u?/o7.!> The tableshows that the autocorrelations are positive for u} for all lags between 1 and 15. In the case of« |e. some of the autocorrelations are positive and some are negative. They are all much smallerin magnitude than the autocorrelations for u}.
The GARCH model appears to have done a good job in explaining the data. For a morescientific test, we can use what is known as the Ljung—Box statistic. If a certain series has mobservations the Ljung-Box statistic is
Km » wentkat
where 7, is the autocorrelation for a lag of & and
m—2"6 nk
8 Hor a series x;, the autocorrelation with a lag of k is the coefficient of correlation between x; and x;4,4.382 CHAPTER 17
Table 17.2 Autocorrelations before and after the useof a GARCH model
Time Autocorrelation Autocorrelationlag for u? for u?/o?1 0.072 0.0042 0.041 —0.0053 0.057 0.0084 0.107 0.0035 0.075 0.0166 0.066 0.0087 0.019 —0.0338 9.085 0.0129 0.054 0.01010 0.030 —0.02311 0.038 —0.00412 0.038 —0.02113 0.057 —0.00114 0.040 0.002i) 0.007 —0,028
For k = 15, zero autocorrelation can be rejected with 95% confidence when the Ljung—Box statisticis greater than 25.
From Table 17.2, the Ljung—Box statistic for the ur series is about 123. This is strong evidence ofautocorrelation. For the uw fo? series the Ljung-Box statistic is 8.2, suggesting that the auto-correlation has been largely removed by the GARCH model.
17.6 USING GARCH(1,1) TO FORECAST FUTURE VOLATILITY
 
Substituting y = | — a — f in equation (17.8), the variance rate estimated at the end of day n—1for day n is
5, = (1a = PV, +auy 1 + Bonsso thaton — Vp = (un) — Vi) + Boni — Vi)
On day n+ in the future, we have2 — fy 2Snyk — Vi = Un ge-1 — Ve) + Blonye-1 — Vi)The expected value of Usk is Oo yk: Hence,
E(ojse — Vz) = (0 + B)E(on 44-1 — Vi)Estimating Volatilities and Correlations 383
whete £ denotes the expected value. Using this equation repeatedly yields
Elon se — Vi) = (@ + BY (on — Vi)or(onyx) = Vi + (@+ B02 — Vi) (37.13)
This equation forecasts the volatility on day n +k using the information available at the end ofday n— 1. In the EWMA model, w + 8 = 1 and equation (17.13) shows that the expected futurevariance rate equals the current variance rate. When a + f < 1, the final term in the equationbecomes progressively smaller as k increases. Figure 17.2 shows the expected path followed by thevariance rate for situations where the current variance rate is different from V,. As mentionedearlier, the variance rate exhibits mean reversion with a reversion level of V, and a reversion rate of1 —a@— B. Our forecast of the future variance rate tends towards V; as we look further and furtherahead. This analysis emphasizes the point that we must have a + 8 < | for a stable GARCH(, 1)process. When a + 6 > 1, the weight given to the long-term average variance is negative and theprocess is ‘‘mean fleeing” rather than ‘“‘mean reverting”’.
In the yen-dollar exchange rate example considered earlier a+ fB=0.9602 andV, = 0.00004422. Suppose that our estimate of the current variance rate per day is 0.00006. (Thiscorresponds to a volatility of 0.77% per day.) In 10 days the expected variance rate is
0.00004422 + 0.9602!°(0.00006 — 0.00004422) = 0.00005473
The expected volatility per day is 0.74%, which is only marginally below the current volatility of0.77% per day. However, the expected variance rate in 100 days is
0.00004422 + 0.9602'(0.00006 — 0.00004422) = 0.00004449The expected volatility per day is 0.67%, very close the long-term volatility.
Variancerate
 
 
Time Time
(a) )
Figure 17.2 Expected path for the variance rate when (a) current variance rate is above long-termvariance rate and (b) current variance rate is below long-term variance rate384 CHAPTER 17
Volatility Term Structures
Consider an option lasting between day n and day n + N. We can use equation (17.13) to calculatethe expected variance rate during the life of the option as
1 N-1 3wee E(o7 44) (17.14)
The longer the life of the option, the closer this is to V,.
As we discussed in Chapter 15, the market prices of different options on the same asset are oftenused to calculate a volatility term structure. This is the relationship between the implied volatilitiesof the options and their maturities. Equation (17.14) can be used to estimate a volatility termstructure based on the GARCH(1, 1} model. The square root of the expression in equation (17.14)
is an estimate of the volatility appropriate for valuing the N-day option. The estimated volatility
term structure is not usually the same as the actual volatility term structure. However, as we willshow, it is often used to predict the way that the actual volatility term structure will respond tovolatility changes.
When the current volatility is above the long-term volatility, the GARCH{(I, 1) model estimates adownward-sloping volatility term structure. When the current volatility is below the long-termvolatility, it estimates an upward-sloping volatility term structure. Table 17.3 shows the results ofusing equation (17.14) to estimate a volatility term structure for options on the yen-dollarexchange rate when the current estimate of the variance rate is 0.00006. For the purposes of thistable, the daily volatilities have been annualized by multiplying them by /252. Consider, forexample, a 10-day option. The estimated variance rate is 0.00006 for the first day. Similarcalculations to those given earlier in this section show that the estimated variance rate is
0.00004422 + 0.9602(0.00006 — 0.00004422) = 0.00005937for the second day,0.00004422 + 0.96027(0.00006 — 0.00004422) = 0.00005879
for the third day, and so on. The average variance rate over 10 days is7q(0.00006 + 0.00005937 + 0.00005879 + - - -) = 0.00005745The estimated volatility is therefore V0.00005745 x ¥252 = 12.03%.
Impact of Volatility Changes
Table 17.4 shows the effect of a volatility change on options of varying maturities for our yen-dollar exchange rate example. We assume as before that the current variance rate is 0.00006 perday. This corresponds to a daily volatility of 0.7746% and a volatility of 0.7746./252 = 12.30% peryear. We will refer to this as the instantaneous volatility. A 1% increase changes the instantaneous
Table 17.3 Yen-dollar volatility term structure predicted from GARCH(I, 1)
Ostion life (dave) 6 an Pa an 0GOption life (days) 10 30 56 100 500
 
Option volatility (% per annum) 12.03 11.61 11.35 11.01 10.65Estimating Volatilities and Correlations 385
Table 17.4 Impact of 1% change in the instantaneous volatility predictedfrom GARCH(1. 1)
 
Option life (days) 10 30 = 50—s«s100—S 500Option volatility now (%) 12.03 11.61 11.35 11.01 10.65Option volatility after change (%) 12.89 12.25 11.83 11.29 10.71Increase in volatility (%) 0.86 0.64 0.48 0.28 0.06
volatility to 13.30% per year. The daily volatility becomes 0.1330//252 = 0.84% and the currentvariance rate per day becomes 0.00847 = 0.00007016.
Consider the calculation of the 10-day volatility after the 1% increase in the instantaneousvolatility. The variance rate is 0.00007016 for the first day. Equation (17.13) gives an estimatedvariance rate of
0.00004422 + 0.9602(0.00007016 — 0.00004422) = 0.00006913
for the second day,0.00004422 + 0.96027(0.00007016 — 0.00004422) = 0.00006814
for the third day, and so on. The average variance rate over 10 days is, from equation (17.14),74(0.00007016 + 0.00006913 + 0.000068 14 + - - -) = 0.00006597
The estimated volatility is therefore //0.00006597 x /252 = 12.89%. The increase in volatility fordifferent maturities is shown in the third row of Table 17.4.
Many financial institutions use analyses such as this when determining the exposure of theirbooks to volatility changes. Rather than consider an across-the-board increase of 1% in impliedvolatilities when calculating vega, they relate the size of the volatility increase that is considered tothe maturity of the option. Based on Table 17.4, a 0.86% volatility increase would be consideredfor a 10-day option, a 0.64% increase for a 30-day option, a 0.48% increase for a 50-day option,and so on.
17.7_ CORRELATIONS
 
 
The discussion so far has centered on the estimation and forecasting of volatility. As explained inChapter 16, correlations also play a key role in the calculation of VaR. In this section, we showhow correlation estimates can be updated in a similar way to volatility estimates.
The correlation between two variables X and Y can be defined as
cov(X, Y)oyoy
where oy and oy arc the standard deviation of X and Y and cov(X, Y) is the covariance between Xand Y. The covariance between X and F is defined as
EU(X — tx)Y — by)386 CHAPTER 17
where jry and jy are the means of X and Y, and F denotes the expected value. Although it is easierto develop intuition about the meaning of a correlation than it is for a covariance, it is covariancesthat are the fundamental variables of our analysis.'*Define x; and y; as the percentage changes in X and Y between the end of day / — 1 and the endof day i:x, = eX y = MedXi} Yi-y
where X; and Y; are the values of X and Y at the end of day i. We also define:
o,,,: Daily volatility of variable X, estimated for day n
xn
oy,,: Daily volatility of variable Y, estimated for day n
cov, : Estimate of covariance between daily changes in X and Y, calculated on day nOur estimate of the correlation between X and Y on day n is
cov,
FxnFy,n
 
Using an equal-weighting scheme and assuming that the means of x; and y, are zero, equation (17.3)shows that we can estimate the variance rates of X and Y from the most recent m observations as
1
> If. » If.On = mann Fn = maw Yn-iA similar estimate for the covariance between X and Y is1COVn =D Mat Yas (17.15)i=l
One alternative is an EWMA model similar to equation (17.7). The formula for updating thecovariance estimate is then
cov, = ACov,_| +(1 _ A)Xn-1 Ya-1
A similar analysis to that presented for the EWMA volatility model shows that the weights given toobservations on the x;y,’s decline as we move back through time. The lower the value of A, thegreater the weight that is given to recent observations.
Example 17.3 Suppose that 4=0.95 and that the estimate of the correlation between twovariables X and Y on day n — 1 is 0.6. Suppose further that the estimate of the volatilities for theX and Y on day n— 1 are 1% and 2%, respectively. From the relationship between correlation andcovariance, the estimate of the covariance between the X and Y on day n— 1 is
0.6 x 0.01 x 0.02 = 0.00012
4 An analogy here is that variance rates were the fundamental variables for the EWMA and GARCH schemes infirst part of this chapter, even though volatilities are easier to understand.Estimating Volatilities and Correlations 387
 
Suppose that the percentage changes in X and Y on day n — 1 are 0.5% and 2.5%, respectively. Thevariance and covariance for day n would be updated as follows:
Oz = 0.95 x 0.01? +.0.05 x 0.005? = 0.00009625
55 = 0.95 x 0.02? + 0.05 x 0.025? = 0.00041125
cov, = 0.95 x 0.00012 + 0.05 x 0.005 x 0.025 = 0.00012025
 
The new volatility of X is /0.00009625 = 0.981% and the new volatility of Y is /0.00041125 =2.028%. The new coefficient of correlation between X and Y is
0.00012025
0.00981 x 0.02028 0°04
GARCH models can also be used for updating covariance estimates and forecasting the futurelevel of covariances. For example, the GARCH(1, 1) model for updating a covariance is
COV, = W+ AX, 1 Vy) + BCOV,_}crm average covariance is w/(1—a— f). Formulas similar to those im equa-tions (17.13) and (17.14) can be developed for forecasting future covariances and calculating theaverage covariance during the life of an option.!°
Consistency Condition for Covariances
Once all the variances and covariances have been calculated, a variance-covariance matrix can beconstructed. When i # j, the (7, j) element of this matrix shows the covariance between variable iand variable j. When i = j, it shows the variance of variable i.
Not all variance-covariance matrices are internally consistent. The condition for an N x Nyariance-covariance matrix, Q, to be internally consistent is
w'Qw > 0 (17.16)
for all N x 1 vectors w, where w" is the transpose of w. A matrix that satisfies this property isknown as positive semidefinite.To understand why the condition in equation (17.16) must hold, suppose that
w’ =[wy, Wr... W,)-
The expression w" Qw is the variance of a portfolio consisting of w; of market variable i. As such, itmust be positive or zero.
To ensure that a positive-semidefinite matrix is produced, variances and covariances should becalculated consistently. For example, if variances are calculated by giving equal weight to the lastm data items, the same should be done for covariances. If variances are updated using an EWMAmodel with A = 0.94, the same should be done for covariances.
'S An extension of the ideas in this chapter is to multivariate GARCH models where an entire variance-covariancematrix is updated in a consistent way. See R. Engle and J. Mezrich. “GARCH for Groups,” RISK, August 1996,pp. 36-40, for a discussion of alternative approaches.388 CHAPTER 17
 
An example of a variance-covariance matrix that is not internally consistent is
1 0 090 1 0.90.9 0.9 1
The variance of each variable is 1.0, and so the covariances are also coefficients of correlation. Thefirst variable is highly correlated with the third variable and the second variable is highly correlatedwith the third variable. However, there is no correlation at all between the first and secondvariables. This seems strange. When we set w equal to (1,1,—1), we find that the condition inequation (17.16) is not satisfied, proving that the matrix is not positive semidefinite.'®
SUMMARY
Most popular option pricing models, such as Black-Scholes, assume that the volatility of theunderlying asset is constant. This assumption is far from perfect. In practice, the volatility of anasset, like the asset’s price, is a stochastic variable. Unlike the asset price, it is not directly observable.This chapter has discussed schemes for attempting to keep track of the current level of volatility.
We define u; as the percentage change in a market variable between the end of day i — 1 and theend of day i. The variance rate of the market variable (i.¢., the square of its volatility) is calculatedas a weighted average of the u?. The key feature of the schemes that have been discussed here is thatthey do not give equal weight to the observations on the u?. The more recent an observation, thegreater the weight assigned to it. In the EWMA model and the GARCH(1, 1) model, the weightsassigned to observations decrease exponentially as the observations become older. TheGARCH(I, 1) model differs from the EWMA model in that some weight is also assigned to thelong-run average variance rate. Both the EWMA and GARCH(1, 1) models have structures thatenable forecasts of the future level of variance rate to be produced relatively easily.
Maximum likelihood methods are usually used to estimate parameters in GARCH(1, 1) andsimilar models from historical data. These methods involve using an iterative procedure todetermine the parameter values that maximize the chance or likelihood that the historical datawill occur. Once its parameters have been determined, a model can be judged by how well itremoves autocorrelation from the w?.
For every model that is developed to track variances, there is a corresponding model that can bedeveloped to track covariances. The procedures described here can therefore be used to update thecomplete variance-covariance matrix used in value-at-risk calculations.
SUGGESTIONS FOR FURTHER READING
 
Bollerslev, T., “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics,31 (1986), 307-27.‘© It can be shown that the condition for a 3 x 3 matrix of correlations to be internally consistent is
biz + pis + P33 — 2p121304 < 1
where ;; is the coefficient of correlation between variables i and j.Estimating Volatilities and Correlations 389
Cumby, R., S. Figlewski, and J. Hasbrook, ‘‘Forecasting Volatilities and Correlations with EGARCHModels,” Journal of Derivatives, 1, no. 2 (Winter 1993), 51-63.
Engle, R. F., “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UKInflation,” Econometrica, 50 (1982), 987-1008.
Engle R. F., and J. Mezrich, “Grappling with GARCH,” RISK, September 1995, pp. 112-117.Engle, R. F., and J. Mezrich. “GARCH for Groups,” RISK, August 1996, pp. 36-40.
Engle, R. F., and V. Ng, “Measuring and Testing the Impact of News on Volatility,” Journal ofFinance, 48 (1993), 1749-78.
Ljung, G. M., and G. E. P. Box, “On a Measure of Lack of Fit in Time Series Models,” Biometrica,65 (1978), 297-303.
Nelson, D., “Conditional Heteroscedasticity and Asset Returns: A New Approach,” Econometrica,59 (1990), 347-70.
Noh, J., R. F. Engle, and A. Kane, “Forecasting Volatility and Option Prices of the S&P 500 Index,”Journal of Derivatives, 2 (1994), 17-30.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
 
17.1. Explain the exponentially weighted moving average (EWMA) model for estimating volatilityfrom historical data.
17.2. What is the difference between the exponentially weighted moving average model and theGARCH(I, 1) model for updating volatilities?
17.3. The most recent estimate of the daily volatility of an asset is 1.5% and the price of the asset atthe close of trading yesterday was $30.00. The parameter 4 in the EWMA model is 0.94. Supposethat the price of the asset at the close of trading today is $30.50. How will this cause the volatilityto be updated by the EWMA model?
17.4. A company uses an EWMA model for forecasting volatility. It decides to change the parameter 4from 0.95 to 0.85. Explain the likely impact on the forecasts.
17.5. The volatility of a certain market variable is 30% per annum. Calculate a 99% confidenceinterval for the size of the percentage daily change in the variable.
17.6. A company uses the GARCH(I, 1) model for updating volatility. The three parameters are @, a,and £. Describe the impact of making a small increase in each of the parameters while keepingthe others fixed.
17.7. The most recent estimate of the daily volatility of the USD-GBP exchange rate is 0.6% and theexchange rate at 4 p.m. yesterday was 1.5000. The parameter 4 in the EWMA model is 0.9.Suppose that the exchange rate at 4 p.m. today proves to be 1.4950. How would the estimate ofthe daily volatility be updated?
17.8. Assume that S&P 500 at close of trading yesterday was 1,040 and the daily volatility of theindex was estimated as 1% per day at that time. The parameters in a GARCH(I, 1) model arew = 0.000002, a = 0.06, and 6 =0.92. If the level of the index at close of trading today is1,060, what is the new volatility estimate?390
17.9.
17.10.
17.11.
17.12.
17.13.
17.14.
CHAPTER 17
Suppose that the current daily volatilities of asset A and asset B are 1.6% and 2.5%, respectively.The prices of the assets at close of trading yesterday were $20 and $40 and the estimate of thecoefficient of correlation between the returns on the two assets made at that time was 0.25. Theparameter 4 used in the EWMA model is 0.95.a. Calculate the current estimate of the covariance between the assets.b. On the assumption that the prices of the assets at close of trading today are $20.5 and $40.5,update the correlation estimate.
The parameters of a GARCH(I, 1) model are estimated as w = 0.000004, a = 0.05, and 6 = 0.92.What is the long-run average volatility and what is the equation describing the way that thevariance rate reverts to its long-run average? If the current volatility is 20% per year, what is theexpected volatility in 20 days? ’
Suppose that the current daily volatilities of asset X and asset Y are 1.0% and 1.2%,respectively. The prices of the assets at close of trading yesterday were $30 and $50 and theestimate of the coefficient of correlation between the returns on the two assets made at this timewas 0.50. Correlations and volatilities are updated using a GARCH(1, 1) model. The estimates ofthe model’s parameters are a = 0.04 and 8 = 0.94. For the correlation # = 0.000001, and for thevolatilities » = 0.000003. If the prices of the two assets at close of trading today are $31 and $51,how is the correlation estimate updated?
Suppose that the daily volatility of the FTSE 100 stock index (measured in pounds sterling) is1.8% and the daily volatility of the USD-GBP exchange rate is 0.9%. Suppose further that thecorrelation between the FTSE 100 and the USD-GBP exchange rate is 0.4. What is the volatilityof the FTSE 100 when it is translated to U.S. dollars? Assume that the USD-GBP exchange rateis expressed as the number of U.S. dollars per pound sterling. (Hint: When Z = XY, thepercentage daily change in Z is approximately equal to the percentage daily change in X plus thepercentage daily change in Y.)
Suppose that in Problem 17.12 the correlation between the S&P 500 index (measured in dollars)and the FTSE 100 index (measured in sterling) is 0.7, the correlation between the S&P 500 index(measured in dollars) and the USD-GBP exchange rate is 0.3, and the daily volatility of theS&P 500 index is 1.6%. What is the correlation between the S&P 500 index (measured in dollars)and the FTSE 100 index when it is translated to dollars? ({7iné: For three variables X, Y, and Z,the covariance between X + Y and Z equals the covariance between X and Z plus the covariancebetween Y and Z.)
Show that the GARCH(1, 1) model
o2 =a +auz_, + or_,in equation (17.9) is equivalent to the stochastic volatility modeldV =a(V, — V)dt+éVdz
where time is measured in days, V is the variance of the asset price, and
 
a=l1-a-f, Vi elk E=av2
 
 
What is the stochastic volatility model when time is measure in years? (Hint: The variable u,,_, isthe return on the asset price in time 8. It can be assumed to be normally distributed with meanzero and standard deviation o,,_,. It follows that the means of wy and uy are ony and 3041,respectively.)Estimating Volatilities and Correlations 391
 
f
ASSIGNMENT QUESTIONS
 
 
17.15.
17.16.
17.17.
17.18.
Suppose that the current price of gold at close of trading yesterday was $300 and its volatilitywas estimated as 1.3% per day. The price at the close of trading today is $298. Update thevolatility estimate using
a. The EWMA model with A = 0.94
b. The GARCH(1, 1) model with # = 0.000002, a = 0.04, and B = 0.94Suppose that in Problem 17.15 the price of silver at the close of trading yesterday was $8, itsvolatility was estimated as 1.5% per day, and its correlation with gold was estimated as 0.8, Theprice of silver at the close of trading today is unchanged at $8. Update the volatility of silver andthe correlation between silver and gold using the two models in Problem 17.14. In practice, is the@ parameter likely to be the same for gold and silver?
An Excel spreadsheet containing 500 days of daily data on a number of different exchange ratesand stock indices can be downloaded from the author’s Web site:www.rotman.utoronto.ca/“hullChoose one exchange rate and one stock index. Estimate the value of A in the EWMA modelthat minimizes the value ofYo _ ByL
where v; is the variance forecast made at the end of day i— 1 and 8; is the variance calculated
from data between day i and day i + 25. Use the Solver tool in Excel. Set the variance forecast at
the end of the first day equal to the square of the return on that day to start the EWMA
calculations.
Suppose that the parameters ina GARCH(1, 1) model are a = 0.03, 6 = 0.95, and w = 0.000002.a. What is the long-run average volatility?b. If the current volatility is 1.5% per day, what is your estimate of the volatility in 20, 40,and 60 days?
.