N (de = 66.44
Thus a 1% (0.01) increase in volatility (from 25% to 26%) increases the value of the option byapproximately 0.6644 (= 0.01 x 66.44).
Vega
Stock priceK
Figure 14.11 Variation of vega with stock price for an option318 CHAPTER 14
Calculating vega from the Black-Scholes model and its extensions may seem strange because oneof the assumptions underlying Black-Scholes is that volatility is constant. It would be theoreticallymore correct to calculate vega from a model in which volatility is assumed to be stochastic.However, it turns out that the vega calculated from a stochastic volatility model is very similar tothe Black-Scholes vega so the practice of calculating vega from a model in which volatility isconstant works reasonably well.!!
Gamma neutrality protects against large changes in the price of the underlying asset betweenhedge rebalancing. Vega neutrality protects for a variable o. As might be expected, whether it isbest to use an available traded option for vega or gamma hedging depends on the time betweenhedge rebalancing and the volatility of the volatility.”
14.9 RHO
 
 
The rho of a portfolio of options is the rate of change of the value of the portfolio with respect tothe interest rate:
rho = —or
It measures the sensitivity of the value of a portfolio to interest rates. For a European call optionon a non-dividend-paying stock,rho = KTe~" N(d))
where d, is defined as in equation (12.20). For a European put option,tho = —KTe~’ N(—d,)
These same formulas apply to European call and put options on stocks and stock indices payingknown dividend yields when d, is as in equation (13.4).
Example14.8 Consider a four-month put option on a stock index. The current value of the indexis 305, the strike price is 300, the dividend yield is 3% per annum, the risk-free interest rate is 8% perannum, and the volatility of the index is 25% per annum. In this case 5) = 305, K = 300, g = 0.03,r= 0.08, o = 0.25, and T = 4/12. The option’s rho is
—KTe—"" N(—dy) = —42.6
This means that for a 1% (0.01) change in the risk-free interest rate (from 8% to 9%) the value of theoption decreases by 0.426 (= 0.01 x 42.6).
In the case of currency options, there are two rhos corresponding to the two interest rates. The rhocorresponding to the domestic interest rate is given by the formulas already presented, with d, as in
"! See J. C. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance,42 (June 1987), 281-300; J. C. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a StochasticVolatility,” Advances in Futures and Options Research, 3 (1988), 27-61.
"2 For a discussion of this issue, see J. C. Hull and A. White, “Hedging the Risks from Writing Foreign CurrencyOptions,” Journal of International Money and Finance, 6 (June 1987), 131-52.The Greek Letters 319
equation (13.9). The rho corresponding to the foreign interest rate for a European call on acurrency isrho = —Te~7? SyN(d,)and for a European put it isrho = Te“ SyN(—d;)where d, is given by equation (13.9).For a European call futures option rho is —cT and for a European put futures option rho is — pT,where c and p are the European call and put futures option prices, respectively.
14.10 HEDGING IN PRACTICE
 
 
In an ideal world, traders working for financial institutions would be able to rebalance theirportfolios very frequently in order to maintain a zero delta, a zero gamma, a zero vega, and so on.In practice, this is not possible. When managing a large portfolio dependent on a single underlying- asset, traders usually zero out delta at least once a day by trading the underlying asset. Unfortu-nately, a zero gamma and a zero vega are less easy to achieve because it is difficult to find optionsor other nonlinear derivatives that can be traded in the volume required at competitive prices. Inmost cases, gamma and vega are monitored. When they get too large in a positive or negativedirection, either corrective action is taken or trading is curtailed.
There are big economies of scale in being an options trader. As noted earlier, maintaining deltaneutrality for an individual option on, say, the S&P 500 by trading daily would be prohibitivelyexpensive. But it is realistic to do this for a portfolio of several hundred options on the S&P 500.This is because the cost of daily rebalancing (either by trading the stocks underlying the index orby trading index futures) is covered by the profit on many different trades.
In many markets financial institutions find that the majority of their trades are sales of call andput options to their clients. Short calls and short puts have negative gammas and negative vegas. Itfollows that, as time goes by, both the gamma and vega of a financial institution’s portfolio tend tobecome progressively more negative. Traders working for the financial institution are then alwayslooking for ways to buy options (i.e., acquire positive gamma and vega) at competitive prices.There is one aspect of an options portfolio that mitigates this problem somewhat. Options areoften close to the money when they are first sold, so that they have relatively high gammas andvegas. But after some time has elapsed, they tend to become deep out of the money or deep in themoney. In such cases their gammas and vegas become very small and of little consequence. The" worst situation for an options trader is that written options remain very close to the money right
up until maturity. :
14.11 SCENARIO ANALYSIS
 
 
In addition to monitoring risks such as delta, gamma, and vega, option traders often also carry outa scenario analysis. The analysis involves calculating the gain or loss on their portfolio over aspecified period under a variety of different scenarios. The time period chosen is likely to dependon the liquidity of the instruments. The scenarios can be either chosen by management orgenerated by a model.
Consider a bank with a portfolio of options on a foreign currency. There are two main variables320 CHAPTER 14Table 14.5 Profit or loss realized in two weeks under different scenarios(millions of dollars)
Exchange rate
Volatility 0.94 0.96 0.98 1.00 1.02 1.04 1.06
 
 
8% +102 +55 +25 +6 -10 —34 —8010% +80 +40 +17 +2 —14 —38 —8512% +60 +25 +9 -2 —18 -42 —90
on which the value of the portfolio depends. These are the exchange rate and the exchange ratevolatility. Suppose that the exchange rate is currently 1.0000 and its volatility is 10% per annum.The bank could calculate a table such as Table 14.5 showing the profit or loss experienced during atwo-week period under different scenarios. This table considers seven different exchange rates andthree different volatilities. Because a one-standard-deviation move in the exchange rate during atwo-week period is about 0.02, the exchange rate moves considered are approximately one, two,and three standard deviations.
In Table 14.5 the greatest loss is in the lower right corner of the table. The loss corresponds tothe volatility increasing to 12% and the exchange rate moving up to 1.06. Usually the greatest lossin a table such as this occurs at one of the corners, but this is not always so. Consider, for example,the situation where a bank’s portfolio consists of a reverse butterfly spread (see Section 9.2). The
greatest loos wi alt ba aesorianacd if the exchange rate gta ere 7groatest loss will be exporionced if the exchange rate stays where it is.
14.12 PORTFOLIO INSURANCE
 
A portfolio manager is often interested in acquiring a put option on his or her portfolio. Thisprovides protection against market declines while preserving the potential for a gain if the marketdoes well. One approach (discussed in Chapter 13) is to buy put options on a market index such asthe S&P 500. An alternative is to create the option synthetically.
Creating an option synthetically involves maintaining a position in the underlying asset (orfutures on the underlying asset) so that the delta of the position is equal to the delta of the requiredoption. The position necessary to create an option synthetically is the reverse of that necessary tohedge it. This is because the procedure for hedging an option involves the creation of an equal andopposite option synthetically.
There are two reasons why it may be more attractive for the portfolio manager to create therequired put option synthetically than to buy it in the market. The first is that options markets donot always have the liquidity to absorb the trades that managers of large funds would like to carryout. The second is that fund managers often require strike prices and exercise dates that aredifferent from those available in exchange-traded options markets.
The synthetic option can be created from trading the portfolio or from trading in index futurescontracts. We first examine the creation of a put option by trading the portfolio. Recall that thedelta of a European put on the portfolio is
A=e[N(d)— 1] (14.8)The Greek Letters 321
where, with our usual notation,_ In(So/K) + (¢—q +07 /2)ToJ/T
So is the value of the portfolio, K is the strike price, r is the risk-free rate, ¢ is the dividend yield onthe portfolio, o is the volatility of the portfolio, and T is the life of the option.
To create the put option synthetically, the fund manager should ensure that at any given time aproportion
dq
e Tl — N(A)I
of the stocks in the original portfolio has been sold and the proceeds invested in riskless assets. Asthe value of the original portfolio declines, the delta of the put given by equation (14.8) becomesmore negative and the proportion of the original portfolio sold must be increased. As the value of theoriginal portfolio increases, the delta of the put becomes less negative and the proportion of theoriginal portfolio sold must be decreased (i.e., some of the original portfolio must be repurchased).Using this strategy to create portfolio insurance means that at any given time funds are divided” between the stock portfolio on which insurance is required and riskless assets. As the value of thestock portfolio increases, riskless assets are sold and the position in the stock portfolio is increased.As the value of the stock portfolio declines, the position in the stock portfolio is decreased andniskless assets are purchased. The cost of the insurance arises from the fact that the portfoliomanager is always selling after a decline in the market and buying after a rise in the market.
Example 14.9 A portfolio is worth $90 million. To protect against market downturns, themanagers of the portfolio require a six-month European put option on the portfolio with a strikeprice of $87 million. The risk-free rate is 9% per annum, the dividend yield is 3% per annum, and thevolatility of the portfolio is 25% per annum. The S&P 500 index stands at 900. As the portfolio isconsidered to mimic the S&P 500 fairly closely, one alternative is to buy 1,000 put option contractson the S&P 500 with a strike price of 870. Another alternative is to create the option synthetically. Inthis case, Sy = 90 million, K = 87 million, r = 0.09, g = 0.03, o = 0.25, and T = 0.5, so that
1n(90/87) + (0.09 — 0.03 + 0.257/2)0.50.25./0.5
and the delta of the required option is initially
= 0.4499
 
e~"[N(d,) — 1] = —0.3215
This shows that 32.15% of the portfolio should be sold initially to match the delta of the requiredoption. The amount of the portfolio sold must be monitored frequently. For example, if the value ofthe portfolio reduces to $88 million after one day, the delta of the required option changes to —0.3679and a further 4.64% of the original portfolio should be sold. If the value of the portfolio increases to$92 million, the delta of the required option changes to —0.2787 and 4.28% of the original portfolioshould be repurchased.
Use of Index Futures
Using index futures to create options synthetically can be preferable to using the underlying stocksbecause the transactions costs associated with trades in index futures are generally lower than thoseassociated with the corresponding trades in the underlying stocks. The dollar amount of thefutures contracts shorted as a proportion of the value of the portfolio should from equations (14.3)322 CHAPTER 14
and (14.8) beee OMT Ny =e" Me™ [1 — MA)
where 7* is the maturity time of the futures contract. If the portfolio is worth K, times the indexand each index futures contract is on K> times the index, the number of futures contracts shorted atany given time should be
«op —eT* K,fO MeN — Ma2
Example 14.10 Suppose that in the previous example futures contracts on the S&P 500 maturingin nine months are used to create the option synthetically. In this case, initially T = 0.5, T* = 0.75,K, = 100, 000, Kz = 250, and d, = 0.4499, so that the number of futures contracts shorted should be
. + KDe TT — Nd Ie = 123.22
or 123 rounding to the nearest whole number. As time passes and index changes, the position infutures contracts must be adjusted.
Up to now we have assumed that the portfolio mirrors the index. As discussed in Section 13.3, thehedging scheme can be adjusted to deal with other situations. The strike price for the options usedshould be the expected level of the market index when the portfolio’s value reaches its insuredvalue. The number of index options used should be f times the number of options that would berequired if the portfolio had a beta of 1.0. The volatility of portfolio can be assumed to be its betatimes the volatility of an appropriate well-diversified index.
Creating put options on the index synthetically does not work well if the volatility of the indexchanges rapidly or if the index exhibits large jumps. On Monday, October 19, 1987, the Dow JonesIndustrial Average dropped by more than 20%. Portfolio managers who had insured themselves bybuying put options in the exchange-traded or over-the-cou
Those who had chosen to create put options synthetically found that they were unable to sell eitherstocks or index futures fast enough to protect their position.
  
er market survived this crash well.
Brady Commission Report
The report of the Brady Commission on the October 19, 1987, crash provides interesting insightsinto the effect of portfolio insurance on the market at that time.!? The Brady Commissionestimated that $60 billion to $90 billion of equity assets were under portfolio insurance administra-tion in October 1987. During the period Wednesday, October 14, 1987, to Friday, October 16,1987, the market declined by about 10%, with much of this decline taking place on Fridayafternoon. The decline should have generated at least $12 billion of equity or index futures salesas a result of portfolio insurance schemes.'* In fact, less than $4 billion were sold, which meansthat portfolio insurers approached the following week with huge amounts of selling already
'3 See “Report of the Presidential Task Force on Market Mechanisms,” January 1988.
'4 To put this in perspective, on Monday, October 19, 1987, all previous records were broken when 604 millionshares worth $21 billion were traded on the New York Stock Exchange. Approximately $20 billion of S&P 500futures contracts were traded on that day.The Greek Letters 323
dictated by their models. The Brady Commission estimated that on Monday, October 19 sellprograms by three portfolio insurers accounted for almost 10% of the sales on the New YorkStock Exchange, and that portfolio insurance sales amounted to 21.3% of all sales in index futuresmarkets. It seems likely that portfolio insurance caused some downward pressure on the market.
Because the market declined so fast and the stock exchange systems were overloaded, manyportfolio insurers were unable to execute the trades generated by their models and failed to obtainthe protection they required. Needless to say, the popularity of portfolio insurance schemes basedon dynamic trading in stocks and futures has declined considerably since October 1987.
1413. STOCK MARKET VOLATILITY
 
We have already considered the issue of whether volatility is caused solely by the arrival of newinformation or whether trading itself generates volatility. Portfolio insurance schemes such as those
just described have the potential to increase volatility. When the market declines, they cause' portfolio managers either to sell stock or to sell index futures contracts. Either action mayaccentuate the decline. The sale of stock is liable to drive down the market index further in adirect way. The sale of index futures contracts is liable to drive down futures prices. This createsselling pressure on stocks via the mechanism of index arbitrage (see Chapter 3), so that the marketindex is liable to be driven down in this case as well. Similarly, when the market rises, the portfolioinsurance schemes cause portfolio managers either to buy stock or to buy futures contracts. Thismay accentuate the rise.
In addition to formal portfolio insurance schemes, we can speculate that many investorsconsciously or subconsciously follow portfolio insurance schemes of their own. For example, aninvestor may be inclined to enter the market when it is rising but will sell when it is falling to limitthe downside risk.
Whether portfolio insurance schemes (formal or informal) affect volatility depends on howeasily the market can absorb the trades that are generated by portfolio insurance. If portfolioinsurance trades are a very small fraction of all trades, there is likely to be no effect. As portfolioinsurance becomes more popular, it is liable to have a destabilizing effect on the market.
SUMMARY
 
 
Financial institutions offer a variety of option products to their clients. Often the options do notcorrespond to the standardized products traded by exchanges. The financial institutions are thenfaced with the problem of hedging their exposure. Naked and covered positions leave them subjectto an unacceptable level of risk. One course of action that is sometimes proposed is a stop-lossstrategy. This involves holding a naked position when an option is out of the money andconverting it to a covered position as soon as the option moves into the money. Althoughsuperficially attractive, the strategy does not work at all well.
The delta of an option, A, is the rate of change of its price with respect to the price of theunderlying asset. Delta hedging involves creating a position with zero delta (sometimes referredto as a delta-neutral position). Because the delta of the underlying asset is 1.0, one way ofhedging is to take a position of —A in the underlying asset for each long option being hedged.324 CHAPTER 14
The delta of an option changes over time. This means that the position in the underlying assethas to be frequently adjusted.
Once an option position has been made delta neutral, the next stage is often to look at itsgamma. The gamma of an option is the rate of change of its delta with respect to the price of theunderlying asset. It is a measure of the curvature of the relationship between the option price andthe asset price. The impact of this curvature on the performance of delta hedging can be reducedby making an option position gamma neutral. If T is the gamma of the position being hedged, thisreduction is usually achieved by taking a position in a traded option that has a gamma of —T.
Delta and gamma hedging are both based on the assumption that the volatility of the underlyingasset is constant. In practice, volatilities do change over time. The vega of an option or an optionportfolio measures the rate of change of its value with respect to volatility. A trader who wishes tohedge an option position against volatility changes can make the position vega neutral. As with theprocedure for creating gamma neutrality, this usually involves taking an offsetting position in atraded option. If the trader wishes to achieve both gamma and vega neutrality, two traded optionsare usually required.
Two other measures of the risk of an option position are theta and rho. Theta measures the rateof change of the value of the position with respect to the passage of time, with all else remainingconstant. Rho measures the rate of change of the value of the position with respect to the short-term interest rate, with all else remaining constant.
In practice, option traders usually rebalance their portfolios at least once a day to maintain deltaneutrality. It is usually not feasible to maintain gamma and vega neutrality on a regular basis.Typically a trader monitors these measures. If they get too large, either corrective action is taken ortrading is curtailed.
Portfolio managers are sometimes interested in creating put options synthetically for thepurposes of insuring an equity portfolio. They can do so either by trading the portfolio or bytrading index futures on the portfolio. Trading the portfolio involves splitting the portfoliobetween equities and risk-free securities. As the market declines, more is invested in risk-freesecurities. As the market increases, more is invested in equities. Trading index futures involveskeeping the equity portfolio intact and selling index futures. As the market declines, more indexfutures are sold; as it rises, fewer are sold. The strategy works well in normal market conditions. OnMonday, October 19, 1987, when the Dow Jones Industrial Average dropped by more than500 points, it worked badly. Portfolio insurers were unable to sell either stocks or index futuresfast enough to protect their positions. As a result the popularity of such schemes has declinedsharply.
SUGGESTIONS FOR FURTHER READING
On hedging option positions
Boyle, P. P., and D. Emanuel, “‘Discretely Adjusted Option Hedges,” Journal of Financial Economics,8 (1980), 259-82.
Figlewski, S., “Options Arbitrage in Imperfect Markets,” Journal of Finance, 44 (December 1989),1289-1311.
Galai, D., “The Components of the Return from Hedging Options against Stocks,” Journal of Business,56 (January 1983), 45-54.The Greek Letters 325
Hull, J. C., and A. White, ‘Hedging the Risks from Writing Foreign Currency Options,” Journal ofInternational Money and Finance, 6 (June 1987), 131-52.
On portfolio insurance
Asay, M., and C. Edelberg, “Can a Dynamic Strategy Replicate the Returns on an Option?” Journal ofFutures Markets, 6 (Spring 1986), 63-70.
Bookstaber, R., and J. A. Langsam, ‘Portfolio Insurance Trading Rules,” Journal of Futures Markets,8 (February 1988), 15-31.
Etzioni, E. S., ““Rebalance Disciplines for Portfolio Insurance,” Journal of Portfolio Insurance, 13 (Fall1986), 59-62.
Leland, H. E., “Option Pricing and Replication with Transactions Costs,” Journal of Finance,40 (December 1985), 1283-1301.
Leland, H. E., ““Who Should Buy Portfolio Insurance?”’ Journal of Finance, 35 (May 1980), 581-94.
Rubinstein, M., “Alternative Paths for Portfolio Insurance,” Financial Analysts Journal, 41 (July/August1985), 42-52.
Rubinstein, M., and II. E. Leland, “Replicating Options with Positions in Stock and Cash,” [nancialAnalysts Journal, 37 (July/August 1981), 63-72.
Tilley, J. A., and G. O. Latainer, “A Synthetic Option Framework for Asset Allocation,” FinancialAnalysts Journal, 41 (May/June 1985), 32-41.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
 
14.1. Explain how a stop-loss hedging scheme can be implemented for the writer of an out-of-the-money call option. Why does it provide a relatively poor hedge?
14.2. What does it mean to assert that the delta of a call option is 0.7? How can a short position in1,000 options be made delta neutral when the delta of each option is 0.7?
14.3. Calculate ihe delia of an ai-ihe-money six-monih European cali opiion on a non-dividend-paying stock when the risk-free interest rate is 10% per annum and the stock price volatility is25% per annum.
14.4. What does it mean to assert that the theta of an option position is —0.1 when time is measured inyears? If a trader feels that neither a stock price nor its implied volatility will change, what typeof option position is appropriate?
14.5. What is meant by the gamma of an option position? What are the risks in the situation where thegamma of a position is large and negative and the delta 1s zero?
14.6. “The procedure for creating an option position synthetically is the reverse of the procedure forhedging the option position.” Explain this statement.
14.7. Why did portfolio insurance not work well on October 19, 1987?
14.8, The Black-Scholes price of an out-of-the-money call option with an exercise price of $40 is $4.A trader who has written the option plans to use a stop-loss strategy. The trader’s plan is to buy
at $40.10 and to sell at $39.90. Estimate the expected number of times the stock will be boughtor sold.326
14.9.
14.10.
14.11,
14.12,
14.13.
14.14,
14.15.
14.16.
14.17.
14.18.
CHAPTER 14
Suppose that a stock price is currently $20 and that a call option with an exercise price of $25 iscreated synthetically using a continually changing position in the stock. Consider the followingtwo scenarios:
a. Stock price increases steadily from $20 to $35 during the life of the option
b. Stock price oscillates wildly, ending up at $35Which scenario would make the synthetically created option more expensive? Explain youranswer.
What is the delta of a short position in 1,000 European call options on silver futures? Theoptions mature in eight months, and the futures contract underlying the option matures in ninemonths. The current nine-month futures price is $8 per ounce, the exercise price of the options is$8, the risk-free interest rate is 12% per annum, and the volatility of silver is 18% per annum.
In Problem 14.10, what initial position in nine-month silver futures is necessary for deltahedging? If silver itself is used, what is the initial position? If one-year silver futures are used,what is the initial position? Assume no storage costs for silver.
A company uses delta hedging to hedge a portfolio of long positions in put and call options onacurrency. Which of the following would give the most favorable result?
a. A virtually constant spot rate
b. Wild movements in the spot rateExplain your answer.
Repeat Problem 14.12 for a financial institution with a portfolio of short positions in put andcall options for a currency.
A financial institution has just sold 1,000 seven-month European call options on the Japaneseyen. Suppose that the spot exchange rate is 0.80 cents per yen, the exercise price is 0.81 cents peryen, the risk-free interest rate in the United States is 8% per annum, the risk-free interest rate inJapan is 5% per annum, and the volatility of the yen is 15% per annum. Calculate the delta,gamma, vega, theta, and rho of the financial institution’s position. Interpret each number.
Under what circumstances is it possible to make a European option on a stock index bothgamma neutral and vega neutral by adding a position in one other European option?
A fund manager has a well-diversified portfolio that mirrors the performance of the S&P 500 andis worth $360 million. The value of the S&P 500 is 1,200, and the portfolio manager would liketo buy insurance against a reduction of more than 5% in the value of the portfolio over the nextsix months. The risk-free interest rate is 6% per annum. The dividend yield on both the portfolioand the S&P 500 is 3%, and the volatility of the index is 30% per annum.a. If the fund manager buys traded European put options, how much would the insurance cost?b. Explain carefully alternative strategics open to the fund manager involving traded Europeancall options, and show that they lead to the same result.c. If the fund manager decides to provide insurance by keeping part of the portfolio in risk-freesecurities, what should the initial position be?d. If the fund manager decides to provide insurance by using nine-month index futures, whatshould the initial position be?
Repeat Problem 14.16 on the assumption that the portfolio has a beta of 1.5. Assume that thedividend yield on the portfolio is 4% per annum.
Show by substituting for the various terms in equation (14.7) that the equation is true for:a. A single European call option on a non-dividend-paying stockb. A single European put option on a non-dividend-paying stockc. Any portfolio of European put and call options on a non-dividend-paying stockThe Greek Letters 327
 
14.19.
14.20.
14.21.
14.22.
14.23.
What is the equation corresponding to equation (14.7) for (a) a portfolio of derivatives on acurrency and (b) a portfolio of derivatives on a futures contract?
Suppose that $70 billion of equity assets are the subject of portfolio insurance schemes. Assumethat the schemes are designed to provide insurance against the value of the assets declining bymore than 5% within one year. Making whatever estimates you find necessary, use theDerivaGem software to calculate the value