Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

WDS'09 Proceedings of Contributed Papers, Part I, 148 153, 2009. ISBN 978-80-7378-101-9 MATFYZPRESS Volatility Modelling L. Jarešová Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. In this paper the Black Scholes implied volatility and the GARCH volatility are compared. The implied volatility is used in the form of volatility indices. Empirical study of the performance of GARCH in forecasting the implied volatility shows that the GARCH volatility can be used to forecast the implied volatility and is better than the historical volatility. Two different volatility indices (VIX, VDAX) of two different market indices (SPX index in United States and DAX index in Germany) were considered in the empirical study. Introduction The word volatility means rapid or unexpected changes. During the last decades the meaning has become more diversified according to who uses it. For traders volatility means opportunity, because with more volatile markets there is a higher probability of bigger profits whereas with low volatility trading resembles dull sport. For risk managers volatility means danger, because with more volatile markets there is also a higher probability of bigger losses, connected with the fact that mathematical models widely used for risk management could underestimate the risk in the portfolio. For financial theorists volatility often means some parameter or process in mathematical models that has to be chosen to fit the real world properly. Since volatility is directly unobservable, it is no wonder that volatility modelling is a very big challenge. Nowadays the number of volatility models is enormous and is still increasing. Market volatility is currently one of the most investigated phenomena in finance. The intention of this paper is to compare the implied Black Scholes volatility with the GARCH volatility. The Black Scholes Volatility In the well known Black Scholes (BS) model [Black and Scholes, 1973] it is assumed that the price of an underlying asset S t follows the geometric Brownian motion given by ds t = µs t dt + σs t dw t, (1) where drift µ is expected return on the asset, volatility σ measures the variability around µ and W t is the standard Brownian motion. Important implication of this assumption is that the logarithm of the ending price at time T > t is equal to ln(s T ) = ln(s t ) + (µ σ 2 /2)(T t) + σ T t ɛ, (2) where ɛ is the standard normal random variable. This means that S T given S t is log normally distributed. Further it follows that the logarithmic returns r t = ln(s t /S t1 ) are normally distributed. This result is however not supported by empirical data studies. Financial time series of logarithmic returns exhibit typical properties [Franke et al., 2007]. Returns r t have in general a leptokurtic distribution, the kurtosis is mostly much greater than 3 that clearly rejects the normal distribution. Typical shape of the distribution is highly peaked and fat-tailed that is characteristic to mixtures of distributions with different variances. Time series of returns tend to form clusters. This suggests that the volatility (or variance) is autocorrelated whilst the time series of returns itself are often uncorrelated. 148

Assuming additionally that markets are perfect (world with no arbitrage opportunities, no costs and taxes, no restrictions on short selling and with all securities perfectly divisible) and asset yield r and volatility σ are known and constant, the fair price of a European call option in the BS model is given by (assuming no dividend yield) c = S N(d 1 ) K e r(t t) N(d 2 ), (3) where N(d) is the standard normal distribution function, S is the spot price of the underlying asset, K is the strike price and ( ) d 1 = ln S/(K e r(t t) ) σ + σ T t, d 2 = d 1 σ T t. (4) T t 2 All variables except for volatility are directly observable in the market. But the volatility is a very fundamental parameter that affects the option value c a lot and is crucial for the correct risk evaluation. More on option pricing and practical risk assessment can be found in [Hull, 2008] or [Wilmott, 2006]. Option prices can be, contrary to volatility, directly observed in the market. Using BS equation in the opposite way we can calculate the volatility that yields a theoretical value of the option c equal to the current market price of that option. Such volatility is called the implied volatility. Implied volatility in general depends (in contrast to BS assumption) on the strike price K and the expiration of the option T t (time to maturity). The collection of all such implied volatilities is known as the volatility surface. For more understanding of volatility surfaces look in [Gatheral, 2006] or [Rebonato, 2004]. Volatility Indices The implied volatility contains important information about the current investor sentiment in the market that from principle is not included in the historical prices of the asset. Every mathematical model that uses just the historical prices is therefore disadvantaged. The question is how big is this disadvantage and what mistake can one make by just using the historical prices of the asset. Implied volatility is a very good financial indicator of the fear in the market since it often signifies financial turmoil. Some exchanges have transformed this information in volatility indices. The most known index is the VIX index of the CBOE 1 launched in 2003 (data begins 1990). The CBOE utilizes a wide variety of strike prices of options on the S&P 500 index (SPX index, the core index for U.S. equities) in order to obtain the estimator of 30-day expected volatility. More details about the calculation are described in the methodology [CBOE and Goldman Sachs]. VDAX-NEW index (launched in 2005 as a successor for VDAX launched in 1996, data begins 1992) is an analogous index of the Deutsche Börse in Germany based on the prices of options on the German DAX index. In last years the volatility indices are becoming more and more popular and new indices emerge. In 2004 exchange traded volatility futures started and in 2006 even exchange traded options on volatility started (both on the VIX index, today are available futures and options on more volatility indices). GARCH Models The ARCH (AutoRegressive Conditional Heteroskedasticity) model was first introduced by [Engle, 1982] and generalized to GARCH (Generalized ARCH) by [Bollerslev, 1986]. Models of these types are nowadays very popular and many other generalizations were derived. Summary 1 Chicago Board Options Exchange 149

of them can be found in [Bollerslev, 2008] (more than 100 models). New frontiers for research are summarized in [Engle, 2002] Process {r t }, t Z is a (strong) GARCH(p, q), if E[r t F t 1 ] = 0 (the conditional mean is unpredictable) and Var[r t F t 1 ] = σ 2 t (the conditional variance is time dependent), where q p σt 2 = ω + α i rt i 2 + β j σt j 2 (5) i=1 j=1 and Z t = r t /σ t are i.i.d. random variables. F t 1 denotes the σ-algebra generated by the historical returns, α i and β j are real coefficients. ARCH(q) is a special type of GARCH(p, q) where β j = 0 for j = 1,..., p. Sufficient condition for σt 2 0 is ω, α i, β j 0. If E[rt 2 ] = σ 2 <, then the unconditional variance satisfies σ 2 = ω/(1 α i β j ). Persistence of GARCH model is defined as α i + β j and is less than 1, if the unconditional variance exists. Financial data usually has high persistence (near 1) and ω near 0. If we additionally assume E[rt 4 ] = c <, then {η t = rt 2 σt 2 } is a white noise WN(0, σ 2 ) and {rt 2 } is an ARMA(max(p, q), p) process (ARMA representation of a GARCH process). GARCH(1, 1) In the following the conditional variance has the form σt 2 = ω + αrt 1 2 + βσ2 t 1. If the unconditional variance exists and Z t = r t /σ t are standard normal random variables, then E[rt 4 ] <, if it holds that 3α 2 + 2αβ + β 2 < 1. The kurtosis is then given as Kurt(r t ) = E[r4 t ] (E[rt 2 = 3 + 6α 2 ])2 1 β 2 3. (6) 2αβ 3α2 Hence the distribution of returns that follow the GARCH(1, 1) is leptokurtic even if the variables Z t = r t /σ t are standard normal. Table 1. GARCH(1, 1) estimated parameters. Index Market ˆω ˆα ˆβ Persistence Volatility p.a. Kurt(rt ) kurt(z t ) SPX USA 0.000 0.076 0.916 0.992 18% 15.129 4.633 *** *** *** DAX Germany 0.000 0.102 0.884 0.987 22% 13.675 4.219 ** *** *** P-value: 0 *** 0.1% ** 1% * 5%. 10% The parameters of the GARCH(1, 1) fitted on the mentioned indices SPX and DAX are in the Table 1. Estimation was done on 5 year history till 23.4.2009 in R using the fgarch library. For information about R see [R Development Core Team, 2006]. Volatility p.a. is the annualized estimated volatility given by ω/(1 Persistence) 260. Volatility Indices vs. GARCH volatility On the Figure 1 there is a comparison of the index returns, volatility indices and the estimated volatility ˆσ t = ˆω + ˆαr t 1 2 + ˆβˆσ t 1 2. It is very interesting that the graph with volatility index and with the corresponding GARCH estimated volatility are very similar. It is important to realize that the volatility indices are based on the actual prices of traded options and the GARCH model takes into account only the historical returns. Another interesting observation is that the path of volatility index in USA (VIX) is very similar to the path of volatility index in Germany (VDAX). These two indices are indeed very 150

SPX Index and volatility VIX Index DAX Index and volatility VDAX Index GARCH volatility 20 40 60 80 GARCH volatility 10 20 30 40 50 60 70 80 VIX index SPX returns 10 20 30 40 50 60 70 80 0.10 0.05 0.00 0.05 0.10 VDAX index DAX returns 10 20 30 40 50 60 70 0.05 0.00 0.05 0.10 1992 1994 1996 1998 2000 2002 2004 2006 2008 Time 1992 1994 1996 1998 2000 2002 2004 2006 2008 Time Figure 1. Logarithmic returns time series of SPX and DAX indices, volatility indices VIX and VDAX and estimated volatility σ t by a GARCH(1, 1) model. dependent, the correlation of the values is 0.86 and the correlation of absolute changes in the values is 0.38. But the index VIX is based on US equities and the index DAX on different German equities. This shows that the perception of fear among investors is global and that different parts of the world are definitely not independent. Performance of GARCH in Forecasting Volatility For the estimation shown in Figure 1, however, the whole time series of SPX and DAX indices from 1.1.1992 to 23.4.2009 were used. But on each day only the history is available, so this estimation, although with very nice results, is a little bit cheating. For modelling and application it is important, how successful GARCH is in forecasting volatility using only the historical information available today. In the following empirical study t is equal to working days from 3.5.2002 to 24.4.2009, σ t denotes the value of volatility index. For each t, the 3 year history available at t is taken and the parameters ω, α and β are estimated; ˆσ t 2 = ω + ˆαr t 1 2 + ˆβˆσ t 1 2 is the estimated GARCH volatility on the day t, ˆσ t 2 (1) = ω + ˆαr t 2 + ˆβˆσ t 2 is the forecast of GARCH volatility one day after t. As a benchmark for comparison the historical 3 month volatility s t (standard deviation of daily log-returns during the last 3 months) is used. The historical volatility is often used in practice when the implied volatility is not available. To make the values comparable, all of the mentioned volatility estimators except for volatility index were annualized by the factor 260. In the following analysis the annualized values are used without the change of notation. The aim of the study is to explain the value of volatility index σ t. For the evaluation the linear regression models of the form σ t = Intercept + Coefficient V olatility estimator + Error term (7) were used, where volatility estimators are taken as ˆσ t, ˆσ t (1), s t and σ t 1 (the value of the volatility index on the previous day). 151

Table 2. Comparison of the regression models used to predict the todays volatility. Index Model Intercept Coefficient s.e. R 2 SPX index σ t 1 + ˆσ t 5.59% 0.85 3.85% 88.43% vol. index VIX σ t 1 + ˆσ t (1) 5.52% 0.85 3.72% 89.21% σ t 1 + s t 6.06% 0.82 4.49% 84.24% σ t 1 + σ t 1 0.27% 0.99 1.80% 97.47% DAX index σ t 1 + ˆσ t 5.89% 0.78 4.14% 86.71% vol. index VDAX σ t 1 + ˆσ t (1) 5.81% 0.78 4.03% 87.44% σ t 1 + s t 5.97% 0.77 4.63% 83.36% σ t 1 + σ t 1 0.21% 0.99 1.51% 98.23% The results are shown in the Table 2, where the short notation for linear regression model (7) is used (1 stands for a constant included in the model). The best predictor of today s volatility is the value of the volatility index yesterday, explains 97% and 98% of the variability in the volatility index. GARCH estimated volatility today ˆσ t explains 88% and 87% of the variability in the volatility index. Even better in the prediction of the volatility today σ t is the GARCH forecast of tomorrows volatility ˆσ t (1) that explains 89% and 87% of the variability in the volatility index. The historical volatility is in both cases the worst predictor and explains 84% and 83% of the variability in the volatility index. Closer to Normality It is well known that normal distribution is not very appropriate for financial time series of returns, but it is often used in practice because of its simplicity and nice statistical properties. The main problem in using normal distribution to model returns is that in normal world it is almost nothing going on farther than 3 standard deviations, but in finance, there are quite a lot of outliers suggesting that the tails have to be fat. Using the normal distribution in quantitative methods can therefore induce a false feeling of safety. The typical shape of the distribution of returns is characteristic to mixture of distribution of different variances. Therefore it is reasonable to standardize the returns with the timedependent volatility to make the shape of the distribution more normal. How it works is shown on the Figure 2, where the time series from 3.5.2002 to 24.4.2009 were used. On the left two graphs, there are QQ-plots of index log-returns. The kurtosis of SPX index returns is 11.22 and the kurtosis of DAX index returns is 7.62. If these returns are standardized by the estimated GARCH volatility ˆσ t or the volatility from volatility indices, then the QQ-plot becomes more linear and the kurtosis becomes significantly smaller (4.18 (4.08) for SPX (DAX) returns standardizes by GARCH volatility and 3.53 (3.44) for SPX (DAX) returns standardizes by GARCH volatility). These values are closer to the value 3 of the kurtosis of the normal distribution, but they are still high enough to reject normality in statistical tests. We can see from QQ-plots, that the improvement in the shape of the distribution to normal distribution is large. Conclusion In this paper two types of volatility were compared. First type was the implied volatility that is the consequence of wide usage of the imperfect BS model. Implied volatility contains the information about risk anticipated in the market and was used in the form of volatility indices. Second type was the estimated GARCH volatility that uses only the information available in the historical prices. In the empirical study it was shown that implied volatility and GARCH estimated volatility are very similar and that GARCH volatility is better in forecasting the implied market volatility than the commonly used historical volatility. This can be used in applications in cases, where 152

Figure 2. QQ plots of index log-returns, log-returns divided by estimated GARCH volatility and log-returns divided by the volatility from volatility indices. the implied volatility is not available, but is needed to price some derivatives or to evaluate the risk. References Black, F. and Scholes, M., The pricing of options and corporate liabilities, The Journal of Political Economy, 81, 637 654, 1973. Bollerslev, T., Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307 327, 1986. Bollerslev, T., Glossary to ARCH (GARCH), Tech. Rep. 49, CREATES, research paper available at ftp://ftp.econ.au.dk/creates/rp/08/rp08 49.pdf, 2008. CBOE and Goldman Sachs, The CBOE volatility index - VIX, calculation methodology is available at http://www.cboe.com/micro/vix/vixwhite.pdf. Engle, R., New frontiers for ARCH models, J. Appl. Econ., 17, 425 446, 2002. Engle, R. F., Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation, Econometrica, 50, 987 1007, 1982. Franke, J., Härdle, W. K., and Hafner, C. M., Statistics of Financial Markets: An Introduction, Springer, 2007. Gatheral, J., The Volatility Surface, John Wiley and Sons, 2006. Hull, J. C., Options, Futures, and Other Derivatives (7th Ed.), Prentice Hall, 2008. R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, URL http://www.r-project.org, ISBN 3-900051-07-0, 2006. Rebonato, R., Volatility and Correlation, John Wiley and Sons, 2004. Wilmott, P., Paul Wilmott on Quantitative Finance (2nd Ed.), John Wiley and Sons, 2006. 153