47 32.50
339
A similar calculation can be made for other strike prices and a volatility smile constructed. Theresults of doing this are shown in Table 15.3, and the volatility smile is shown in Figure 15.6.’ Itturns out that we are in the opposite situation to that of Figure 15.1. At-the-money options havehigher volatilities than either out-of-the-money or in-the-money options.
15.7 EMPIRICAL RESEARCH
 
 
A number of problems arise in carrying out empirical research to test the Black-Scholes andother option pricing models.® The first problem is that any statistical hypothesis about howoptions are priced has to be a joint hypothesis to the effect that (1) the option pricing formula is
A Impliedvolatility
40+30+
20-7
 
40
50
60
Strike price
Figure 15.6 Volatility smile for situation in Table 15.3
” In this case, the smile is a frown!
® See the end-of-chapter references for citations to the studies reviewed in this section.340 CHAPTER 15
correct and (2) markets are efficient. If the hypothesis is rejected, it may be the case that (1) isuntrue, (2) is untrue, or both (1) and (2) are untrue. A second problem is that the stock pricevolatility is an unobservable variable. One approach is to estimate the volatility from historicalstock price data. Altcrnatively, implied volatilities can be used in some way. A third problem forthe researcher is to make sure that data on the stock price and option price are synchronous. Forexample, if the option is thinly traded, it is not likely to be acceptable to compare closing optionprices with closing stock prices. The closing option price might correspond to a trade at 1:00 p.m.,whereas the closing stock price corresponds to a trade at 4:00 p.m.
Black and Scholes (1972) and Galai (1977) have tested whether it is possible to make excessreturns above the risk-free rate of interest by buying options that are undervalued by the market(relative to the theoretical price) and selling options that are overvalued by the market (relative tothe theoretical price). Black and Scholes used data from the over-the-counter options market whereoptions are dividend protected. Galai used data from the Chicago Board Options Exchange(CBOE) where options are not protected against the effects of cash dividends. Galai used Black’sapproximation as described in Section 12.13 to incorporate the effect of anticipated dividends intothe option price. Both studies showed that, in the absence of transactions costs, significant excessreturns over the risk-free rate can be obtained by buying undervalued options and selling over-valued options. However, it is possible that these excess returns are available only to marketmakers and that, when transactions costs are considered, they vanish.
A number of researchers have chosen to make no assumptions about the behavior of stock pricesand have tested whether arbitrage strategies can be used to make a riskless profit in options markets.Garman (1976) provides a computational procedure for finding any arbitrage possibilities that existin a given situation, Onc frequently cited study by Klemkosky and Resnick (1979) tests whether therelationship in equation (8.8) is ever violated. It concludes that some small arbitrage profits arepossible from using the relationship. These are due mainly to the overpricing of American calls.
Chiras and Manaster (1978) carried out a study using CBOE data to compare a weightedimplied volatility from options on a stock at a point in time with the volatility calculated fromhistorical data. They found that the former provide a much better forecast of the volatility of thestock price during the life of the option. We can conclude that option traders are using more than
just historical data when determining future volatilities. Chiras and Manaster also tested to seegust oistorica: Gata when Getermining mature yOrtiities. Umiras ang Wwaanaster aiso tested to see
whether it was possible to make above-average returns by buying options with low impliedvolatilities and selling options with high implied volatilities. The strategy showed a profit of10% per month. The Chiras and Manaster study can be interpreted as providing good supportfor the Black-Scholes model and showing that the CBOE was inefficient in some respects.
MacBeth and Merville (1979) tested the Black-Scholes model using a different approach. Theylooked at different call options on the same stock at the same time and compared the volatilitiesimplied by the option prices. The stocks chosen were AT&T, Avon, Kodak, Exxon, IBM, andXerox, and the time period considered was the year 1976. They found that implied volatilities tendedto be relatively high for in-the-money options and relatively low for out-of-the-money options. Arelatively high implied volatility is indicative of a relatively high option price, and a relatively lowimplied volatility is indicative of a relatively low option price. Therefore, if it is assumed that Black—Scholes prices at-the-money options correctly, it can be concluded that out-of-the-money (highstrike price) call options are overpriced by Black-Scholes and in-the-money (low strike price) calloptions are underpriced by Black-Scholes. These effects become more pronounced as the time tomaturity increases and the degree to which the option is in or out of the money increases. MacBethand Merville’s results are consistent with Figure 15.3. The results were confirmed by Lauterbachand Schultz (1990) in a later study concerned with the pricing of warrants.Volatility Smiles 341
Rubinstein has done a great deal of research similar to that of MacBeth and Merville. No clear-cut pattern emerged from his early research, but the research in his 1994 paper and joint 1996paper with Jackwerth gives results consistent with Figure 15.3. Options with low strike prices havemuch higher volatilities than those with high strike prices. As mentioned previously in the chapter,leverage and the resultant negative correlation between volatility and stock price may partiallyaccount for the finding. It is also possible that investors fear a repeat of the crash of 1987.
A number of authors have researched the pricing of options on assets other than stocks. Forexample, Shastri and Tandon (June 1986) and Bodurtha and Courtadon (1987) have examined themarket prices of currency options; in another paper, Shastri and ‘landon (December 1986) haveexamined the market prices of futures options; and Chance (1986) has examined the market prices ofindex options.
In most cases, the mispricing by Black-Scholes is not sufficient to present profitable opportun-ities to investors when transactions costs and bid—offer spreads are taken into account. Whenprofitable opportunities are sought, it is important to bear in mind that, even for a market maker,some time must elapse between a profitable opportunity being identified and action being taken.This delay, even if it is only to the next trade, can be sufficient to eliminate the protitableopportunity.
SUMMARY
 
The Black-Scholes model and its extensions assume that the probability distribution of theunderlying asset at any given future time is lognormal. This assumption is not the one made bytraders. They assume the probability distribution of an equity price has a heavier left tail and a lessheavy right tail than the lognormal distribution. They also assume that the probability of anexchange rate has a heavier right tail and a heavier left tail than the lognormal distribution.
Traders use volatility smiles to allow for nonlognormality. The volatility smile defines therelationship between the implied volatility of an option and its strike price. For equity options,the volatility smile tends to be downward sloping. This means that out-of-the-money puts and in-the-money calls tend to have high implied volatilities, whereas out-of-the-money calls and in-the-money puts tend to have low implied volatilities. For foreign currency options, the volatility smileis U-shaped. Deep-out-of-the-money and deep-in-the-money options have higher implied volatil-ities than at-the-money options.
Often traders also use a volatility term structure. The implied volatility of an option thendepends on its life. When volatility smiles and volatility term structures are combined, theyproduce a volatility surface. This defines implied volatility as a function of both the strike price
and the time to maturity.
SUGGESTIONS FOR FURTHER READING
 
Bakshi, G., C. Cao, and Z. Chen, “Empirical Performance of Alternative Option Pricing Models,”Journal of Finance, $2, no. 5 (December 1997), 2004-49.
Black, F., “How to Use the Holes in Black-Scholes,” RISK, March 1988.
Black, F., and M. Scholes, “The Valuation of Option Contracts and a Test of Market Efficiency,”Journal of Finance, 27 (May 1972), 399-418.342 CHAPTER 15
Bodurtha, J. N., and G. R. Courtadon, “Tests of an American Option Pricing Model on the ForeignCurrency Options Market,” Journal of Financial and Quantitative Anatysis, 22 (June 1987), 153-68.
Breeden, M., and R. Litzenberger, “Prices of State Contingent Claims Implicit in Option Prices,”Journal of Business, 51 (1978), 621-51.
Chance, D. M., “Empirical Tests of the Pricing of Index Call Options,” Advances in Futures and OptionsResearch, 1, pt. A (1986), 141-66.
Chiras, D., and S. Manaster, “The Information Content of Option Prices and a Test of MarketEfficiency,” Journal of Financial Economics, 6 (September 1978), 213-34.
Acien CY, Jou! nea! Ecol nics, & (september IY ots
Derman, E., “Regimes of Volatility,” RISK, April 1999, pp. 55-59.
Dumas, B., J. Fleming, and R. E. Whaley, “Implied Volatility Functions: Empirical Tests,” Journal ofFinance 53, 6 (December 1998), 2059-2106.
Galai, D., ‘Tests of Market Efficiency and the Chicago Board Options Exchange.” Journal of Business,50 (April 1977), 167-97.
Garman, M., “An Algebra for Evaluating Hedge Portfolios,” Journal of Financial Economics, 3 (October1976), 403-27.
Harvey, C. R., and R. E. Whaley, “S&P 100 Index Option Volatility,” Journal of Finance, 46 (1991),1551-61.
Harvey, C. R., and R. E. Whaley, “Market Volatility Prediction and the Efficiency of the S&P 100Index Option Market,” Journal of Financial Economics, 31 (1992), 43-73.
Harvey, C. R., and R. E. Whaley, “Dividends and S&P 100 Index Option Valuations,” Journal ofFutures Markets, 12 (1992), 123-37.
Jackwerth, J. C., and M. Rubinstein, “Recovering Probability Distributions from Option Prices,”Journal of Finance, 51 (December 1996), 1611-31.
Klemkosky, R. C., and B. G. Resnick, “Put-Call Parity and Market Efficiency,” Journal of Finance,34 (December 1979), 1141-55.
Lauterbach, B., and P. Schultz, “Pricing Warrants: An Empirical Study of the Black-Scholes Modeland Its Alternatives,” Journal of Finance, 4, no. 4 (September 1990), 1181-1210.
MacBeth, J. D., and L. J. Merville, “An Empirical Examination of the Black-Scholes Call OptionPricing Model,” Journal of Finance, 34 (December 1979), 1173-86.
Melick, W. R., and C. P. Thomas, “Recovering an Asset’s Implied Probability Density Function from
Option Prices: An Application to Crude Oil during the Gulf Crisis,” Journal of Financial andQuantitative Analysis, 32, no. 1 (March 1997), 91-115.
Rubinstein, M., “Nonparametric Tests of Alternative Option Pricing Models Using All ReportedTrades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 throughAugust 31, 1978,” Journal of Finance, 40 (June 1985), 455-80.
Rubinstein, M., “Implied Binomial Trees,” Journal of Finance, 49, 3 (July 1994), 771-818.
Shastri, K., and K. Tandon, “An Empirical Test of a Valuation Model for American Options onFutures Contracts,” Journal of Financial and Quantitative Analysis, 21 (December 1986), 377-92.
Shastri, K., and K, Tandon, “Valuation of Foreign Currency Options: Some Empirical Tests,” Journalof Financial and Quantitative Analysis, 21 (June 1986), 145-60.
Xu, X., and S. J. Taylor, “The Term Structure of Volatility Implied by Foreign Exchange Options,”Journal of Financial and Quantitative Analysis, 29 (1994), 57-74.Volatility Smiles 343
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
15.2.
15.3.
15.4.
15,5.
15.6.
15.7.
15.8.
15.9,
15.10.
15.11.
15.12.
15.13.
. What pattern of implied volatilities is likely to be observed when
a. Both tails of the stock price distribution are less heavy than those of the lognormaldistribution?b. The right tail is heavier, and the left tail is less heavy, than that of a lognormal distribution?
What pattern of implied volatilities is likely to be observed for six-month options when thevolatility is uncertain and positively correlated to the stock price?
What pattern of implied volatilities is likely to be caused by jumps in the underlying asset price? Isthe pattern likely to be more pronounced for a six-month option than for a three-month option?
A call and put option have the same strike price and time to maturity. Show that the differencebetween their prices should be the same for any option pricing model.
Explain carefully why Figure 15.4 is consistent with Figure 15.3.
The market price of a European call is $3.00 and its Black-Scholes price is $3.50, TheBlack-Scholes price of a European put option with the same strike price and time to maturityis $1.00. What should the market price of this option be? Explain the reasons for your answer.
A stock price is currently $20. Tomorrow. news is expected to be announced that will citherincrease the price by $5 or decrease the price by $5. What are the problems in usingBlack-Scholes to value one-month options on the stock?
What are the major problems in testing a stock option pricing model empirically?
Suppose that a central bank’s policy is to allow an exchange rate to fluctuate between 0.97 and1,03. What pattern of implied volatilities for options on the exchange rate would you expect to see?
Option traders sometimes refer to deep-out-of-the-money options as being options on volatility.Why do you think they do this?
A European call option on a certain stock has a strike price of $30, a time to maturity of oneyear, and an implied volatility of 30%. A European put option on the same stock has a strikeprice of $30, a time to maturity of one year, and an implied volatility of 33%. What is thearbitrage opportunity open to a trader? Does the arbitrage work only when the lognormalassumption underlying Black-Scholes holds? Explain carefully the reasons for your answer.
Suppose that the result of a major lawsuit affecting Microsoft is due to be announced tomorrow.Microsoft’s stock price is currently $60. If the ruling is favorable to Microsoft, the stock price isexpected to jump to $75. If it is unfavorable, the stock is expected to jump to $50. What is therisk-neutral probability of a favorable ruling? Assume that the volatility of Microsoft’s stock willbe 25% for six months after the ruling if the ruling is favorable and 40% if it is unfavorable. UseDerivaGem to calculate the relationship between implied volatility and strike price for six-monthEuropean options on Microsoft today. Microsoft does not pay dividends. Assume that the six-month risk-free rate is 6%. Consider call options with strike prices of 30, 40, 50, 60, 70, and 80.
An exchange rate is currently 0.8000. The volatility of the exchange rate is quoted as 12% andinterest rates in the two countries are the same. Using the lognormal assumption, estimate theprobability that the exchange rate in three months will be (a) less than 0.7000, (b) between 0.7000and 0.7500, (c) between 0.7500 and 0.8000, (d) between 0.8000 and 0.8500, {e) between 0.8500and 0.9000, and (f) greater than 0.9000. Based on the volatility smile usually observed in themarket for exchange rates, which of these estimates would you expect to be too low and whichwould you expect to be too high?344
15.14.
15.15.
CHAPTER 15
A stock price is $40. A six-month European call option on the stock with a strike price of $30has an implied volatility of 35%. A six-month European call option on the stock with a strikeprice of $50 has an implied volatility of 28%. The six-month risk-free rate is 5% and nodividends are expected. Explain why the two implied volatilities are different. Use DerivaGem tocalculate the prices of the two options. Use put—call parity to calculate the prices of six-monthEuropean put options with strike prices of $30 and $50. Use DerivaGem to calculate the impliedvolatilities of these two put options.
“The Black-Scholes model is used by traders as an interpolation tool.” Discuss this view.
ASSIGNMENT QUESTIONS
 
be4nhocn
15.17,
15.18.
15.19.
15.20.
15.21.
A ? ; o outstand;. A company’s stock is sculing for $4. The company has no outstandi
compan ack is scllins for 94. The company n
the liquidation value of the company to be at least $300,000 a:outstanding. What volatility smile would you expect to see?
A company is currently awaiting the outcome of a major lawsuit. This is expected to be knownwithin one month. The stock price is currently $20. If the outcome is positive, the stock price isexpected to be $24 at the end of one month. If the outcome is negative, it is expected to be $18 atthis time. The one-month risk-free interest rate is 8% per annum.
a. What is the risk-neutral probability of a positive outcome?
b. What are the values of one-month call options with strike prices of $19, $20, $21, $22,
and $23?c. Use DerivaGem to calculate a volatility smile for one-month call options.d. Verify that the same volatility smile is obtained for one-month put options.
A futures price is currently $40. The risk-free interest rate is 5%. Some news is expectedtomorrow that will cause the volatility over the next three months to be either 10% or 30%.There is a 60% chance of the first outcome and a 40% chance of the second outcome. UseDerivaGem to calculate a volatility smile for three-month options.
Data for a number of foreign currencies are provided on the author’s Web site:www.rotman.utoronto.ca/~hullChoose a currency and use the data to produce a table similar to Table 15.1.
Data for a number of stock indices are provided on the author’s Web site:www.rotman.utoronto.ca/” hull
Choose an index and test whether a three standard deviation down movement happens more
often than a three standard deviation up movement.
Consider a European call and a European put with the same strike price and time to maturity.Show that they change in value by the same amount when the volatility increases from a level a,to a new level co, within a short period of time. (Hint: Use put-call parity.)Volatility Smiles 345
APPENDIX 15A
Determining Implied Risk-Neutral Distributions from Volatility SmilesThe price of a European call option on an asset with strike price K and maturity T is given byOOc= etl (Sr — K)g(Sr) dSSp=Kwhere r is the interest rate (assumed constant), Sy is the asset price at time T, and g is the risk-
neuttal probability density function of S;. Differentiating once with respect to K, we obtain
dc
OO— _p,7TBeane | Sr dsr
Differentiating again with respect to K gives
ae -rame "g(K)This shows that the probability density function, g, is given by
Pe
@
aK) =e?
 
aK2o&
This result, which is from Breeden and Litzenberger (1978), allows risk-neutral probabilitydistributions to be estimated from volatility smiies. Suppose that c,, c2, and c; are the prices ofEuropean call options with maturity T and strike prices are K — 6, K, and K +4, respectively.Assuming 6 is small, an estimate of g(K) is
rey ty — 2cyoo”
In Chapter 14 we examined measures such as delta, gamma, and vega for describing different aspectsof the risk in a portfolio consisting of options and other financial assets. A financial institutionusually calculates each of these measures each day for every market variable to which it is exposed.Often there are hundreds, or even thousands, of these market variables. A delta~gamma—vegaanalysis therefore leads to a huge number of different risk measures being produced each day. Thesetisk measures provide valuable information for a trader who is responsible for managing the part ofthe financial institution’s portfolio that is dependent on a particular market variable, but they are oflimited use to senior management.
Value at risk (VaR) is an attempt to provide a single number summarizing the total risk in aportfolio of financial assets for senior management. It has become widely used by corporatetreasurers and fund managers as well as by financial institutions. Central bank regulators also useVaR in determining the capital a bank is required to keep to reflect the market risks it is bearing.!
In this chapter we explain the VaR measure and describe the two main approaches forcalculating it. These are the historical simulation approach and the model-building approach. Bothare widely used by both financial and nonfinancial companies. There is no general agreement onwhich of the two is better.
CHAPTER 16VA
vi LU
 
IIE AT RC I
‘Xx mn
ckI
IN
16.1 THE VaR MEASURE
 
 
When using the value-at-risk measure, the manager in charge of a portfolio of financial instru-ments is interested in making a statement of the following form:
“We are X percent certain that we will not lose more than V dollars in the next N days.”
and X, the confidence level. It is the loss level over N days that the manager is X% certain will notbe exceeded.
In calculating a bank’s capital for market risk, regulators use N = 10 and X = 99. This meansthat they focus on the loss level over a 10-day period that is expected to be exceeded only 1% of thetime. The capital they require the bank to keep is at least three times this VaR measure.
' For a discussion of this, see P. Jackson, D. J. Maude, and W. Perraudin, “Bank Capital and Value at Risk,”Journal of Derivatives, 4, no. 3 (Spring 1997), 73-90.
? To be more precise, the market risk capital required for a particular bank is k times the 10-day 99% VaR, wherethe multiplier & is chosen by the regulators and is at least 3.0.
346Value at Risk 347
 
   
(100 — X)%
 
VaR
Figure 16.1 Calculation of VaR from the probability distribution of changes in the portfolio value;confidence level is X%
In general, when N days is the time horizon and X% is the confidence Jevel, VaR is the losscorresponding to the (100 — X)th percentile of the distribution of the change in the value of theportfolio over the next N days. For example, when N = 5 and X = 97, it is the third percentile ofthe distribution of changes in the value of the portfolio over the next five days. Figure 16.1illustrates VaR for the situation where the change in the value of the portfolio is approximatelynormally distributed.
VaR is an attractive measure because it is easy to understand. In essence, it asks the simplequestion “How bad can things get?” This is the question all senior managers want answered. Theyare very comfortable with the idea of compressing all the Greek letters for all the market variablesunderlying the portfolio into a single number.
If we accept that it is useful to have a single number to describe the risk of a portfolio, aninteresting question is whether VaR is the best alternative. Some researchers have argued that VaRmay tempt traders to choose a portfolio with a return distribution similar to that in Figure 16.2.The portfolios in Figures 16.1 and 16.2 have the same VaR, but the portfolio in Figure 16.2 ismuch riskier because potential losses are much larger.
A measure that deals with the problem we have just mentioned is Conditional VaR (C-VaR).°Whereas VaR asks the question “‘How bad can things get?”, C-VaR asks “If things do get bad,how much can we expect to lose?” C-VaR is the expected loss during an N-day period conditional
_—
Figure 16.2 Alternative situation to Figure 16.1; VaR is the same, but the potential loss is larger
 
 
3 This measure was suggested by P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent Measures of Risk,”Mathematical Finance, 9 (1999), 203-28. These authors define certain properties that a good risk measure shouldhave and show that the standard VaR measure does not have all of them.348 CHAPTER 16
that we are in the (100 — X)% left tail of the distribution. For example, with X = 99 and N = 10,C-VaR is the average amount we lose over a 10-day period assuming that a 1% worst-case eventoccurs.
In spite of its weaknesses, VaR (not C-VaR) is the most popular measure of risk among bothregulators and senior management. We will therefore devote most of the rest of this chapter to howit can be measured.
The Time Horizon
In theory, VaR has two parameters. These are N, the time horizon measured in days, and X, theconfidence interval. In practice, analysts almost invariably set N = 1 in the first instance. This isbecause there is not enough data to estimate directly the behavior of market variables over periodsof time longer than one day. The usual assumption is
N-day VaR = |-day VaR x /N
This formula is exactly true when the changes in the value of the portfolio on successive days haveindependent identical normal distributions with mean zero. In other cases it is an approximation.
We mentioned earlier that regulators require a bank’s capital to be at least three times the 10-day99% VaR. Given the way a 10-day VaR is calculated, this capital level is, to all intents andpurposes, 3 x //10 = 9.49 times the I-day 99% VaR.
16.2 HISTORICAL SIMULATION
 
 
Historical simulation is one popular way of estimating VaR. It involves using past data in a verydirect way as a guide to what might happen in the future. Suppose that we wish to calculate VaRfor a portfolio using a one-day time horizon, a 99% confidence level, and 500 days of data. Thefirst step is to identify the market variables affecting the portfolio. These will typically be exchangerates, equity prices, interest rates, and so on. We then collect data on the movements in thesemarket variables over the most recent 500 days. This provides us with 500 alternative scenarios forwhat can happen between today and tomorrow. Scenario | is where the percentage changes in the
Table 16.1 Data for VaR historical simulation calculation
 
Market Market MarketDay variable 1 variable 2 on variable N0 20.33 0.1132 to 65.371 20.78 0.1159 vee 64.912 21.44 0.1162 vee 65.023 20.97 0.1184 oe 64.90498 25.72 0.1312 oo 62.22
499 25.75 0.1323 oe 61.99500 25.85 0.1343 oe 62.10Value at Risk 349
Table 16.2 Scenarios generated for tomorrow (Day 501) usingdata in Table 16.1
Scenario Market Market Market Portfolio valuenumber variable 1 variable 2 --- yariable N (8 millions)1 26.42 0.1375 --- 61.66 23.712 26.67 0.1346 --- 62.21 23.123 25.28 0.1368  --- 61.99 22.94499 25.88 0.1354 --- 61.87 23.63500 25.95 0.1363 --- 62.21 22.87
values of all variables are the same as they were on the first day for which we have collected data;scenario 2 is where they are the same as on the second day for which we have data; and so on. Foreach scenario we calculate the dollar change in the value of the portfolio between today andtomorrow. This defines a probability distribution for daily changes in the value