It is important for a financial institution to monitor the volatilities of the market

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1 CHAPTER 10 Volatlty It s mportant for a fnancal nsttuton to montor the volatltes of the market varables (nterest rates, exchange rates, equty prces, commodty prces, etc.) on whch the value of ts portfolo depends. Ths chapter descrbes the procedures t can usetodoths. The chapter starts by explanng how volatlty s defned. It then examnes the common assumpton that percentage returns from market varables are normally dstrbuted and presents the power law as an alternatve. After that t moves on to consder models wth mposng names such as exponentally weghted movng average (EWMA), autoregressve condtonal heteroscedastcty (ARCH), and generalzed autoregressve condtonal heteroscedastcty (GARCH). The dstnctve feature of these models s that they recognze that volatlty s not constant. Durng some perods, volatlty s relatvely low, whle durng other perods t s relatvely hgh. The models attempt to keep track of varatons n volatlty through tme DEFINITION OF VOLATILITY A varable s volatlty, σ, s defned as the standard devaton of the return provded by the varable per unt of tme when the return s expressed usng contnuous compoundng. (See Appendx A for a dscusson of compoundng frequences.) When volatlty s used for opton prcng, the unt of tme s usually one year, so that volatlty s the standard devaton of the contnuously compounded return per year. When volatlty s used for rsk management, the unt of tme s usually one day so that volatlty s the standard devaton of the contnuously compounded return per day. Defne S as the value of a varable at the end of day. The contnuously compounded return per day for the varable on day s ln S S 1 Ths s almost exactly the same as S S 1 S 1 201

2 202 MARKET RISK An alternatve defnton of daly volatlty of a varable s therefore the standard devaton of the proportonal change n the varable durng a day. Ths s the defnton that s usually used n rsk management. EXAMPLE 10.1 Suppose that an asset prce s $60 and that ts daly volatlty s 2%. Ths means that a one-standard-devaton move n the asset prce over one day would be or $1.20. If we assume that the change n the asset prce s normally dstrbuted, we can be 95% certan that the asset prce wll be between = $57.65 and = $62.35 at the end of the day. If we assume that the returns each day are ndependent wth the same varance, the varance of the return over T days s T tmes the varance of the return over one day. Ths means that the standard devaton of the return over T days s T tmes the standard devaton of the return over one day. Ths s consstent wth the adage uncertanty ncreases wth the square root of tme. EXAMPLE 10.2 Assume as n Example 10.1 that an asset prce s $60 and the volatlty per day s 2%. The standard devaton of the contnuously compounded return over fve days s 5 2 or 4.47%. Because fve days s a short perod of tme, ths can be assumed to be the same as the standard devaton of the proportonal change over fve days. A one-standard-devaton move would be = If we assume that the change n the asset prce s normally dstrbuted, we can be 95% certan that the asset prce wll be between = $54.74 and = $65.26 at the end of the fve days. Varance Rate Rsk managers often focus on the varance rate rather than the volatlty. The varance rate s defned as the square of the volatlty. The varance rate per day s the varance of the return n one day. Whereas the standard devaton of the return n tme T ncreases wth the square root of tme, the varance of ths return ncreases lnearly wth tme. If we wanted to be pedantc, we could say that t s correct to talk about the varance rate per day, but volatlty s per square root of day. Busness Days vs. Calendar Days One ssue s whether tme should be measured n calendar days or busness days. As shown n Busness Snapshot 10.1, research shows that volatlty s much hgher on busness days than on non-busness days. As a result, analysts tend to gnore weekends and holdays when calculatng and usng volatltes. The usual assumpton s that there are 252 days per year.

3 Volatlty 203 BUSINESS SNAPSHOT 10.1 What Causes Volatlty? It s natural to assume that the volatlty of a stock or other asset s caused by new nformaton reachng the market. Ths new nformaton causes people to revse ther opnons about the value of the asset. The prce of the asset changes and volatlty results. However, ths vew of what causes volatlty s not supported by research. Wth several years of daly data on an asset prce, researchers can calculate: 1. The varance of the asset s returns between the close of tradng on one day and the close of tradng on the next day when there are no ntervenng nontradng days. 2. The varance of the asset s return between the close of tradng on Frday and the close of tradng on Monday. The second s the varance of returns over a three-day perod. The frst s a varance over a one-day perod. We mght reasonably expect the second varance to be three tmes as great as the frst varance. Fama (1965), French (1980), and French and Roll (1986) show that ths s not the case. For the assets consdered, the three research studes estmate the second varance to be 22%, 19%, and 10.7% hgher than the frst varance, respectvely. At ths stage you mght be tempted to argue that these results are explaned by more news reachng the market when the market s open for tradng. But research by Roll (1984) does not support ths explanaton. Roll looked at the prces of orange juce futures. By far the most mportant news for orange juce futures s news about the weather, and ths s equally lkely to arrve at any tme. When Roll compared the two varances for orange juce futures, he found that the second (Frday-to-Monday) varance s only 1.54 tmes the frst (one-day) varance. The only reasonable concluson from all ths s that volatlty s, to a large extent, caused by tradng tself. (Traders usually have no dffculty acceptng ths concluson!) Assumng that the returns on successve days are ndependent and have the same standard devaton, ths means that or σ yr =σ day 252 σ day = σ yr 252 showng that the daly volatlty s about 6% of annual volatlty.

4 204 MARKET RISK Jan 04 Jan 05 Jan 06 Jan 07 Jan 08 Jan 09 Jan 10 Jan 11 Jan 12 Jan 13 Jan 14 FIGURE 10.1 The VIX Index, January 2004 to August IMPLIED VOLATILITIES Although rsk managers usually calculate volatltes from hstorcal data, they also try and keep track of what are known as mpled volatltes. The one parameter n the Black Scholes Merton opton prcng model that cannot be observed drectly s the volatlty of the underlyng asset prce (see Appendx E). The mpled volatlty of an opton s the volatlty that gves the market prce of the opton when t s substtuted nto the prcng model. The VIX Index The CBOE publshes ndces of mpled volatlty. The most popular ndex, the VIX, s an ndex of the mpled volatlty of 30-day optons on the S&P 500 calculated from a wde range of calls and puts. 1 Tradng n futures on the VIX started n 2004 and tradng n optons on the VIX started n A trade nvolvng optons on the S&P 500 s a bet on both the future level of the S&P 500 and the volatlty of the S&P 500. By contrast, a futures or optons contract on the VIX s a bet only on volatlty. One contract s on 1,000 tmes the ndex. EXAMPLE 10.3 Suppose that a trader buys an Aprl futures contract on the VIX when the futures prce s $18.5 (correspondng to a 30-day S&P 500 volatlty of 18.5%) and closes out the contract when the futures prce s $19.3 (correspondng to an S&P 500 volatlty of 19.3%). The trader makes a gan of $800. Fgure 10.1 shows the VIX ndex between January 2004 and August Between 2004 and md-2007, t tended to stay between 10 and 20. It reached 30 durng 1 Smlarly, the VXN s an ndex of the volatlty of the NASDAQ 100 ndex and the VXD s an ndex of the volatlty of the Dow Jones Industral Average.

5 Volatlty 205 TABLE 10.1 Percentage of Days When Absolute Sze of Daly Exchange Rate Moves Is Greater Than One, Two,, Sx Standard Devatons (S.D. = standard devaton of percentage daly change) Real Normal World (%) Model (%) > 1 S.D > 2 S.D > 3 S.D > 4 S.D > 5 S.D > 6 S.D the second half of 2007 and a record 80 n October and November 2008 after the falure of Lehman Brothers. By early 2010, t had returned to more normal levels, but n May 2010 t spked at over 45 because of the European soveregn debt crss. In August 2011, t ncreased agan because of market uncertantes. Sometmes market partcpants refer to the VIX ndex as the fear ndex ARE DAILY PERCENTAGE CHANGES IN FINANCIAL VARIABLES NORMAL? When confdence lmts for the change n a market varable are calculated from ts volatlty, a common assumpton s that the change s normally dstrbuted. Ths s the assumpton we made n Examples 10.1 and In practce, most fnancal varables are more lkely to experence bg moves than the normal dstrbuton would suggest. Table 10.1 shows the results of a test of normalty usng daly movements n 12 dfferent exchange rates over a 10-year perod. 2 The frst step n the producton of the table s to calculate the standard devaton of daly percentage changes n each exchange rate. The next stage s to note how often the actual percentage changes exceeded one standard devaton, two standard devatons, and so on. These numbers are then compared wth the correspondng numbers for the normal dstrbuton. Daly percentage changes exceed three standard devatons on 1.34% of the days. The normal model for returns predcts that ths should happen on only 0.27% of days. Daly percentage changes exceed four, fve, and sx standard devatons on 0.29%, 0.08%, and 0.03% of days, respectvely. The normal model predcts that we should hardly ever observe ths happenng. The table, therefore, provdes evdence to support the exstence of the fact that the probablty dstrbutons of changes n exchange rates have heaver tals than the normal dstrbuton. When returns are contnuously compounded, the return over many days s the sum of the returns over each of the days. If the probablty dstrbuton of the return n a day were the same non-normal dstrbuton each day, the central lmt theorem 2 Ths table s based on J. C. Hull and A. Whte, Value at Rsk When Daly Changes n Market Varables Are Not Normally Dstrbuted, Journal of Dervatves 5, no. 3 (Sprng 1998): 9 19.

6 206 MARKET RISK BUSINESS SNAPSHOT 10.2 Makng Money from Foregn Currency Optons Black, Scholes, and Merton n ther opton prcng model assume that the underlyng asset s prce has a lognormal dstrbuton at a future tme. Ths s equvalent to the assumpton that asset prce changes over short perods, such as one day, are normally dstrbuted. Suppose that most market partcpants are comfortable wth the assumptons made by Black, Scholes, and Merton. You have just done the analyss n Table 10.1 and know that the normal/lognormal assumpton s not a good one for exchange rates. What should you do? The answer s that you should buy deep-out-of-the-money call and put optons on a varety of dfferent currences and wat. These optons wll be relatvely nexpensve and more of them wll close n-the-money than the Black Scholes Merton model predcts. The present value of your payoffs wll on average be much greater than the cost of the optons. In the md-1980s, a few traders knew about the heavy tals of foregn exchange probablty dstrbutons. Everyone else thought that the lognormal assumpton of the Black Scholes Merton was reasonable. The few traders who were well nformed followed the strategy we have descrbed and made lots of money. By the late 1980s, most traders understood the heavy tals and the tradng opportuntes had dsappeared. of statstcs would lead to the concluson that the return over many days s normally dstrbuted. In fact, the returns on successve days are not dentcally dstrbuted. (One reason for ths, whch wll be dscussed later n ths chapter, s that volatlty s not constant.) As a result, heavy tals are observed n the returns over relatvely long perods as well as n the returns observed over one day. Busness Snapshot 10.2 shows how you could have made money f you had done an analyss smlar to that n Table 10.1 n 1985! Fgure 10.2 compares a typcal heavy-taled dstrbuton (such as the one for foregn exchange) wth a normal dstrbuton that has the same mean and standard devaton. 3 The heavy-taled dstrbuton s more peaked than the normal dstrbuton. In Fgure 10.2, we can dstngush three parts of the dstrbuton: the mddle, the tals, and the ntermedate parts (between the mddle and the tals). When we move from the normal dstrbuton to the heavy-taled dstrbuton, probablty mass shfts from the ntermedate parts of the dstrbuton to both the tals and the mddle. If we are consderng the percentage change n a market varable, the heavy-taled dstrbuton has the property that small and large changes n the varable are more lkely than they would be f a normal dstrbuton were assumed. Intermedate changes are less lkely. 3 Kurtoss measures the sze of a dstrbuton s tals. A leptokurtc dstrbuton has heaver tals than the normal dstrbuton. A platykurtc dstrbuton has less heavy tals than the normal dstrbuton; a dstrbuton wth the same szed tals as the normal dstrbuton s termed mesokurtc.

7 Volatlty 207 Heavy-taled Normal FIGURE 10.2 Comparson of Normal Dstrbuton wth a Heavy-Taled Dstrbuton The two dstrbutons have the same mean and standard devaton THE POWER LAW The power law provdes an alternatve to assumng normal dstrbutons. The law asserts that, for many varables that are encountered n practce, t s approxmately true that the value of the varable, v, has the property that when x s large Prob(v > x) = Kx α (10.1) where K and α are constants. The equaton has been found to be approxmately true for varables v as dverse as the ncome of an ndvdual, the sze of a cty, and the number of vsts to a webste n a day. EXAMPLE 10.4 Suppose that we know from experence that α=3for a partcular fnancal varable and we observe that the probablty that v > 10 s Equaton (10.1) gves 0.05 = K 10 3 so that K = 50. The probablty that v > 20 can now be calculated as The probablty that v > 30 s and so on = =

8 208 MARKET RISK TABLE 10.2 Values Calculated from Table 10.1 x ln(x) Prob(v > x) ln[prob(v > x)] Equaton (10.1) mples that ln[prob(v > x)] =lnk αlnx We can therefore do a quck test of whether t holds by plottng ln[prob(v > x)] aganst ln x. In order to do ths for the data n Table 10.1, defne the v as the number of standard devatons by whch an exchange rate changes n one day. The values of ln(x) and ln[prob(v > x)] are calculated n Table The data n Table 10.2 s plotted n Fgure Ths shows that the logarthm of the probablty of the exchange rate changng by more than x standard devatons s approxmately lnearly dependent on ln x for x 3. Ths s evdence that the power law holds for ths data. Usng data for x = 3, 4, 5, and 6, a regresson analyss gves the best-ft relatonshp as ln[prob(v > x)] = ln(x) ln[prob(v>x)] ln(x) FIGURE 10.3 Log-Log Plot for Probablty that Exchange Rate Moves By More than a Certan Number of Standard Devatons. v s the exchange rate change measured n standard devatons.

9 Volatlty 209 showng that estmates for K and α are as follows: K = e = and α= An estmate for the probablty of a change greater than 4.5 standard devatons s = An estmate for the probablty of a change greater than seven standard devatons s = We examne the power law more formally and explan better procedures for estmatng the parameters when we consder extreme value theory n Chapter 13. We also consder ts use n the assessment of operatonal rsk n Chapter MONITORING DAILY VOLATILITY Defne σ n as the volatlty per day of a market varable on day n, as estmated at the end of day n 1. The varance rate, whch, as mentoned earler, s defned as the square of the volatlty, s σ 2 n. Suppose that the value of the market varable at the end of day s S. Defne u as the contnuously compounded return durng day (between the end of day 1 and the end of day ) so that u =ln One approach to estmatng σ n s to set t equal to the standard devaton of the u s. When the most recent m observatons on the u are used n conjuncton wth the usual formula for standard devaton, ths approach gves: S S 1 σ 2 n = 1 m 1 m (u n u) 2 (10.2) =1 where u s the mean of the u s: u = 1 m m u n =1 EXAMPLE 10.5 Table 10.3 shows a possble sequence of stock prces. Suppose that we are nterested n estmatng the volatlty for day 21 usng 20 observatons on the u so that n = 21 and m = 20. In ths case, u = and the estmate of the standard devaton of the daly return calculated usng equaton (10.2) s 1.49%.

10 210 MARKET RISK TABLE 10.3 Data for Computaton of Volatlty Closng Stock Prce Relatve Daly Return Day Prce (dollars) S S 1 u =ln(s S 1 ) For rsk management purposes, the formula n equaton (10.2) s usually changed n a number of ways: 1. As explaned n Secton 10.1, u s defned as the percentage change n the market varable between the end of day 1 and the end of day so that u = S S 1 S 1 (10.3) Ths makes very lttle dfference to the values of u that are computed. 2. u s assumed to be zero. The justfcaton for ths s that the expected change n a varable n one day s very small when compared wth the standard devaton of changes m 1 s replaced by m. Ths moves us from an unbased estmate of the volatlty to a maxmum lkelhood estmate, as we explan n Secton Ths s lkely to be the case even f the varable happened to ncrease or decrease qute fast durng the m days of our data.

11 Volatlty 211 These three changes allow the formula for the varance rate to be smplfed to σ 2 n = 1 m where u s gven by equaton (10.3). m u 2 (10.4) n =1 EXAMPLE 10.6 Consder agan Example When n = 21 and m = 20, m u 2 n = so that equaton (10.4) gves σ 2 n =1 = = and σ n = or 1.46%. Ths s only a lttle dfferent from the result n Example Weghtng Schemes Equaton (10.4) gves equal weght to each of u 2 n 1, u2 n 2,, and u2 n m. Our objectve s to estmate σ n, the volatlty on day n. It therefore makes sense to gve more weght to recent data. A model that does ths s m σ 2 n = α u 2 (10.5) n =1 The varable α s the amount of weght gven to the observaton days ago. The α s are postve. If we choose them so that α < α j when > j, less weght s gven to older observatons. The weghts must sum to unty, so that m α = 1 =1 An extenson of the dea n equaton (10.5) s to assume that there s a long-run average varance rate and that ths should be gven some weght. Ths leads to the model that takes the form m σ 2 n =γv L + α u 2 (10.6) n where V L s the long-run varance rate and γ s the weght assgned to V L. Because the weghts must sum to unty, we have m γ+ α = 1 =1 =1

12 212 MARKET RISK Ths s known as an ARCH(m) model. It was frst suggested by Engle. 5 The estmate of the varance s based on a long-run average varance and m observatons. The older an observaton, the less weght t s gven. Defnng ω=γv L, the model n equaton (10.6) can be wrtten m σ 2 n =ω+ α u 2 (10.7) n In the next two sectons, we dscuss two mportant approaches to montorng volatlty usng the deas n equatons (10.5) and (10.6). = THE EXPONENTIALLY WEIGHTED MOVING AVERAGE MODEL The exponentally weghted movng average (EWMA) model s a partcular case of the model n equaton (10.5) where the weghts, α, decrease exponentally as we move back through tme. Specfcally, α +1 =λα where λ s a constant between zero and one. It turns out that ths weghtng scheme leads to a partcularly smple formula for updatng volatlty estmates. The formula s σ 2 n =λσ2 n 1 + (1 λ)u2 n 1 (10.8) The estmate, σ n, of the volatlty for day n (made at the end of day n 1) s calculated from σ n 1 (the estmate that was made at the end of day n 2 of the volatlty for day n 1) and u n 1 (the most recent daly percentage change). To understand why equaton (10.8) corresponds to weghts that decrease exponentally, we substtute for σ 2 to get n 1 σ 2 n =λ[λσ2 n 2 + (1 λ)u2 n 2 ] + (1 λ)u2 n 1 or σ 2 n = (1 λ)(u2 n 1 +λu2 n 2 ) +λ2 σ 2 n 2 Substtutng n a smlar way for σ 2 n 2 gves σ 2 n = (1 λ)(u2 n 1 +λu2 n 2 +λ2 u 2 n 3 ) +λ3 σ 2 n 3 5 See R. F. Engle, Autoregressve Condtonal Heteroscedastcty wth Estmates of the Varance of U.K. Inflaton, Econometrca 50 (1982): Robert Engle won the Nobel Prze for Economcs n 2003 for hs work on ARCH models.

13 Volatlty 213 Contnung n ths way, we see that m σ 2 n = (1 λ) λ 1 u 2 n +λm σ 2 n m =1 For a large m, thetermλ m σ 2 n m s suffcently small to be gnored so that equaton (10.8) s the same as equaton (10.5) wth α = (1 λ)λ 1. The weghts for the u declne at rate λ as we move back through tme. Each weght s λ tmes the prevous weght. EXAMPLE 10.7 Suppose that λ s 0.90, the volatlty estmated for a market varable for day n 1s 1% per day, and durng day n 1 the market varable ncreased by 2%. Ths means that σ 2 n 1 = = and u 2 n 1 = = Equaton (10.8) gves σ 2 n = = The estmate of the volatlty for day n, σ n, s, therefore, or 1.14% per day. Note that the expected value of u 2 n 1 s σ2 or In ths example, the n 1 realzed value of u 2 s greater than the expected value, and as a result our volatlty n 1 estmate ncreases. If the realzed value of u 2 had been less than ts expected value, n 1 our estmate of the volatlty would have decreased. The EWMA approach has the attractve feature that the data storage requrements are modest. At any gven tme, we need to remember only the current estmate of the varance rate and the most recent observaton on the value of the market varable. When we get a new observaton on the value of the market varable, we calculate a new daly percentage change and use equaton (10.8) to update our estmate of the varance rate. The old estmate of the varance rate and the old value of the market varable can then be dscarded. The EWMA approach s desgned to track changes n the volatlty. Suppose there s a bg move n the market varable on day n 1 so that u 2 s large. From n 1 equaton (10.8) ths causes our estmate of the current volatlty to move upward. The value of λ governs how responsve the estmate of the daly volatlty s to the most recent daly percentage change. A low value of λ leads to a great deal of weght beng gven to the u 2 n 1 when σ n s calculated. In ths case, the estmates produced for the volatlty on successve days are themselves hghly volatle. A hgh value of λ (.e., a value close to 1.0) produces estmates of the daly volatlty that respond relatvely slowly to new nformaton provded by the daly percentage change. The RskMetrcs database, whch was orgnally created by JPMorgan and made publcly avalable n 1994, used the EWMA model wth λ=0.94 for updatng daly volatlty estmates. The company found that, across a range of dfferent market varables, ths value of λ gves forecasts of the varance rate that come closest to the

14 214 MARKET RISK realzed varance rate. 6 In 2006, RskMetrcs swtched to usng a long memory model. Ths s a model where the weghts assgned to the u 2 as ncreases declne n less fast than n EWMA THE GARCH(1,1) MODEL We now move on to dscuss what s known as the GARCH(1,1) model proposed by Bollerslev n The dfference between the EWMA model and the GARCH(1,1) model s analogous to the dfference between equaton (10.5) and equaton (10.6). In GARCH(1,1), σ 2 n s calculated from a long-run average varance rate, V L,aswell as from σ n 1 and u n 1. The equaton for GARCH(1,1) s σ 2 n =γv L +αu 2 n 1 +βσ2 n 1 (10.9) where γ s the weght assgned to V L, α s the weght assgned to u 2, and β s the n 1 weght assgned to σ 2. Because the weghts must sum to one: n 1 γ+α+β=1 The EWMA model s a partcular case of GARCH(1,1) where γ=0, α=1 λ, and β=λ. The (1,1) n GARCH(1,1) ndcates that σ 2 n s based on the most recent observaton of u 2 and the most recent estmate of the varance rate. The more general GARCH(p, q) model calculates σ 2 n from the most recent p observatons on u2 and the most recent q estmates of the varance rate. 8 GARCH(1,1) s by far the most popular of the GARCH models. Settng ω=γv L, the GARCH(1,1) model can also be wrtten σ 2 n =ω+αu2 n 1 +βσ2 n 1 (10.10) 6 See JPMorgan, RskMetrcs Montor, Fourth Quarter, We wll explan an alternatve (maxmum lkelhood) approach to estmatng parameters later n the chapter. The realzed varance rate on a partcular day was calculated as an equally weghted average of the u 2 on the subsequent 25 days. (See Problem ) 7 See T. Bollerslev, Generalzed Autoregressve Condtonal Heteroscedastcty, Journal of Econometrcs 31 (1986): Other GARCH models have been proposed that ncorporate asymmetrc news. These models are desgned so that σ n depends on the sgn of u n 1. Arguably, the models are more approprate than GARCH(1,1) for equtes. Ths s because the volatlty of an equty s prce tends to be nversely related to the prce so that a negatve u n 1 should have a bgger effect on σ n than the same postve u n 1. For a dscusson of models for handlng asymmetrc news, see D. Nelson, Condtonal Heteroscedastcty and Asset Returns: A New Approach, Econometrca 59 (1990): and R. F. Engle and V. Ng, Measurng and Testng the Impact of News on Volatlty, Journal of Fnance 48 (1993):

15 Volatlty 215 Ths s the form of the model that s usually used for the purposes of estmatng the parameters. Once ω, α, and β have been estmated, we can calculate γ as 1 α β. The long-term varance V L can then be calculated as ω γ. For a stable GARCH(1,1) model, we requre α+β< 1. Otherwse the weght appled to the long-term varance s negatve. EXAMPLE 10.8 Suppose that a GARCH(1,1) model s estmated from daly data as σ 2 n = u2 n σ2 n 1 Ths corresponds to α=0.13, β=0.86, and ω= Because γ=1 α β, t follows that γ=0.01 and because ω=γv L, t follows that V L = In other words, the long-run average varance per day mpled by the model s Ths corresponds to a volatlty of = or 1.4% per day. Suppose that the estmate of the volatlty on day n 1 s 1.6% per day so that σ 2 n 1 = = and that on day n 1 the market varable decreased by 1% so that u 2 n 1 = = Then: σ 2 n = = The new estmate of the volatlty s, therefore, = or 1.53% per day. The Weghts Substtutng for σ 2 n equaton (10.10) we obtan n 1 σ 2 n =ω+αu2 n 1 +β(ω+αu2 n 2 +βσ2 n 2 ) or Substtutng for σ 2 we get n 2 σ 2 n =ω+βω+αu2 n 1 +αβu2 n 2 +β2 σ 2 n 2 σ 2 n =ω+βω+β2 ω+αu 2 n 1 +αβu2 n 2 +αβ2 u 2 n 3 +β3 σ 2 n 3 Contnung n ths way, we see that the weght appled to u 2 n s αβ 1. The weghts declne exponentally at rate β. The parameter β can be nterpreted as a decay rate. It s smlar to λ n the EWMA model. It defnes the relatve mportance of the observatons on the u n determnng the current varance rate. For example, f β=0.9, u 2 n 2 s only 90% as mportant as u2 n 1 ; u2 n 3 s 81% as mportant as u2 ; and so on. n 1 The GARCH(1,1) model s the same as the EWMA model except that, n addton to

16 216 MARKET RISK assgnng weghts that declne exponentally to past u 2, t also assgns some weght to the long-run average varance rate CHOOSING BETWEEN THE MODELS In practce, varance rates do tend to be pulled back to a long-run average level. Ths s the mean reverson phenomenon dscussed n Secton 7.5. The GARCH(1,1) model ncorporates mean reverson whereas the EWMA model does not. GARCH(1,1) s, therefore, theoretcally more appealng than the EWMA model. In the next secton, we wll dscuss how best-ft parameters ω, α, and β n GARCH(1,1) can be estmated. When the parameter ω s zero, the GARCH(1,1) reduces to EWMA. In crcumstances where the best-ft value of ω turns out to be negatve, the GARCH(1,1) model s not stable and t makes sense to swtch to the EWMA model MAXIMUM LIKELIHOOD METHODS It s now approprate to dscuss how the parameters n the models we have been consderng are estmated from hstorcal data. The approach used s known as the maxmum lkelhood method. It nvolves choosng values for the parameters that maxmze the chance (or lkelhood) of the data occurrng. We start wth a very smple example. Suppose that we sample 10 stocks at random on a certan day and fnd that the prce of one of them declned durng the day and the prces of the other nne ether remaned the same or ncreased. What s our best estmate of the proporton of stock prces that declned durng the day? The natural answer s 0.1. Let us see f ths s the result gven by maxmum lkelhood methods. Suppose that the probablty of a prce declne s p. The probablty that one partcular stock declnes n prce and the other nne do not s p(1 p) 9. (There s a probablty p that t wll declne and 1 p that each of the other nne wll not.) Usng the maxmum lkelhood approach, the best estmate of p s the one that maxmzes p(1 p) 9. Dfferentatng ths expresson wth respect to p and settng the result equal to zero, t can be shown that p = 0.1 maxmzes the expresson. The maxmum lkelhood estmate of p s therefore 0.1, as expected. Estmatng a Constant Varance In our next example of maxmum lkelhood methods, we consder the problem of estmatng a varance of a varable X from m observatons on X when the underlyng dstrbuton s normal. We assume that the observatons are u 1, u 2,, u m and that the mean of the underlyng normal dstrbuton s zero. Denote the varance by v. The lkelhood of u beng observed s the probablty densty functon for X when X = u.thss ( ) 1 u 2 exp 2πv 2v

17 Volatlty 217 The lkelhood of m observatons occurrng n the order n whch they are observed s [ m =1 1 2πv exp ( )] u 2 2v (10.11) Usng the maxmum lkelhood method, the best estmate of v s the value that maxmzes ths expresson. Maxmzng an expresson s equvalent to maxmzng the logarthm of the expresson. Takng logarthms of the expresson n equaton (10.11) and gnorng constant multplcatve factors, t can be seen that we wsh to maxmze or [ ] m ln(v) u2 v =1 m ln(v) m =1 u 2 v (10.12) Dfferentatng ths expresson wth respect to v and settng the result equaton to zero, t can be shown that the maxmum lkelhood estmator of v s 1 m m u 2 =1 Ths maxmum lkelhood estmator s the one we used n equaton (10.4). The correspondng unbased estmator s the same wth m replaced by m 1. Estmatng EWMA or GARCH(1,1) We now consder how the maxmum lkelhood method can be used to estmate the parameters when EWMA, GARCH(1,1), or some other volatlty updatng procedure s used. Defne v =σ 2 as the varance estmated for day. Assume that the probablty dstrbuton of u condtonal on the varance s normal. A smlar analyss to the one just gven shows the best parameters are the ones that maxmze [ m =1 1 2πv exp ( )] u 2 2v Takng logarthms we see that ths s equvalent to maxmzng [ ] m ln(v ) u2 v =1 (10.13)

18 218 MARKET RISK TABLE 10.4 Estmaton of Parameters n GARCH(1,1) Model for S&P 500 between July 18, 2005 and August 13, 2010 Date Day S u v =σ 2 ln(v ) u 2 v 18-Jul Jul Jul Jul Jul Jul Aug Aug Aug , Tral Estmates of GARCH parameters ω α β Ths s the same as the expresson n equaton (10.12), except that v s replaced by v. It s necessary to search teratvely to fnd the parameters n the model that maxmze the expresson n equaton (10.13). The spreadsheet n Table 10.4 ndcates how the calculatons could be organzed for the GARCH(1,1) model. The table analyzes data on the S&P 500 between July 18, 2005, and August 13, The numbers n the table are based on tral estmates of the three GARCH(1,1) parameters: ω, α, and β. The frst column n the table records the date. The second column counts the days. The thrd column shows the S&P 500 at the end of day, S. The fourth column shows the proportonal change n the exchange rate between the end of day 1 and the end of day. Thssu = (S S 1 ) S 1. The ffth column shows the estmate of the varance rate, v =σ 2, for day made at the end of day 1. On day three, we start thngs off by settng the varance equal to u 2 2.On subsequent days, equaton (10.10) s used. The sxth column tabulates the lkelhood measure, ln(v ) u 2 v. The values n the ffth and sxth columns are based on the current tral estmates of ω, α, and β. We are nterested n choosng ω, α, and β to maxmze the sum of the numbers n the sxth column. Ths nvolves an teratve search procedure The data and calculatons can be found at www-2.rotman.utoronto.ca/ hull/rmfi4e/ GarchExample. 10 As dscussed later, a general purpose algorthm such as Solver n Mcrosoft s Excel can be used.

19 Volatlty 219 In our example, the optmal values of the parameters turn out to be ω= α= β= and the maxmum value of the functon n equaton (10.13) s 10, (The numbers shown n Table 10.4 are actually those calculated on the fnal teraton of the search for the optmal ω, α, and β.) The long-term varance rate, V L, n our example s ω 1 α β = = The long-term volatlty s or % per day. Fgures 10.4 and 10.5 show the S&P 500 ndex and the GARCH(1,1) volatlty durng the fve-year perod covered by the data. Most of the tme, the volatlty was less than 2% per day, but volatltes over 5% per day were experenced durng the credt crss. (Very hgh volatltes are also ndcated by the VIX ndex see Fgure 10.1.) Jul-05 Jul-06 Jul-07 Jul-08 Jul-09 Jul-10 FIGURE 10.4 S&P 500 Index: July 18, 2005, to August 13, 2010

20 220 MARKET RISK Jul-05 Jul-06 Jul-07 Jul-08 Jul-09 Jul-10 FIGURE 10.5 GARCH(1,1) Daly Volatlty of S&P 500 Index: July 18, 2005, to August 13, 2010 An alternatve more robust approach to estmatng parameters n GARCH(1,1) s known as varance targetng. 11 Ths nvolves settng the long-run average varance rate, V L, equal to the sample varance calculated from the data (or to some other value that s beleved to be reasonable). The value of ω then equals V L (1 α β) and only two parameters have to be estmated. For the data n Table 10.4, the sample varance s , whch gves a daly volatlty of %. Settng V L equal to the sample varance, the values of α and β that maxmze the objectve functon n equaton (10.13) are and , respectvely. The value of the objectve functon s 10, , only margnally below the value of 10, obtaned usng the earler procedure. When the EWMA model s used, the estmaton procedure s relatvely smple. We set ω=0, α=1 λ, and β=λ, and only one parameter, λ, has to be estmated. In the data n Table 10.4, the value of λ that maxmzes the objectve functon n equaton (10.13) s and the value of the objectve functon s 10, For both GARCH(1,1) and EWMA, we can use the Solver routne n Excel to search for the values of the parameters that maxmze the lkelhood functon. The routne works well provded we structure our spreadsheet so that the parameters we are searchng for have roughly equal values. For example, n GARCH(1,1) we could let cells A1, A2, and A3 contan ω 10 5,10α, and β. We could then set B1=A1/100,000, B2=A2/10, and B3=A3. We would then use B1, B2, and B3 for the calculatons, but we would ask Solver to calculate the values of A1, A2, and A3 that 11 See R. Engle and J. Mezrch, GARCH for Groups, Rsk (August 1996):

21 Volatlty 221 maxmze the lkelhood functon. Sometmes, Solver gves a local maxmum, so a number of dfferent startng values for the parameter should be tested. How Good Is the Model? The assumpton underlyng a GARCH model s that volatlty changes wth the passage of tme. Durng some perods, volatlty s relatvely hgh; durng other perods, t s relatvely low. To put ths another way, when u 2 s hgh, there s a tendency for u 2 +1, u2 +2, to be hgh; when u2 s low there s a tendency for u 2 +1, u 2, to be low. We can test how true ths s by examnng the autocorrelaton +2 structure of the u 2. Let us assume that the u 2 do exhbt autocorrelaton. If a GARCH model s workng well, t should remove the autocorrelaton. We can test whether t has done ths by consderng the autocorrelaton structure for the varables u 2 σ2. If these show very lttle autocorrelaton, our model for σ has succeeded n explanng autocorrelatons n the u 2. Table 10.5 shows results for the S&P 500 data. The frst column shows the lags consdered when the autocorrelaton s calculated. The second column shows autocorrelatons for u 2 ; the thrd column shows autocorrelatons for u2 σ2.12 The table shows that the autocorrelatons are postve for u 2 for all lags between 1 and 15. In the case of u 2 σ2, some of the autocorrelatons are postve and some are negatve. They are much smaller n magntude than the autocorrelatons for u 2. The GARCH model appears to have done a good job n explanng the data. For a more scentfc test, we can use what s known as the Ljung-Box statstc. 13 If a certan seres has m observatons the Ljung-Box statstc s m K w k c 2 k k=1 where c k s the autocorrelaton for a lag of k, K s the number of lags consdered, and w k = m + 2 m k For K = 15, zero autocorrelaton can be rejected wth 95% confdence when the Ljung-Box statstc s greater than 25. From Table 10.5, the Ljung-Box Statstc for the u 2 seres s about 1,566. Ths s strong evdence of autocorrelaton. For the u 2 σ2 seres, the Ljung-Box statstc s 21.7, suggestng that the autocorrelaton has been largely removed by the GARCH model. 12 For a seres x, the autocorrelaton wth a lag of k s the coeffcent of correlaton between x and x +k. 13 See G. M. Ljung and G. E. P. Box, On a Measure of Lack of Ft n Tme Seres Models, Bometrca 65 (1978):

22 222 MARKET RISK TABLE 10.5 Autocorrelatons Before and After the Use of a GARCH Model Tme Autocorr Autocorr Lag for u 2 for u 2 σ USING GARCH(1,1) TO FORECAST FUTURE VOLATILITY The varance rate estmated at the end of day n 1 for day n, when GARCH(1,1) s used, s σ 2 n = (1 α β)v L +αu2 n 1 +βσ2 n 1 so that On day n + t n the future, we have σ 2 n V L =α(u2 n 1 V L ) +β(σ2 n 1 V L ) σ 2 n+t V L =α(u2 n+t 1 V L ) +β(σ2 n+t 1 V L ) The expected value of u 2 n+t 1 s σ2 n+t 1. Hence, E[σ 2 n+t V L ] = (α+β)e[σ2 n+t 1 V L ] where E denotes expected value. Usng ths equaton repeatedly yelds E[σ 2 n+t V L ] = (α+β)t (σ 2 n V L )

23 Volatlty 223 Varance rate Varance rate V L V L Tme Tme (a) FIGURE 10.6 Expected Path for the Varance Rate When (a) Current Varance Rate Is above Long-Term Varance Rate and (b) Current Varance Rate Is below Long-Term Varance Rate (b) or E[σ 2 n+t ] = V L + (α+β) t (σ 2 n V L) (10.14) Ths equaton forecasts the volatlty on day n + t usng the nformaton avalable at the end of day n 1. In the EWMA model, α+β=1 and equaton (10.14) shows that the expected future varance rate equals the current varance rate. When α+β< 1, the fnal term n the equaton becomes progressvely smaller as t ncreases. Fgure 10.6 shows the expected path followed by the varance rate for stuatons where the current varance rate s dfferent from V L. As mentoned earler, the varance rate exhbts mean reverson wth a reverson level of V L and a reverson rate of 1 α β.our forecast of the future varance rate tends toward V L as we look further and further ahead. Ths analyss emphaszes the pont that we must have α+β< 1 for a stable GARCH(1,1) process. When α+β> 1, the weght gven to the long-term average varance s negatve and the process s mean fleeng rather than mean revertng. For the S&P 500 data consdered earler, α+β= and V L = Suppose that our estmate of the current varance rate per day s (Ths corresponds to a volatlty of 1.732% per day.) In 10 days, the expected varance rate s ( ) = The expected volatlty per day s or 1.72%, stll well above the long-term volatlty of 1.44% per day. However, the expected varance rate n 500 days s ( ) = and the expected volatlty per day s 1.45%, very close to the long-term volatlty.

24 224 MARKET RISK Volatlty Term Structures Suppose t s day n. Defne and so that equaton (10.14) becomes V(t) = E(σ 2 n+t ) a =ln 1 α+β V(t) = V L + e at [V(0) V L ] V(t) s an estmate of the nstantaneous varance rate n t days. The average varance rate per day between today and tme T s 1 T 0 T V(t)dt = V L + 1 e at [V(0) V at L ] As T ncreases, ths approaches V L. Defne σ(t) as the volatlty per annum that should be used to prce a T-day opton under GARCH(1,1). Assumng 252 days per year, σ(t) 2 s 252 tmes the average varance rate per day, so that { σ(t) 2 = 252 V L + 1 } e at [V(0) V at L ] (10.15) The relatonshp between the volatltes of optons and ther maturtes s referred to as the volatlty term structure. The volatlty term structure s usually calculated from mpled volatltes, but equaton (10.15) provdes an alternatve approach for estmatng t from the GARCH(1,1) model. Although the volatlty term structure estmated from GARCH(1,1) s not the same as that calculated from mpled volatltes, t s often used to predct the way that the actual volatlty term structure wll respond to volatlty changes. When the current volatlty s above the long-term volatlty, the GARCH(1,1) model estmates a downward-slopng volatlty term structure. When the current volatlty s below the long-term volatlty, t estmates an upward-slopng volatlty term structure. In the case of the S&P 500 data, a =ln( ) = and TABLE 10.6 S&P 500 Volatlty Term Structure Predcted from GARCH(1,1) Opton lfe (days) Opton volatlty (% per annum)

25 Volatlty 225 TABLE 10.7 GARCH(1,1) Impact of 1% Increase n the Instantaneous Volatlty Predcted from Opton lfe (days) Increase n volatlty (%) V L = Suppose that the current varance rate per day, V(0), s estmated as per day. It follows from equaton (10.15) that [ σ(t) 2 = ] e t ( ) T where T s measured n days. Table 10.6 shows the volatlty per year for dfferent values of T. Impact of Volatlty Changes Equaton (10.15) can be wrtten as { σ(t) 2 = 252 V L + 1 ( )} e at σ(0) 2 at 252 V L When σ(0) changes by Δσ(0), σ(t) changes by approxmately 1 e at at σ(0) Δσ(0) (10.16) σ(t) Table 10.7 shows the effect of a volatlty change on optons of varyng maturtes for our S&P 500 data. We assume as before that V(0) = so that σ(0) = = 27.50%. The table consders a 100-bass-pont change n the nstantaneous volatlty from 27.50% per year to 28.50% per year. Ths means that Δσ(0) = 0.01 or 1%. Many fnancal nsttutons use analyses such as ths when determnng the exposure of ther books to volatlty changes. Rather than consder an across-the-board ncrease of 1% n mpled volatltes when calculatng vega, they relate the sze of the volatlty ncrease that s consdered to the maturty of the opton. Based on Table 10.7, a 0.97% volatlty ncrease would be consdered for a 10-day opton, a 0.92% ncrease for a 30-day opton, a 0.87% ncrease for a 50-day opton, and so on. SUMMARY In rsk management, the daly volatlty of a market varable s defned as the standard devaton of the percentage daly change n the market varable. The daly varance rate s the square of the daly volatlty. Volatlty tends to be much hgher on tradng days than on nontradng days. As a result, nontradng days are gnored n

26 226 MARKET RISK volatlty calculatons. It s temptng to assume that daly changes n market varables are normally dstrbuted. In fact, ths s far from true. Most market varables have dstrbutons for percentage daly changes wth much heaver tals than the normal dstrbuton. The power law has been found to be a good descrpton of the tals of many dstrbutons that are encountered n practce. Ths chapter has dscussed methods for attemptng to keep track of the current level of volatlty. Defne u as the percentage change n a market varable between the end of day 1 and the end of day. The varance rate of the market varable (that s, the square of ts volatlty) s calculated as a weghted average of the u 2. The key feature of the methods that have been dscussed here s that they do not gve equal weght to the observatons on the u 2. The more recent an observaton, the greater the weght assgned to t. In the EWMA model and the GARCH(1,1) model, the weghts assgned to observatons decrease exponentally as the observatons become older. The GARCH(1,1) model dffers from the EWMA model n that some weght s also assgned to the long-run average varance rate. Both the EWMA and GARCH(1,1) models have structures that enable forecasts of the future level of varance rate to be produced relatvely easly. Maxmum lkelhood methods are usually used to estmate parameters n GARCH(1,1) and smlar models from hstorcal data. These methods nvolve usng an teratve procedure to determne the parameter values that maxmze the chance or lkelhood that the hstorcal data wll occur. Once ts parameters have been determned, a model can be judged by how well t removes autocorrelaton from the u 2. The GARCH(1,1) model can be used to estmate a volatlty for optons from hstorcal data. Ths analyss s often used to calculate the mpact of a shock to volatlty on the mpled volatltes of optons of dfferent maturtes. FURTHER READING On the Causes of Volatlty Fama, E. F. The Behavor of Stock Market Prces. Journal of Busness 38 (January 1965): French, K. R. Stock Returns and the Weekend Effect. Journal of Fnancal Economcs 8 (March 1980): French, K. R., and R. Roll. Stock Return Varances: The Arrval of Informaton and the Reacton of Traders. Journal of Fnancal Economcs 17 (September 1986): Roll, R. Orange Juce and Weather. Amercan Economc Revew 74, no. 5 (December 1984): On GARCH Bollerslev, T. Generalzed Autoregressve Condtonal Heteroscedastcty. Journal of Econometrcs 31 (1986): Cumby, R., S. Fglewsk, and J. Hasbrook. Forecastng Volatltes and Correlatons wth EGARCH Models. Journal of Dervatves 1, no. 2 (Wnter 1993): Engle, R. F. Autoregressve Condtonal Heteroscedastcty wth Estmates of the Varance of U.K. Inflaton. Econometrca 50 (1982): Engle, R. F., and J. Mezrch. Grapplng wth GARCH. Rsk (September 1995):

27 Volatlty 227 Engle, R. F., and V. Ng. Measurng and Testng the Impact of News on Volatlty. Journal of Fnance 48 (1993): Nelson, D. Condtonal Heteroscedastcty and Asset Returns; A New Approach. Econometrca 59 (1990): Noh, J., R. F. Engle, and A. Kane. Forecastng Volatlty and Opton Prces of the S&P 500 Index. Journal of Dervatves 2 (1994): PRACTICE QUESTIONS AND PROBLEMS (ANSWERS AT END OF BOOK) 10.1 The volatlty of an asset s 2% per day. What s the standard devaton of the percentage prce change n three days? 10.2 The volatlty of an asset s 25% per annum. What s the standard devaton of the percentage prce change n one tradng day? Assumng a normal dstrbuton wth zero mean, estmate 95% confdence lmts for the percentage prce change n one day Why do traders assume 252 rather than 365 days n a year when usng volatltes? 10.4 What s mpled volatlty? What does t mean f dfferent optons on the same asset have dfferent mpled volatltes? 10.5 Suppose that observatons on an exchange rate at the end of the past 11 days have been , , , , , , , , , , and Estmate the daly volatlty usng both approaches n Secton The number of vstors to webstes follows the power law n equaton (10.1) wth α=2. Suppose that 1% of stes get 500 or more vstors per day. What percentage of stes get (a) 1,000 and (b) 2,000 or more vstors per day? 10.7 Explan the exponentally weghted movng average (EWMA) model for estmatng volatlty from hstorcal data What s the dfference between the exponentally weghted movng average model and the GARCH(1,1) model for updatng volatltes? 10.9 The most recent estmate of the daly volatlty of an asset s 1.5% and the prce of the asset at the close of tradng yesterday was $ The parameter λ n the EWMA model s Suppose that the prce of the asset at the close of tradng today s $ How wll ths cause the volatlty to be updated by the EWMA model? A company uses an EWMA model for forecastng volatlty. It decdes to change the parameter λ from 0.95 to Explan the lkely mpact on the forecasts Assume that an ndex at close of tradng yesterday was 1,040 and the daly volatlty of the ndex was estmated as 1% per day at that tme. The parameters n a GARCH(1,1) model are ω= , α=0.06, and β=0.92. If the level of the ndex at close of tradng today s 1,060, what s the new volatlty estmate? The most recent estmate of the daly volatlty of the dollar sterlng exchange rate s 0.6% and the exchange rate at 4:00 p.m. yesterday was The parameter λ n the EWMA model s 0.9. Suppose that the exchange rate at

28 228 MARKET RISK 4:00 p.m. today proves to be How would the estmate of the daly volatlty be updated? A company uses the GARCH(1,1) model for updatng volatlty. The three parameters are ω, α, and β. Descrbe the mpact of makng a small ncrease n each of the parameters whle keepng the others fxed The parameters of a GARCH(1,1) model are estmated as ω= , α=0.05, and β=0.92. What s the long-run average volatlty and what s the equaton descrbng the way that the varance rate reverts to ts long-run average? If the current volatlty s 20% per year, what s the expected volatlty n 20 days? Suppose that the daly volatlty of the FTSE 100 stock ndex (measured n pounds sterlng) s 1.8% and the daly volatlty of the dollar sterlng exchange rate s 0.9%. Suppose further that the correlaton between the FTSE 100 and the dollar sterlng exchange rate s 0.4. What s the volatlty of the FTSE 100 when t s translated to U.S. dollars? Assume that the dollar sterlng exchange rate s expressed as the number of U.S. dollars per pound sterlng. (Hnt: When Z = XY, the percentage daly change n Z s approxmately equal to the percentage daly change n X plus the percentage daly change n Y.) Suppose that GARCH(1,1) parameters have been estmated as ω= , α=0.04, and β=0.94. The current daly volatlty s estmated to be 1%. Estmate the daly volatlty n 30 days Suppose that GARCH(1,1) parameters have been estmated as ω= , α=0.04, and β=0.94. The current daly volatlty s estmated to be 1.3%. Estmate the volatlty per annum that should be used to prce a 20-day opton. FURTHER QUESTIONS Suppose that observatons on a stock prce (n dollars) at the end of each of 15 consecutve days are as follows: 30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 30.9, 30.5, 31.1, 31.3, 30.8, 30.3, 29.9, 29.8 Estmate the daly volatlty usng both approaches n Secton Suppose that the prce of an asset at close of tradng yesterday was $300 and ts volatlty was estmated as 1.3% per day. The prce at the close of tradng today s $298. Update the volatlty estmate usng (a) The EWMA model wth λ=0.94 (b) The GARCH(1,1) model wth ω= , α=0.04, and β= An Excel spreadsheet contanng over 900 days of daly data on a number of dfferent exchange rates and stock ndces can be downloaded from the author s webste: www-2.rotman.utoronto.ca/ hull/data. Choose one exchange rate and one stock ndex. Estmate the value of λ n the EWMA model that mnmzes the value of (v β ) 2