1. What, according to the Black-Scholes option valuation model, is the relationship be- tween the value of a call option and each of the following?
a) Risk as measured by the variability of the underlying stock’s return b) Interest rates
c) The term of the option (i.e., the length of time to expiration)
2. According to Black-Scholes option valuation and put–call parity, what will happen to the value of a put option if interest rates decline?
3. How may the Black-Scholes option valuation model be used to determine the risk as- sociated with the underlying stock?
4. An investor sells a stock short in July and its price declines in November—the posi- tion has generated a profit. However, the individual does not want to close the posi- tion and realize the profit during this tax year. Instead, the investor wants to maintain the position until January so the gain will be taxed next year. How can the hedge ra- tio be used to reduce the risk associated with the price of the stock rising while still transferring the gain to the next tax year?
5. An investor expects the price of a stock to remain stable and writes a straddle. What is this individual’s risk exposure? How may the investor close the position?
6. You expect the price of a stock to decline but do not want to sell the stock short and run the risk that the price of the stock may rise dramatically. How could you use a bear spread strategy to take advantage of your expectation of a lower stock price?
7. You sell a stock short. How can you use an option to reduce your risk of loss should the price of the stock rise?
8. How do collars, the hedge ratio, the protective call, and the protective put help inves- tors manage risk?
9. If you thought a stock was fairly valued and its price would not change, how could you use a straddle to take advantage of your valuation? If you follow this strategy and the stock’s price does not remain stable, have you increased your risk exposure?
10. The Black-Scholes valuation model shows that higher interest rates result in higher call option valuations. Do these higher interest rates and call valuations imply that put values will also rise?