of the stock or futures contracts that theadministrators of the portfolio insurance schemes will attempt to sell if the market falls by23% in a single day.
Does a forward contract on a stock index have the same delta as the corresponding futurescontract? Explain your answer.
A bank’s position in options on the dollar-euro exchange rate has a delta of 30,000 and agamma of —80,000. Explain how these numbers can be interpreted. The exchange rate (dollarsper euro) is 0.90. What position would you take to make the position delta neutral? After a shortperiod of time, the exchange rate moves to 0.93. Estimate the new delta. What additional trade isnecessary to keep the position delta neutral? Assuming the bank did set up a delta-neutralposition originally, has it gained or lost money from the exchange rate movement?
Use the put-call parity relationship to derive, for a non-dividend-paying stock, the relationshipbetween:
a. The delta of a European call and the delta of a European put
b. The gamma of a European call and the gamma of a European put
c. The vega of a European call and the vega of a European put
d. The theta of a European call and the theta of a European put
 
14.24.
14.25.
Consider a one-year European call option on a stock when the stock price is $30, the strike priceis $30, the risk-free rate is 5%, and the volatility is 25% per annum. Use the DerivaGemsoftware to calculate the price, delta, gamma, vega, theta, and rho of the option. Verify thatdelta is correct by changing the stock price to $30.1 and recomputing the option price. Verifythat gamma is correct by recomputing the delta for the situation where the stock price is $30.1.Carry oul similar calculations to verify that vega, theia, and rho are correct, Use the DerivaGemApplications Builder functions to plot the option price, delta, gamma, vega, theta, and rhoagainst the stock price for the stock option.
A financial institution has the following portfolio of over-the-counter options on sterling:
Type Position Delta of option Gamma of option Vega of option
 
Call —1,000 0.50 2.2 1.8Call —500 0.80 0.6 0.2Put —2,000 —0.40 1.3 0.7Call —500 0.70 18 1.4
A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.a. What position in the traded option and in sterling would make the portfolio both gammaneutral and delta neutral?b. What position in the traded option and in sterling would make the portfolio both veganeutral and delta neutral?328
CHAPTER 14
 
14.26.
14.27.
14.28,
14.29.
14,30.
Consider again the situation in Problem 14.24. Suppose that a second traded option with a deltaof 0.1, a gamma of 0.5, and a vega of 0.6 is available. How could the portfolio be made delta,gamma, and vega neutral?
A deposit instrument offered by a bank guarantees that investors will receive a return during asix-month period that is the greater of (a) zero and (b) 40% of the return provided by a marketindex. An investor is planning to put $100,000 in the instrument. Describe the payoff as anoption on the index. Assuming that the risk-free rate of interest is 8% per annum, the dividendyield on the index is 3% per annum, and the volatility of the index is 25% per annum, is theproduct a good deal for the investor?
The formula for the price of a European call futures option in terms of the futures price, Fy, isfrom Chapter 13e=e [FyM(d\) — KN(@2)]whereIn(Fo/K) +.0°T/2oVT ‘and K,r, T, and o are the strike price, interest rate, time to maturity, and volatility, respectively.
a. Prove that FyN’(d;) = KN (do).
b. Prove that the delta of the call price with respect to the futures price is e"? N(d)).
c. Prove that the vega of the call price is Fy. /TN (de.
d. Prove the formula for the rho of a call futures option given at the end of Section 14.9. Thedelta, gamma, theta, and vega of a call futures option are the same as those for a calloption on a stock paying dividends at rate g, with g replaced by r and Sp replaced by /.Explain why the same is not true of the rho of a call futures option.
d= d, =d, —oVT
Use DerivaGem to check that equation (14.7) is satisfied for the option considered in Se ction 14.1,
(Note: DerivaGem produces a value of theta “ner calendar day”. The theta in equation (4, Tis
ote: Derivafsem produces a value of theta “per calendar da! ine theta fps
“per year’’.)
Use the DerivaGem Application Builder functions to reproduce Table 14.2. (Note that inTable 14.2 the stock position is rounded to the nearest 100 shares.) Calculate the gamma andtheta of the position each week. Calculate the change in the value of the portfolio each week and
check whether equation (4. 0) Is approximately satisfied. Uvole: DerivaGent produces a value oftheta “per calendar day”. The theta in equation (14.6) is “per year”’.)The Greek Letters 329
APPENDIX 14A
Taylor Series Expansions and Hedge Parameters
A Taylor series expansion of the change in the portfolio value in a short period of time shows therole played by different Greek letters. If the volatility of the underlying asset is assumed to beconstant, the value of the portfolio, II, is a function of the asset price S and time t. The Taylorseries expansion gives
an rn
8S? + 4 57° 5S St +++ 14A.1ase Fae Fasae tT (144.1)
an an anoll = — 58 otha5 Ot a te
 
where dIT and 4S are the change in II and S in a small time interval St. Delta hedging eliminates thefirst term on the right-hand side. The second term is nonstochastic. The third term (which is oforder St) can be made zero by ensuring that the portfolio is gamma neutral as well as delta neutral.Other terms are of higher order than 6.
” For a delta-neutral portfolio, the first term on the right-hand side of equation (14A.1) is zero, sothat
when terms of higher order than é¢ are ignored. This is equation (14.6).When the volatility of the underlying asset is uncertain, TI is a function of o, S, and tf.Equation (144.1) then becomes
an, aM, ONBT ag 8 ay be tay 2 a g2
am2 25S +353 50° +--+
 
where So is the change in o in time é¢. In this case, delta hedging eliminates the first term on theright-hand side. The second term is eliminated by making the portfolio vega neutral. The thirdterm is nonstochastic. The fourth term is eliminated by making the portfolio gamma ncutral.Other Greek letters can be (and in practice are) defined to correspond to higher-order terms.CHAPTER 15
VOLATILITY SMILES
How close are the market prices of options to those predicted by Black-Scholes? Do traders reallyuse Black-Scholes when determining a price for an option? Are the probability distributions ofasset prices really lognormal? What research has been carried out to test the validity of theBlack-Scholes formulas? In this chapter we answer these questions. We explain that traders douse the Black-Scholes model—but not in exactly the way that Black and Scholes originallyintended. This is because they allow the volatility used to price an option to depend on its strikeprice and time to maturity.
A plot of the implied volatility of an option as a function of its strike price is known as avolatility smile. In this chapter we describe the volatility smiles that traders use in equity andforeign currency markets. We explain the relationship between a volatility smile and the risk-neutral probability distribution being assumed for the future asset price. We also discuss howoption traders allow volatility to be a function of option maturity and how they use volatilitymatrices as pricing tools. The final part of the chapter summarizes some of the work researchershave carried out to test Black-Scholes.
15.1 PUT-CALL PARITY REVISITED
 
 
Put-call parity, which we explained in Chapter 8, provides a good starting point for understandingvolatility smiles. It is an important relationship between the price, c, of a European call and theprice, p, of a European put:
pte" =c+Ke™ (15.1)
The call and the put have the same strike price, K, and time to maturity, T. The variable Sy is theprice of the underlying asset today, r is the risk-free interest rate for maturity 7, and q is the yieldon the asset. .
A key feature of the put-call parity relationship is that it is based on a relatively simpleno-arbitrage argument. It does not require any assumption about the future probability distribu-tion of the asset price. It is true both when the asset price distribution is lognormal and when it isnot lognormal.
Suppose that, for a particular value of the volatility, ppg and cps are the values of European putand call options calculated using the Black-Scholes model. Suppose further that pmx, and cme atethe market values of these options. Because put—call parity holds for the Black-Scholes model, we
330Volatility Smiles 331
 
must havePus + Soe” = cps + Ke?
Because it also holds for the market prices, we havePmkt + Spe” = Cmkt + KeSubtracting these two equations givesPBS — Pmkt = €BS — Cmkt (45.2)
This shows that the dollar pricing error when the Black-Scholes model is used to price a Europeanput option should be exactly the same as the dollar pricing error when it is used to price aEuropean call option with the same strike price and time to maturity.
Suppose that the implied volatility of the put option is 22%. This means that pgs = Pm, when avolatility of 22% is used in the Black-Scholes model. From equation (15.2), it follows thatps = Cwkt When this volatility is used. The implied volatility of the call is therefore also 22%.This argument shows that the implied volatility of a European call option is always the same as theimplied volatility of a European put option when the two have the same strike price and maturitydate. To put this another way, for a given strike price and maturity, the correct volatility to use inconjunction with the Black-Scholes model to price a European call should always be the same asthat used to price a European put.
This is also approximately true for American options. It follows that when traders refer to thetelationship between implied volatility and strike price, or to the relationship between impliedvolatility and maturity, they do not need to state whether they are talking about calls or puts. Therelationship is the same for both.
Example 15.1 The value of the Australian dollar is $0.60. The risk-free interest rate is 5% perannum in the United States and 10% per annum in Australia. The market price of a European calloption on the Australian dollar with a maturity of one year and a strike price of $0.59 is 0.0236.DerivaGem shows that the implied volatility of the call is 14.5%. For there to be no arbitrage, theput—call parity relationship in equation (15.1) must apply with g equal to the foreign risk-free rate.The price, p, of a European put option with a strike price of $0.59 and maturity of one year thereforesatisfiesp +0.60e7°'*! = 0.0236 + 0.59e0 >!
so that p = 0.0419. DerivaGem shows that, when the put has this price, its implied volatility is also14.5%. This is what we expect from the analysis just given.
15.2 FOREIGN CURRENCY OPTIONS
 
 
 
The volatility smile used by traders to price foreign currency options has the general form shown inFigure 15.1. The volatility is relatively low tor at-the-money options. lt becomes progressivelyhigher as an option moves either in the money or out of the money.
In Appendix 15A we show how to determine the risk-neutral probability distribution for anasset price at a future time from the volatility smile given by options maturing at that time. Werefer to this as the implied distribution. The volatility smile in Figure 15.1 corresponds to theprobability distribution shown by the solid line in Figure 15.2. A lognormal distribution with the332 CHAPTER 15
 
Impliedvolatility
NL
Strike price
 
Figure 15.1 Volatility smile for foreign currency options
same mean and standard deviation as the implied distribution is shown by the dashed line inFigure 15.2. It can be seen that the implied distribution has heavier tails than the lognormaldistribution.!
To see that Figure 15.1 and 15.2 are consistent with each other, consider first a deep-out-of-the-money call option with a high strike price of Ky. This option pays off only if the exchange rateproves to be above K. Figure 15.2 shows that the probability of this is higher for the impliedprobability distribution than for the lognormal distribution. We therefore expect the implieddistribution to give a relatively high price for the option. A relatively high price leads to a relatively
    
\\\\\\
<— Lognormal\
Figure 15.2 Implied distribution and lognormal distribution for foreign currency options
' This is known as kurtosis. Note that, in addition to having a heavier tail, the implied distribution is more“peaked”. Both small and large movements in the exchange rate are more likely than with the lognormaldistribution. Intermediate movements are less likely.Volatility Smiles 333
high implied volatility-—and this is exactly what we observe in Figure 15.1 for the option. The twofigures are therefore consistent with each other for high strike prices. Consider next a deep-out-of-the-money put option with a low strike price of K,. This option pays off only if the exchange rateproves to be below K,. Figure 15.2 shows that the probability of this is also higher for impliedprobability distribution than for the lognormal distribution. We therefore expect the implieddistribution to give a relatively high price, and a relatively high implied volatility, for this optionas well. Again, this is exactly what we observe in Figure 15.1.
Reason for the Smile in Foreign Currency Options
We have just shown that the smile used by traders for foreign currency options implies that theyconsider that the lognormal distribution understates the probability of extreme movements inexchange rates. To test whether they are right, we examined the daily movements in 12 differentexchange rates over a 10-year period. As a first step we calculated the standard deviation of dailypercentage change in each exchange rate. We then noted how often the actual percentage changeexceeded one standard deviation, two standard deviations, and so on. Finally, we calculated howoften this would have happened if the percentage changes had been normally distributed. (Thelognormal model implies that percentage changes are almost exactly normally distributed over aone-day time period.) The results are shown in Table 15.1.?
Daily changes exceed three standard deviations on 1.34% of days. The lognormal model predictsthat this should happen on only 0.27% of days. Daily changes exceed four, five, and six standarddeviations on 0.29%, 0.08%, and 0.03% of days, respectively. The lognormal model predicts thatwe should hardly ever observe this happening. The table therefore provides evidence to support theexistence of heavy tails and the volatility smile used by traders.
Why are exchange rates not lognormally distributed? Two of the conditions for an asset price tohave a lognormal distribution are:
1. The volatility of the asset is constant.2. The price of the asset changes smoothly with no jumps.
In practice, neither of these conditions is satisfied for an exchange rate. The volatility of anTable 15.1 Percent of days when daily exchange rate
Moves are greater than one, two,..., six standarddeviations (S.D. = standard deviation of daily change)
Real Lognormal
world model>18.D. 25.04 31.73>28.D. 5.27 4.55>38.D. 1.34 0.27>4S.D. 0.29 0.01>58.D. 0.08 0.00
>68.D. 0.03 0.00
? This table is taken from J. C. Hull and A. White, “Value at Risk When Daily Changes in Market Variables AreNot Normally Distributed,” Journal of Derivatives, 5, no. 3 (Spring 1998), 9-19.334 CHAPTER 15
exchange rate is far from constant, and exchange rates frequently exhibit jumps. It turns out thatthe effect of both a nonconstant volatility and jumps is that extreme outcomes become morelikely.
The impact of jumps and nonconstant volatility depends on the option maturity. Thepercentage impact of a nonconstant volatility on prices becomes more pronounced as the maturityof the option is increased, but the volatility smile created by the nonconstant volatility usuallybecomes less pronounced, The percentage impact of jumps on both prices and the volatility smilebecomes less pronounced as the maturity of the option is increased. When we look at sufficientlylong dated options, jumps tend to get “averaged out”, so that the stock price distribution whenthere are jumps is almost indistinguishable from the one obtained when the stock price changessmoothly.
15.3 EQUITY OPTIONS
 
The volatility smile used by traders to price equity options (both those on individual stocks andthose on stock indices) has the general form shown in Figure 15.3. This is sometimes referred to asa volatility skew. The volatility decreases as the strike price increases. The volatility used to price a
low-strike-price option (i.c., a deep-out-of-the-moiicy put or a déep-in-the-money call) is signific-antly higher than that used to price a high-strike-price option (i.e., a deep-in-the-money put or adeep-out-of-the-money call).
The volatility smile for equity options corresponds to the implied probability distribution given
by the solid line in Figure 15.4. A lognormal distribution with the same mean and standard
A Impliedvolatility
 
 
Strike price>
Figure 15.3 Volatility smile for equities
? Often the jumps are in response to the actions of central banks.Volatility Smiles 335
 
Figure 15.4 Implied distribution and lognormal distribution for equity options
deviation as the implied distribution is shown by the dotted line. It can be seen that the implieddistribution has a heavier left tail and a less heavy right tail than the lognormal distribution.
To see that Figures 15.3 and 15.4 are consistent with each other, we proceed as for Figures 15.1and 15.2 and consider options that are deep out of the money. From Figure 15.4 a deep-out-of-the-money call with a strike price of A, has a lower price when the implied distribution is usedthan when the lognormal distribution is used. This is because the option pays off only if the stockprice proves to be above Kz, and the probability of this is lower for the implied probabilitydistribution than for the lognormal distribution. Therefore we expect the implied distribution togive a relatively low price for the option. A relatively low price leads to a relatively low impliedvolatility—and this is exactly what we observe in Figure 15.3 for the option. Consider next a deep-out-of-the-money put option with a strike price of K,. This option pays off only if the stock priceproves to be below K). Figure 15.3 shows that the probability of this is higher for impliedprobability distribution than for the lognormal distribution. We therefore expect the implieddistribution to give a relatively high price, and a relatively high implied volatility, for this option.Again, this is exactly what we observe in Figure 15.3.
The Reason for the Smile in Equity Options
One vossible exnlanation for the smile in equity options co.Une possice expianation (Or ine smue in equity options cor
declines in value, the company’s leverage increases. As a result the volatility of its equity increases,making even lower stock prices more likely. As a company’s equity increases in value, leveragedecreases. As a result the volatility of its equity declines, making higher stock prices less likely. Thisargument shows that we can expect the volatility of equity to be a decreasing function of price andis consistent with Figures 15.3 and 15.4.
It is an interesting observation that the pattern in Figure 15.3 for equities has existed only since
the stock market crash of October 1987 Prior to October 1987 implied volatilities were ch lethe SLOCK Market crasn O1 V/CtOoer 1757. or eT o/ iMpues vomutuities were Much 15s
leverage. AS a company’s equity
leverage. As a company’s equit
dependent on strike price. This has led Mark Rubinstein to suggest that one reason for the patternin Figure 15.3 may be “‘crashophobia”’. Traders are concerned about the possibility of anothercrash similar to October 1987, and they price options accordingly. There is some empirical support336 CHAPTER 15
for crashophobia. Whenever the market declines (increases), there is a tendency for the skew inFigure 15.3 to become more (less) pronounced.
15.4 THE VOLATILITY TERM STRUCTURE ANDVOLATILITY SURFACES
 
 
In addition to a volatility smile, traders use a volatility term structure when pricing options. Thismeans that the volatility used to price an at-the-money option depends on the maturity of theoption. Volatility tends to be an increasing function of maturity when short-dated volatilities arehistorically low. This is because there is then an expectation that volatilities will increase. Similarly,volatility tends to be an decreasing function of maturity when short-dated volatilities are historic-ally high. This is because there is then an expectation that volatilities will decrease.
Volatility surfaces combine volatility smiles with the volatility term structure to tabulate thevolatilities appropriate for pricing an option with any strike price and any maturity. An example ofa volatility surfaces that might be used for foreign currency options is shown in Table 15.2.
One dimension of Table 15.2 is strike price; the other is time to maturity. The main body of thetable shows implied volatilities calculated from the Black-Scholes model. At any given time, some ofthe entries in the table are likely to correspond to options for which reliable market data areavailable. The implied volatilities for these options are calculated directly from their market pricesand entered into the table. The rest of the table is determined using linear interpolation.
When a new option has to be valued, financial engineers look up the appropriate volatility in thetable. For example, when valuing a nine-month option with a strike price of 1.05, a financialengineer would interpolate between 13.4 and 14.0 to obtain a volatility of 13.7%. This is thevolatility that would be used in the Black-Scholes formula (or in the binomial tree model, whichwe will discuss further in Chapter 18).
The shape of the volatility smile depends on the option maturity. As illustrated in Table 15.2, thesmile tends to become less pronounced as the option maturity increases. Define T as the time tomaturity and Fy as the forward price of the asset. Some financial engineers choose to define thevolatility smile as the relationship between implied volatility and
1,8VT Fo
Table 15.2 Volatility surface
 
 
Strike price0.90 0.95 1.00 1.05 1.101 month 14.2 13.0 12.0 13.1 14.53 month 14.0 13.0 12.0 13.1 14.26 month 14.1 13.3 12.5 13.4 14.31 year 14.7 14.0 13.5 14.0 14.8
2 year 15.0 14.4 14.0 14.5 15.15 year 14.8 14.6 14.4 14.7 15.0Volatility Smiles , 337
 
rather than as the relationship between the implied volatility and K. The smile is then usually muchless dependent on the time to maturity.*
The Role of the Model
How important is the pricing model if traders are prepared to use a different volatility for everyoption? It can be argued that the Black-Scholes model is no more than a sophisticated interpola-tion tool used by traders for ensuring that an option is priced consistently with the market prices ofother actively traded options. If traders stopped using Black-Scholes and switched to anotherplausible model, the volatility surface would change and the shape of the smile would change. Butarguably, the doliar prices quoted in the market would not change appreciably.
15.5 GREEK LETTERS
 
The volatility smile complicates the calculation of Greek letters. Derman describes a number ofvolatility regimes or rules of thumb than are sometimes assumed by traders. The simplest of these isknown as the sticky strike rule. This assumes that the implied volatility of an option remains constantfrom one day to the next. It means that Greek letters calculated using the Black-Scholes assumptionsare correct provided that the volatility used for an option is its current implied volatility.
A more complicated rule is known as the sticky delta rule. This assumes that the relationship weobserve between an option price and S/K today will apply tomorrow. As the price of theunderlying asset changes the implied volatility of the option is assumed to change to reflect theoption’s moneyness (i.¢., the extent to which it is in or out of the money). If we use the sticky deltarule, the formulas for Greek letters given in the Chapter 14 are uo louger correct. Por example,delta of a call option is given by
dS OGimp OS
where cpg is the Black-Scholes price of the option expressed as a function of the asset price S andthe implied volatility oj). Consider the impact of this formula on the delta of an equity calloption. From Figure 15.3, volatility is a decreasing function of the strike price K. Alternatively itcan be regarded as an increasing function of S/K. Under the sticky delta model, therefore, thevolatility increases as the asset price increases, so that
IGimpas
 
>0
As a result, delta is higher than that given by the Black-Scholes assumptions.
It turns out that the sticky strike and sticky delta rules do not correspond to internally consistentmodels (except when the volatility smile is flat for all maturities). A model that can be made exactlyconsistent with the smiles is known as the implied volatility function model or the implied treemodel. We will explain this model in Chapter 20.
* For a discussion of this approach, see S. Natenberg, Option Pricing and Volatility: Advanced Trading Strategies andTechniques, 2nd edn., McGraw-Hill, New York, 1994; R. Tompkins, Options Analysis: A State of the Art Guide toOptions Pricing, Irwin, Burr Ridge, IL,1994.
5 See E. Derman, “Regimes of Volatility,” R/SK, April 1999, pp. 54-59.338 CHAPTER 15
 
In practice many banks try to ensure that their exposure to the changes in the volatility surfacethat are most commonly observed is reasonably small. One technique for identifying these changesis principal components analysis, which we discuss in Chapter 16.
15.6 WHEN A S! ATED
 
 
Suppose that a stock price is currently $50 and an important news announcement in a few days isexpécted to either increase the stock price by $8 or reduce it by $8. (This announcement couldconcern the outcome of a takeover attempt or the verdict in an important lawsuit.) The probabilitydistribution of the stock price in, say, three months might then consist of a mixture of twolognormal distributions, the first corresponding to favorable news, and the second to unfavorablenews. The situation is illustrated in Figure 15.5. The solid line shows the mixtures-of-lognormalsdistribution for the stock price in three months; the dashed line shows a lognormal distributionwith the same mean and standard deviation as this distribution. Assume that favorable news andunfavorable news are equally likely.® Assume also that after the news (favorable or unfavorable)the volatility will be constant at 20% for three months.
Consider a three-month European call option on the stock with a strike price of $50. We assumethat the risk-free interest rate is 5% per annum. Because the news announcement is expected verysoon, the valuc of the option assuming favorable news can be calculated from the Black-Scholesformula with Sy) = 58, K = 50,r =5%, 0 = 20%, and T = 0.25. It is 8.743. Similarly, the value ofthe option assuming unfavorable news can be calculated from the Black-Scholes formula withSo = 42, K = 50, r =5%, o = 20%, and T = 0.25. It is 0.101. The value of the call option todayshould therefore be
0.5 x 8.743 +.0.5 x 0.101 = 4.422
The implied volatility calculated from this option price is 41.48%.
   
 
Stock price
 
Figure 15.5 Effect of a single large jump: the solid line is the true distribution;the dashed line is the lognormal distribution
® Strictly speaking, we are assuming that the probabilities are equally likely in a risk-neutral world.Volatility Smiles
Table 15.3 Implied volatilities in a situation where an important announcement is imminent
 
Strike Call price if Call price if Call price Impliedprice (8) goad news (3) bad news ($) today ($) volatility (%)
23.435 7.471 15.453 30.9540 18.497 3.169 18.833 35.4645 13.565 0.771 7.168 39.9450 8.743 0.101 4.422 41 4855 4.546 0.008 2.277 39.2760 1.764 0.000 0.882 35.6665 0.494 0.000 0.2