Hedge Ratios and the Black-Scholes Formula
In the last chapter, we considered two investments in FinCorp stock: 100 shares or 1,000 call options. We saw that the call option position was more sensitive to swings in the stock price than was the all-stock position. To analyze the overall exposure to a stock price more precisely, however, it is necessary to quantify these relative sensitivities. A tool that enables us to summarize the overall exposure of portfolios of options with various exercise prices and times to expiration is the hedge ratio. An option’s hedge ratio is the change in the price of an option for a $1 increase in the stock price. A call option, therefore, has a positive hedge ratio and a put option a negative hedge ratio. The hedge ratio is commonly called the option’s delta.
If you were to graph the option value as a function of the stock value, as we have done for a call option in Figure 21.9 , the hedge ratio is simply the slope of the value curve evalu- ated at the current stock price. For example, suppose the slope of the curve at S 0 5 $120 equals .60. As the stock increases in value by $1, the option increases by approximately
$.60, as the figure shows.
For every call option written, .60 share of stock would be needed to hedge the investor’s portfolio. For example, if one writes 10 options and holds six shares of stock, according to the hedge ratio of .6, a $1 increase in stock price will result in a gain of $6 on the stock holdings, whereas the loss on the 10 options written will be 10 3 $.60, an equivalent $6.
The stock price movement leaves total wealth unaltered, which is what a hedged position is intended to do. The investor holding the stock and options in proportions dictated by their relative price movements hedges the portfolio.
12 For a more complete treatment of American put valuation, see R. Geske and H. E. Johnson, “The American Put Valued Analytically,” Journal of Finance 39 (December 1984), pp. 1511–24.
Notice that this value is consistent with put-call parity:
P5C1PV(X)2S0513.70195e2.103.25210056.35
As we noted traders can do, we might then compare this formula value to the actual put price as one step in formulating a trading strategy.
Black-Scholes hedge ratios are particu- larly easy to compute. The hedge ratio for a call is N ( d 1 ), whereas the hedge ratio for a put is N ( d 1 ) 2 1. We defined N ( d 1 ) as part of the Black-Scholes formula in Equation 21.1 . Recall that N ( d ) stands for the area under the standard normal curve up to d. Therefore, the call option hedge ratio must be positive and less than 1.0, whereas the put option hedge ratio is negative and of smaller absolute value than 1.0.
Figure 21.9 verifies the insight that the slope of the call option valuation function is less than 1.0, approaching 1.0 only as the stock price becomes much greater than the exercise price. This tells us that option values change less than one-for-one with changes in stock prices. Why should this be? Suppose an option is so far in the money that you are absolutely certain it will be exercised. In that case, every dollar increase in the stock price would increase the option value by $1. But if there is a reasonable chance the call option will expire out of the money, even after a moderate stock price gain, a $1 increase in the stock price will not necessarily increase the ultimate payoff to the call; therefore, the call price will not respond by a full dollar.
The fact that hedge ratios are less than 1.0 does not contradict our earlier observation that options offer leverage and are sensitive to stock price movements. Although dollar movements in option prices are less than dollar movements in the stock price, the rate of return volatility of options remains greater than stock return volatility because options sell at lower prices. In our example, with the stock selling at $120, and a hedge ratio of .6, an option with exercise price $120 may sell for $5. If the stock price increases to $121, the call price would be expected to increase by only $.60 to $5.60. The percentage increase in the option value is $.60/$5.00 5 12%, however, whereas the stock price increase is only
$1/$120 5 .83%. The ratio of the percentage changes is 12%/.83% 5 14.4. For every 1%
increase in the stock price, the option price increases by 14.4%. This ratio, the percentage change in option price per percentage change in stock price, is called the option elasticity.
The hedge ratio is an essential tool in portfolio management and control. An example will show why.
Value of a Call (C)
S0
40
20
0
120
Slope 5 .6
Figure 21.9 Call option value and hedge ratio
Example 21.4 Hedge Ratios
Consider two portfolios, one holding 750 IBM calls and 200 shares of IBM and the other holding 800 shares of IBM. Which portfolio has greater dollar exposure to IBM price movements? You can answer this question easily by using the hedge ratio.
Each option changes in value by H dollars for each dollar change in stock price, where H stands for the hedge ratio. Thus, if H equals .6, the 750 options are equivalent to .6 3 750 5 450 shares in terms of the response of their market value to IBM stock price movements. The first portfolio has less dollar sensitivity to stock price change because the 450 share-equivalents of the options plus the 200 shares actually held are less than the 800 shares held in the second portfolio.
This is not to say, however, that the first portfolio is less sensitive to the stock’s rate of return. As we noted in discussing option elasticities, the first portfolio may be of lower total value than the second, so despite its lower sensitivity in terms of total market value, it might have greater rate of return sensitivity. Because a call option has a lower market value than the stock, its price changes more than proportionally with stock price changes, even though its hedge ratio is less than 1.0.
Valuation
T he spreadsheet below can be used to determine option values using the Black-Scholes model. The inputs are the stock price, standard deviation, expiration of the option, exercise price, risk-free rate, and dividend yield.
The call option is valued using Equation 21.1 and the put is valued using Equation 21.3 . For both calls and puts, the dividend-adjusted Black-Scholes formula substitutes Se 2 d T
for S, as outlined on page 731. The model also calculates the intrinsic and time value for both puts and calls.
Further, the model presents sensitivity analysis using the one-way data table. The first workbook presents the analysis of calls while the second workbook presents simi- lar analysis for puts. You can find these spreadsheets at the Online Learning Center at www.mhhe.com/bkm.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Chapter 21- Black-Scholes Option Pricing Call Valuation & Call Time Premiums
Standard deviation (s) Variance (annual, s2) Time to expiration (years, T) Risk-free rate (annual, r) Current stock price (S0) Exercise price (X) Dividend yield (annual, d) d1
d2 N(d1) N(d2)
Black-Scholes call value Black-Scholes put value
Intrinsic value of call Time value of call Intrinsic value of put Time value of put
0.27830 0.07745 0.50 6.00%
$100.00
$105.00 0.00%
0.0029095
—0.193878 0.50116 0.42314
$6.99992
$8.89670
$0.00000 6.99992
$5.00000 3.89670
A B C D
Standard Deviation
Call Option
Value 0.15 0.18 0.20 0.23 0.25 0.28 0.30 0.33 0.35 0.38 0.40 0.43 0.45 0.48 0.50
E
7.000 3.388 4.089 4.792 5.497 6.202 6.907 7.612 8.317 9.022 9.726 10.429 11.132 11.834 12.536 13.236
F G
Standard Deviation
Call Time Value
H I J
Stock Price
Call Option
Value
$60
$65
$70
$75
$80
$85
$90
$95
$100
$105
$110
$115
$120
$125
$130
$135.00 K
7.000 0.017 0.061 0.179 0.440 0.935 1.763 3.014 4.750 7.000 9.754 12.974 16.602 20.572 24.817 29.275 33.893
L M
Stock Price
Call Time Value
$60
$65
$70
$75
$80
$85
$90
$95
$100
$105
$110
$115
$120
$125
$130
$135 N
7.000 0.017 0.061 0.179 0.440 0.935 1.763 3.014 4.750 7.000 9.754 7.974 6.602 5.572 4.817 4.275 3.893 0.150
0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500
7.000 3.388 4.089 4.792 5.497 6.202 6.907 7.612 8.317 9.022 9.726 10.429 11.132 11.834 12.536 13.236 LEGEND:
Enter data Value calculated
See comment
Portfolio Insurance
In Chapter 20, we showed that protec- tive put strategies offer a sort of insur- ance policy on an asset. The protective put has proven to be extremely popular
with investors. Even if the asset price falls, the put conveys the right to sell the asset for the exercise price, which is a way to lock in a minimum portfolio value. With an at-the-money put ( X 5 S 0 ), the maximum loss that can be realized is the cost of the put. The asset can be sold for X, which equals its original value, so even if the asset price falls, the investor’s net loss over the period is just the cost of the put. If the asset value increases, however,
CONCEPT CHECK
8
What is the elasticity of a put option currently sell- ing for $4 with exercise price $120 and hedge ratio 2 .4 if the stock price is currently $122?
upside potential is unlimited.
Figure 21.10 graphs the profit or loss on a protective put position as a function of the change in the value of the underlying asset, P.
While the protective put is a simple and convenient way to achieve portfolio insurance, that is, to limit the worst-case portfo- lio rate of return, there are practi- cal difficulties in trying to insure a portfolio of stocks. First, unless the investor’s portfolio corre- sponds to a standard market index for which puts are traded, a put option on the portfolio will not be available for purchase. And if index puts are used to protect a non-indexed portfolio, tracking error can result. For example, if the portfolio falls in value while the market index rises, the put will fail to provide the intended protection. Tracking error limits the investor’s freedom to pursue active stock selection because such error will be greater as the managed portfolio departs more substantially from the market index.
Here is the general idea behind portfolio insurance programs. Even if a put option on the desired portfolio does not exist, a theoretical option-pricing model (such as the Black- Scholes model) can be used to determine how that option’s price would respond to the portfolio’s value if it did trade. For example, if stock prices were to fall, the put option would increase in value. The option model could quantify this relationship. The net expo- sure of the (hypothetical) protective put portfolio to swings in stock prices is the sum of the exposures of the two components of the portfolio, the stock and the put. The net exposure of the portfolio equals the equity exposure less the (offsetting) put option exposure.
We can create “synthetic” protective put positions by holding a quantity of stocks with the same net exposure to market swings as the hypothetical protective put position. The key to this strategy is the option delta, or hedge ratio, that is, the change in the price of the protective put option per change in the value of the underlying stock portfolio.
Change in Value of Protected Position
Change in Value of Underlying Asset
⫺P
0 0
Cost of Put
Figure 21.10 Profit on a protective put strategy
Example 21.5 Synthetic Protective Put Options
Suppose a portfolio is currently valued at $100 million. An at-the-money put option on the portfolio might have a hedge ratio or delta of 2 .6, meaning the option’s value swings $.60 for every dollar change in portfolio value, but in an opposite direction. Suppose the stock portfolio falls in value by 2%. The profit on a hypothetical protective put position (if the put existed) would be as follows (in millions of dollars):
Loss on stocks: 2% of $100 5 $2.00 Gain on put: .6 3 $2.00 5 1.20
Net loss 5 $ .80
The difficulty with this procedure is that deltas constantly change. Figure 21.11 shows that as the stock price falls, the magnitude of the appropriate hedge ratio increases.
Therefore, market declines require extra hedging, that is, additional conversion of equity into cash. This constant updating of the hedge ratio is called dynamic hedging (alterna- tively, delta hedging).
Dynamic hedging is one reason portfolio insurance has been said to contribute to mar- ket volatility. Market declines trigger additional sales of stock as portfolio insurers strive to increase their hedging. These additional sales are seen as reinforcing or exaggerating market downturns.
In practice, portfolio insurers do not actually buy or sell stocks directly when they update their hedge positions. Instead, they minimize trading costs by buying or selling stock index futures as a substitute for sale of the stocks themselves. As you will see in the next chapter, stock prices and index futures prices usually are very tightly linked by cross-market arbitra- geurs so that futures transactions can be used as reliable proxies for stock transac- tions. Instead of selling equities based on the put option’s delta, insurers will sell an equivalent number of futures contracts. 13
Several portfolio insurers suffered great setbacks during the market crash of October 19, 1987, when the market suffered an unprecedented 1-day loss of about 20%. A description of what
13 Notice, however, that the use of index futures reintroduces the problem of tracking error between the portfolio and the market index.
We create the synthetic option position by selling a proportion of shares equal to the put option’s delta (i.e., selling 60% of the shares) and placing the proceeds in risk-free T-bills. The rationale is that the hypothetical put option would have off- set 60% of any change in the stock portfolio’s value, so one must reduce portfolio risk directly by selling 60% of the equity and putting the proceeds into a risk-free asset. Total return on a synthetic protective put position with $60 million in risk-free investments such as T-bills and $40 million in equity is
Loss on stocks: 2% of $40 5 $.80
1 Loss on bills: 5 0
Net loss 5 $.80
The synthetic and actual protective put positions have equal returns. We conclude that if you sell a proportion of shares equal to the put option’s delta and place the proceeds in cash equivalents, your exposure to the stock market will equal that of the desired protective put position.
0
Value of a Put (P)
S0
Low slope 5 Low hedge ratio
Higher slope ⫽ High hedge ratio
Figure 21.11 Hedge ratios change as the stock price fluctuates
h appened then should let you appreciate the complexities of applying a seemingly straight- forward hedging concept.
1. Market volatility at the crash was much greater than ever encountered before. Put option deltas based on historical experience were too low; insurers underhedged, held too much equity, and suffered excessive losses.
2. Prices moved so fast that insurers could not keep up with the necessary rebalanc- ing. They were “chasing deltas” that kept getting away from them. The futures market also saw a “gap” opening, where the opening price was nearly 10% below the previous day’s close. The price dropped before insurers could update their hedge ratios.
3. Execution problems were severe. First, current market prices were unavailable, with trade execution and the price quotation system hours behind, which made computa- tion of correct hedge ratios impossible. Moreover, trading in stocks and stock futures ceased during some periods. The continuous rebalancing capability that is essential for a viable insurance program vanished during the precipitous market collapse.
4. Futures prices traded at steep discounts to their proper levels compared to reported stock prices, thereby making the sale of futures (as a proxy for equity sales) seem expensive. Although you will see in the next chapter that stock index futures prices normally exceed the value of the stock index, Figure 21.12 shows that on October 19, futures sold far below the stock index level. When some insurers gambled that the futures price would recover to its usual premium over the stock index, and chose to defer sales, they remained underhedged. As the market fell farther, their portfolios experienced substantial losses.
Although most observers at the time believed that the portfolio insurance industry would never recover from the market crash, delta hedging is still alive and well on Wall Street. Dynamic hedges are widely used by large firms to hedge potential losses from options positions. For example, the nearby box notes that when Microsoft ended its employee stock option program and J. P. Morgan purchased many already-issued options
0 10
⫺10
⫺20
⫺30
⫺40
10 11 12 1 2 3 4 10 11 12 1 2 3 4
October 19 October 20
Figure 21.12 S&P 500 cash-to-futures spread in points at 15-minute intervals
Note: Trading in futures contracts halted between 12:15 and 1:05.
Source: The Wall Street Journal. Reprinted by permission of The Wall Street Journal, © 1987 Dow Jones
& Company, Inc. All rights reserved worldwide.
Microsoft, in a shift that could be copied throughout the technology business, said yesterday that it plans to stop issuing stock options to its employees, and instead will pro- vide them with restricted stock.
The deal could portend a seismic shift for Microsoft’s Silicon Valley rivals, and it could well have effects on Wall Street. Though details of the plan still aren’t clear, J. P.
Morgan effectively plans to buy the options from Microsoft employees who opt for restricted stock instead. Employee stock options are granted as a form of compensation and allow employees the right to exchange the options for shares of company stock.
The price offered to employees for the options presum- ably will be lower than the current value, giving J. P. Morgan a chance to make a profit on the deal. Rather than hold- ing the options, and thus betting Microsoft’s stock will rise, people familiar with the bank’s strategy say J. P. Morgan probably will match each option it buys from the company’s employees with a separate trade in the stock market that both hedges the bet and gives itself a margin of profit.
For Wall Street’s so-called rocket scientists who do com- plicated financial transactions such as this one, the strategy behind J. P. Morgan’s deal with Microsoft isn’t particularly unique or sophisticated. They add that the bank has sev- eral ways to deal with the millions of Microsoft options that could come its way.
The bank, for instance, could hedge the options by shorting, or betting against, Microsoft stock. Microsoft has the largest market capitalization of any stock in the mar- ket, and its shares are among the most liquid, meaning it would be easy to hedge the risk of holding those options.
J. P. Morgan also could sell the options to investors, much as they would do with a syndicated loan, thereby spreading the risk. During a conference call with investors, Mr. Ballmer said employees could sell their options to “a third party or set of third parties,” adding that the com- pany was still working out the details with J. P. Morgan and the SEC.
Source: The Wall Street Journal, July 9, 2003. © 2003 Dow Jones &
Company, Inc. All rights reserved.
of Microsoft employees, it was widely expected that Morgan would protect its options position by selling shares in Microsoft in accord with a delta hedging strategy.
Hedging Bets on Mispriced Options
Suppose you believe that the standard deviation of IBM stock returns will be 35% over the next few weeks, but IBM put options are selling at a price consistent with a volatility of 33%. Because the put’s implied volatility is less than your forecast of the stock volatility, you believe the option is underpriced. Using your assessment of volatility in an option- pricing model like the Black-Scholes formula, you would estimate that the fair price for the puts exceeds the actual price.
Does this mean that you ought to buy put options? Perhaps it does, but by doing so, you risk losses if IBM stock performs well, even if you are correct about the volatility. You would like to separate your bet on volatility from the “attached” bet inherent in purchasing a put that IBM’s stock price will fall. In other words, you would like to speculate on the option mispricing by purchasing the put option, but hedge the resulting exposure to the performance of IBM stock.
The option delta can be interpreted as a hedge ratio that can be used for this purpose.
The delta was defined as
Delta5Change in value of option Change in value of stock Therefore, delta is the slope of the option-pricing curve.
This ratio tells us precisely how many shares of stock we must hold to offset our expo- sure to IBM. For example, if the delta is 2 .6, then the put will fall by $.60 in value for every one-point increase in IBM stock, and we need to hold .6 share of stock to hedge each put. If we purchase 10 option contracts, each for 100 shares, we would need to buy 600 shares of stock. If the stock price rises by $1, each put option will decrease in value by
$.60, resulting in a loss of $600. However, the loss on the puts will be offset by a gain on the stock holdings of $1 per share 3 600 shares.