Estimation of Stochastic Volatility Models with Implied. Volatility Indices and Pricing of Straddle Option

Transcription

1 Estimation of Stochastic Volatility Models with Implied Volatility Indices and Pricing of Straddle Option Yue Peng Steven C. J. Simon June 14, 29 Abstract Recent market turmoil has made it clear that modelling market volatility is more than ever a necessity. We calibrate three different stochastic volatility models to five stock market indices (FTSE 1, S&P 5, AEX, CAC 4, and BEL 2) and their respective implied volatility indices (VFTSE, VIX, VAEX, VCAC, and VBEL) using maximumlikelihood estimation. The volatility indices are used as proxies for the instantaneous volatility of the five stock indices. We find that there is a clear difference in goodness of fit for the different models, and that the estimated parameters differ significantly across these five markets. Next, we test whether the difference in statistical goodness-offit between the models is also economically significant by pricing at-the-money forward straddle options, a derivative which provides a good hedge against volatility risk. We find that the choice of dynamics for the stochastic volatility has indeed a clear impact on the price of such a derivative. Keywords: Implied Volatility Index, VIX, maximum-likelihood, CEV model, GARCH model, Heston model, Straddle Option JEL Codes: G13, C22 Corresponding author, ypengq@essex.ac.uk, Center for Computational Finance and Economic Agents, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK. Steven C. J. Simon, steven_cj_simon@yahoo.com, Center for Computational Finance and Economic Agents, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK. 1

2 1 Introduction When considering portfolios of derivatives the evolution of the volatility of the underlying asset is even more important than the change in the level of the underlying. Typically, derivatives are at least as sensitive to the volatility of the underlying assets as the level of the underlying itself. Moreover, volatility risk, also known as vega risk, is more difficult to hedge. Hence modelling the dynamics of volatility has become more and more important. In this paper, we calibrate 3 different stochastic volatility (SV) models (Heston, GARCH and CEV) to 5 stock markets (AEX, BEL 2, CAC 4, FTSE 1, and S&P 5), using the maximum-likelihood approach of Sahalia (27), whereby the (observable) implied volatility index is used as a proxy for the true but unobserved instantaneous volatility of the stock index. The implied volatility index is a market expectation of short-term future volatility, based on prices of short-term options. Whaley (2) mentioned that such a volatility index is a gauge to measure investor s confidence in the market. Combined with its underlying market index, it is widely used to forecast future market movements. The idea of a volatility index based on exchange traded options was first proposed in Gastineau (1977) constructs a volatility index using an average of at-the-money implied volatilities from call options on 14 stocks with three to six months maturities, joint with historical stock volatility. Cox, Ingersoll, and Ross (1985) improve this methodology by several call options on every stock in the index to construct at-the-money implied volatility on a constant time horizon. Brenner and Galai (1989) construct volatility indices for equity, bond and foreign exchange markets from a combination of historical and implied volatility. They find that this combination performs better than when only either of the two types of volatility is used. In Jan 1993, the Chicago Board Option Exchange (CBOE) launched and disseminated a Market Volatility Index (VXO) using the methodology described in Whaley (1993). 1 The VXO is based on 3-day at-the-money implied volatility from S&P 1 index options. In Sep 23, the CBOE introduced a new methodology suggested by Carr and Madan (1998), and Demeter, Derman, Kamal, and Zou (1999), and launched a new volatility index (VIX) based on calendar 3-day implied volatility from S&P 5 index options. This new volatility index can capture the whole volatility skew in the stock market, and gives a more precise 1 It was called VIX in Jan 1993, then it was renamed as VXO after new VIX launched in Sep 23. 2

3 expectation of short term future market volatility. Following these two methodologies, there are many other volatility indices calculated and disseminated in different stock markets. In this paper we use five implied volatility indices. They are VIX of S&P 5 stock market in US, VFTSE of FTSE 1 stock market in UK, VCAC of CAC 4 stock market in France, VAEX of AEX stock market in Netherlands, and VBEL of BEL 2 stock market in Belgium. All these five volatility indices are constructed following the new CBOE methodology for the VIX index. For instance, they all use a maturity time of 3 calendar days. In our paper, the results from calibrating the three stochastic volatility models to these five stock indices show that the Heston model is rejected for all five markets, whereas the GARCH model is only rejected for the S&P 5, the AEX, and the CAC 4 indices. We then test whether the difference in statistical fitting of these three models translates into a significant difference in implied derivative prices. As our focus is on the modelling of volatility, we price at-the-money forward straddle options, introduced by Brenner, Ou, and Zhang (26). In a first step, the results of a sensitivity analysis show that the price of such an option is indeed quite sensitive to the choice between these three different models. Finally, we generate prices for the straddle option for both the FTSE 1 and the S&P 5 indices based on the results of the maximum-likelihood estimation. The results confirm that for S&P 5 index the GARCH and Heston models give significantly different prices to the CEV model, whereas for the FTSE 1 index the prices implied by the GARCH model are quite close to those of the CEV model. That is, the difference in statistical fit translates into a non-trivial difference in option prices. The reminder of our paper is organized as follows: Section 2 briefly introduces the three different stochastic volatility (SV) models. Section 3 discusses the data and descriptive statistics. Section 4 gives the methodology of the maximum likelihood estimation for these three SV model. Section 5 discusses the results of the maximum likelihood estimation for the five stock indices. In Section 6 we introduce the at-the-money forward straddle option, and we calculate prices with the stochastic volatility models calibrated in Section 5. 3

4 2 Three Stochastic Volatility Models It is well known that the volatility of asset prices is not constant. Both local volatility and stochastic volatility models have become popular approaches to deal with time varying volatility. Hull and White (1987) are the first to propose a SV model. In their model, volatility process is assumed to be independent of the asset price process. Here we briefly introduce three different volatility models which are widely used in the current literature, and which are used in this paper. The Heston Model In the model of Heston (1993) the variance of the asset price follows a square-root process, as in the term structure model of Cox, Ingersoll, and Ross (1985). The risk-neutral dynamics of the logarithm of the asset price s t = ln(s t ) and its variance rate v t are: ds t = [r t d t v t /2] dt + (1 ρ 2 )v t dw Q 1 (t) + ρ v t dw Q 1 (t) (1) dv t = δ v ( v v t )dt + k v vt dw Q 2 (t), (2) with v the long run mean of the variance rate, k v its volatility, and δ v its mean-reversion speed. Finally, the instantaneous correlation between the asset process and its variance rate is equal to ρ. W Q means these two stochastic processes are under risk-neutral measure Q. Here both the spot rate r t and the dividend yield d t can be time-dependent, but not stochastic. The Heston variance process is non-negative and mean-reverting. But its conditional distribution function is not known in closed form. Carr and Madan (1999) derive the characteristic function. As such, the transform method can be applied to get the value of European options. As in Ait-Sahalia and Kimmel (27), we assume that the market price of risk of volatility is equal to zero, and that the market price of risk of the stock index is proportional to the stock index itself. Under these assumptions, the physical dynamics are given by: ds t = [ r t d t + [ λ 1 (1 ρ 2 ) 1/2 ] v t ] dt + (1 ρ 2 )v t dw P 1 (t) + ρ v t dw P 1 (t) (3) dv t = δ v ( v v t )dt + k v vt dw P 2 (t), (4) 4

5 where, W P means these two stochastic processes are under risk-averse measure P. GARCH Model Bollerslev (1986) developed the generalized auto regressive conditional heteroscedasticity (GARCH) model, which Nelson (199) and Meddahi (21) apply to dynamic asset pricing. They show that the GARCH (1,1) model is actually equal to a stochastic volatility model. In continuous time the risk-neutral dynamics of the logarithm of the asset s t = ln(s t ) and its variance rate v t are: ds t = [r t d t v t /2] dt + (1 ρ 2 )v t dw Q 1 (t) + ρ v t dw Q 1 (t) (5) dv t = δ v ( v v t )dt + k v v t dw Q 2 (t), (6) Making, again as Ait-Sahalia and Kimmel (27) the same assumption about the market price of risk of both s t and v t as above, the physical dynamics are given by: ds t = [ r t d t + [ λ 1 (1 ρ 2 ) 1/2 ] v t ] dt + (1 ρ 2 )v t dw P 1 (t) + ρ v t dw P 1 (t) (7) dv t = δ v ( v v t )dt + k v v t dw P 2 (t), (8) Constant Elasticity of Variance model Chacko and Viceira (1999), Lewis (2), and Jones (23) assume that the variance rate follows a so-called "constant elasticity of variance" process, already used by Chan, Karolyi, Longstaff, and Sanders (1992) to model interest rate dynamics. The risk-neural dynamics of s t and the variance rate process v t are given by: ds t = [r t d t v t /2] dt + (1 ρ 2 )v t dw Q 1 (t) + ρ v t dw Q 1 (t) (9) dv t = δ v ( v v t )dt + k v v β t dw Q 2 (t), (1) This model includes the previous two model as special cases. The model of Heston (1993) is obtained when β is set equal to.5, whereas the GARCH model is obtained when β = 1. 5

6 Jones (23) shows that the CEV model captures the dynamics of the S&P1 index better than both other models. Again, under the assumption, as made by Ait-Sahalia and Kimmel (27), that the market price of risk of volatility is zero, and the market price of risk of the asset level is proportional to the asset itself, the physical dynamics are given by: ds t = [ r t d t + [ λ 1 (1 ρ 2 ) 1/2 ] v t ] dt + (1 ρ 2 )v t dw P 1 (t) + ρ v t dw P 1 (t)) (11) dv t = δ v ( v v t )dt + k v v β t dw P 2 (t), (12) 3 Data and Descriptive Statistics Our dataset consists of closing levels of five stock market indices and the corresponding implied volatility indices. The stock indices are : AEX (Netherlands), BEL 2 (Belgium), CAC 4, (France), FTSE 1 (UK), and S&P 5 (US). The dataset spans the period January 4, 2 until July 23, 28. The closing levels of the indices, risk-free rates and dividend yields were downloaded from Datastream. The implied volatility index of VIX is available on the website of the CBOE ( and the other four European implied volatility indices, VAEX, VBEL, VCAC and VFTSE, are obtained from the website of Euronext ( Figure 1 shows that the indices of these five stock markets have very similar trends. Figure 2 shows that this is also the case for the corresponding implied volatilities. 2 Although we are not interested in the historical volatility process itself, it would be interesting to compare the evolution of the implied volatility series with their historical counterparts. As Poteshman (2) we use the 3-day annualized realized volatility as a measure for realized volatility: RVt,t+3 = 3 i=3 ( S t+i 1) 365 S 2 t+i 1 i=1 Figure 3 shows that for each of the five stock markets the realized volatility index and the implied volatility index follow rather similar patterns. From Figure 2 and Figure 3 it appears 2 We divide the volatility indices by 1 to make them comparable to the realized volatility. The construction of volatility index refers to the CBOE White Paper. 6

7 that the AEX stock index is the most volatile. Both its implied volatility and realized volatility exceed 6% in the period from mid 22 to early 23. Table 1 and Table 2 give the descriptive statistics for respectively the five implied volatility indices and the corresponding realized volatility series. Both tables suggest that the AEX and the CAC 4 index are the most volatile, as both their implied volatilities and its realized volatilities have the highest maximum, mean and median of the five indices. All implied volatility and realized volatility series have an excess kurtosis of more than three, implying that they are all fat-tailed. Also, for all five series, the mean implied volatility is higher than the mean realized volatility. A possible explanation could be that, compared to the realized volatility, the implied volatility series contain a risk premium. Table 3 gives the correlations between the daily index returns, and Table 4 gives the correlations between the daily levels of the implied volatility indices. From Table 3 we see that both the BEL 2 and the S&P 5 indices are less correlated with the other indices. With the correlation between these two indices themselves being the lowest of all. We find a similar pattern in Table 4 for the implied volatility indices. On each row the correlations involving the implied volatility indices for the BEL 2 and the S&P 5 indices are the two lowest, and the correlation between these two is again the lowest of all. 4 Maximum-Likelihood Estimation Following Ait-Sahalia and Kimmel (27), we calibrate the three stochastic volatility models introduced in Section 2 to the data set described in section 3 with Maximum-Likelihood estimation. The two state variables in either of the three models are the logarithm of the level of stock index price s(t), and the level of the volatility index divided by 1 v(t). The true joint likelihood function for the state variable x t = [s t, v t ] is unknown for each of these three models. Therefore we use closed-form expansions of the joint log-likelihood functions instead of the true (but unknown) log-likelihood functions. Let us introduce the logarithm of the conditional density of x tn, given x tn 1 and the parameter 7

8 vector θ: L ( x tn x tn 1, θ ) = ln [ f ( x tn x tn 1, θ )]. (13) where n = 1...N, which denotes the time steps. t is the initial time of the data series, and t N is the end time of the data series. Ait-Sahalia and Kimmel (27) derive the following closed-form expansion of order k of L ( x tn x tn 1, θ ) : L (k) ( x tn x tn 1, θ ) = M 2 ln(2π t n) 1 2 ln(det[v(x t n ; θ)]) + C( 1) x (x tn x tn 1 ; θ) + t n k i= C (i) x (x tn x tn 1 ; θ) ti n, i! where, v(x tn ) = σ(x tn )σ(x tn ). The order k can be arbitrary. C x (i), i = 1,, 1..., k are unknown coefficients. Each C x (i), can be approximated with a Taylor series in (x tn x tn 1 ) of order l i = 2k 2i. The approximation of the coefficient is denoted by C (l,i) x. Following the results of Ait-Sahalia (21), Ait-Sahalia and Kimmel (27) gives the expressions for the C (l,i) x coefficients of order k = 1 for the Heston, GARCH and CEV models. This allows us to obtain the following closed-form expansion of the logarithm of the conditional density of x(t n ) for each of these three stochastic volatility models. L (k) ( x tn x tn 1, θ ) = M 2 ln(2π t n) 1 2 ln(det[v(x t n ; θ)]) The expressions for the coefficients C (l i,i) x (27). + C(l 1, 1) x (x tn x tn 1 ; θ) + t n k i= C (l i,i) x (x tn x tn 1 ; θ) ti n. i! are given in Appendix A of Ait-Sahalia and Kimmel Finally, the log-likelihood function that is maximized in the calibration is given by: N L (k) ( x tn x tn 1, θ ). n=1 8

9 5 Results Tables 5 to 7 give the results for the three stochastic volatility models. The first column of ) every table gives the estimated parameter values, where L (ˆθ is the value of log likelihood function. The columns Est. give the estimated values of the parameters. The columns s.e. give the corresponding standard errors. With respect to the extend to which the three models perform in fitting the data, we see that the values of the log likelihood function of the GARCH model and the CEV model are fairly close for all five markets. 3. In contrast, the Heston model gives much lower values for the log likelihood function for all markets. Looking at the accuracy of the individual parameter estimates one sees that those parameters that only affect the drift term (δ v, vλ) have very large standard errors. Across all markets and models none of them are significant. In contrast, the parameters that affect the diffusion terms (k v, ρ, β) are always significant. We see that for the Belgian stock index the correlation ρ is much lower and far less significant for all three models, than it is for the other markets. As such, there does not seem to be a clear leverage effect for the Belgian stock index in the period from 4th June 2 to 23rd July 28. As the Heston and the GARCH model are nested in the more general CEV model, testing whether the stock index dynamics are those of the Heston or the GARCH model comes down to testing the hypothesis that β = 1/2 or β = 1, respectively. Table 8 gives the results for the likelihood ratio test and the Wald test for the Heston model, and Table 9 gives the results for the GARCH model. Both tables report the values of the test statistics, as well as p-values (between brackets). We see that, for any meaningfull confidence level, the Heston model is clearly rejected for all five markets. However, the results for the GARCH hypothesis in Table 9 show a different picture. We see that the hypothesis that β = 1 is only rejected for the BEL 2 and the S&P 5 indices. Observe that we have not calculated any p-values for likelihood-ratio test for the (ˆθ) 3 There are several L values in GARCH slightly higher than CEV. This is due to the approximation error of expansions of log likelihood functions for GARCH and CEV, and the error from numerical optimizations of maximum likelihood method. 9

10 AEX, CAC 4, and FTSE 1 indices, as the the likelihood ratio exceeds one in these cases. However, the results of the Wald test clearly suggest that the GARCH hypothesis can not be rejected for these three stock market indices. For the S&P 5 index these results are in line with the results of Ait-Sahalia and Kimmel (27), who find that both the Heston and the GARCH model are rejected for the S&P 5 index and its implied volatility index. Furthermore, we see that the two markets for which the GARCH hypothesis is rejected are also those for which the implied volatility indices are the least correlated with those of the other four markets. Moreover, from the descriptive statistics in Table 1 we see that the implied volatility indices of VBEL and VIX have a level of excess kurtosis and skewness clearly below these of the other three market volatility indices. As such, the fact that the GARCH hypothesis is rejected for the BEL 2 index and the S&P 5 index, is most likely not an artefact of our methodology, but rather a reflection of a real difference is dynamics across the five implied volatility indices. 6 ATM Forward Straddle Option After the CBOE introduced an implied volatility index in 1993, several derivatives on volatility have been proposed in the literature, e.g. futures and options on volatility index, a volatility swap or a variance swap. However, as a volatility index is not traded, such derivatives are difficult to price and hedge. In an early contribution to the literature Whaley (1993) assumes that the volatility index follows a geometric Brownian motion process, and derives closedform expressions for the price of European call options on a volatility index. Grunbichler and Longstaff (1996) assume that the volatility index follows a mean-reverting square-root process, and give closed-form solutions for European call options on the volatility index. Recently, Brenner, Ou, and Zhang (26) suggested another type of volatility derivative: atthe-money forward (ATMF) straddle options (). An, Assaf, and Yang (26) show that the ATMF is essentially equal to an volatility index option, and they find that it gives better hedging results than other hedging strategies. The underlying of the ATMF is an ATMF straddle, which is traded in the market, and with which traders are quite familiar. 1

11 6.1 Construction of ATMF Before discussing the at-the-money-forward straddle option (ATMF ), we introduce the at-the money straddle. A straddle ST t is constructed by a call option Π C t and a put option Π P t with the same strike price K S and the same maturity time T, on the same underlying asset S t :ST t = Π C t + Π P t. At the time of maturity, the pay-off of a straddle is equal to ST T = S T K S. When an investor expects the underlying to change from its current value but he is not sure about the direction of the change, he can go long a straddle. That is, a long position in a straddle is actually a position in the volatility of the underlying asset. Figure 4 shows the pay-off of a straddle. When the options involved are European, we can use the results of Black and Scholes (1973) to obtain the price of a straddle at time t as: ST t = S t (2N(φ 1 ) 1) K S e (rt dt)(t t) (2N(φ 2 ) 1), (14) where: φ 1 = ln(s t/k S ) + ((r t d t ) σ2 )(T t) σ, T t and: φ 2 = ln(s t/k S ) + ((r t d t ) 1 2 σ2 )(T t) σ T t = φ 1 σ T t. Here, r t is the risk-free rate and d t the dividend yield. From equation (14) we see that next to the volatility σ, the value of a straddle is also affected by the following factors: S t, r t and d t. Brenner and Subrahmanyam (1988) show that under the BS assumptions the price of an at-the-money European option is almost linear in the stock price. Therefore, under the BS assumption the price of an ATM straddle is also almost linear in the stock price. That is, we have: ST AT M t = 2e (rt dt)(t t) S t (2N(φ 1 ) 1) = 2e (rt dt)(t t) S t σ T t 2π, (15) 11

12 with: φ 1 = 1 2 σ T t. Dividing equation (15) by the stock price S t, we get: STt AT M /S t = 2e (rt dt)(t t) σ T t, 2π which is only affected by the volatility σ, the interest rate r t and the dividend yield d t. 4 In order to avoid an impact from the interest rate or the dividend yield, we construct an at-the-money forward (ATMF) straddle ST AT MF. For such an ATMF straddle (sold at time t) the strike price is equal to the forward price F t,t. That is: K S = F t,t. The ATMF straddle and the forward F t,t have the same maturity time T. Under the BS assumptions, the price of an ATMF straddle is: AT MF STt = 2S t (2N(φ 1 ) 1) = 2S tσ T t 2π. (16) The price of this instrument is approximately linear in the volatility and the asset price. The next step is to construct a European option on the ATMF straddle: an ATMF straddle option ST O t. The option can be exercised at a future point in time T 1, in which case an ATMF straddle is received in return for the strike price K ST O. This ATMF straddle expires at a future point in time T 2 (T 2 > T 1 ). 6.2 The value of Straddle Option in CEV Model: Sensitivity Analysis Here, we price the straddle option with the CEV model using Monte Carlo simulation, with 5, sample paths and a total number of time steps equal to 252. Our interest mainly goes to the sensitivity of the ATMF straddle to the instantaneous correlation ρ and the variance parameter β. Figures 5, 6, and 7 show the results. Figure 5 shows the sensitivity of the option price to the correlation ρ when β = 1, Figure 6 shows the same sensitivity when β =.8, and the sensitivity to the correlation when β =.5 is shown in Figure 7. 4 In theory, the ATM call or ATM put has the same value and properties as the ATM Straddle. But in reality, they have different transaction costs. Moreover, in reality the option is not always exactly ATM. So only use ATM call or put will cause a more obvious bias in hedging. 12

13 Each of the figures exists out of two rows of plots. The top row shows three plots of option prices for different values for the starting level of the instantaneous variance and the meanreversion speed, for fixed values for β and ρ. The bottom row gives 2D versions of the same 3 plots, which give a clearer view of the different sensitivities. Figures 5 to 7 suggest that both the correlation ρ and the variance parameter, β have a significant impact on the value of the straddle option. Both the range of the straddle prices as well as the shape of the price surface differ strongly across the various combinations for β and ρ. A first, and striking, observation is the effect of a negative instantaneous correlation. For all three assumptions about the value of β, this choice for the value of ρ leads to straddle prices far larger than when ρ is assumed to be zero or strictly positive. For instance, Figure 6 shows that when β =.8, setting ρ equal to.6 gives straddle option prices which are almost always larger than 3, with prices going as high as 12. Whereas for ρ = or ρ =.7, straddle option prices are always below 2. It is well known that the assumption of a positive correlation between the stock index and its volatility is not supported by much empirical evidence. Rather, market data provides strong evidence for the so-called leverage effect: a negative correlation between stock returns and volatility. Also, our own results of the ML estimation provide no evidence of a positive correlation. To see why the hypothesis that ρ > leads to such high prices for the straddle option, we need to go back to the pay-off of the straddle option. The value of the straddle, which is the pay-off of the straddle option, is increasing in both the level of the stock index and the volatility of the index. If these two underlying variables are positively correlated, the price of the straddle is very volatile, and the value of an option on such a straddle is high. If we assume a non-positive correlation between the level of the stock index and its volatility, there is some form of diversification between the two factors driving the value of the straddle. Which leads to a lower volatility for the price of the straddle, and therefore a lower value of the straddle option. Just as the assumption of a negative correlation leads to option prices which stand out from the rest, also the choice of β =.5 (Heston model) seems to be a special case. Both for β = 1, Figure 5, and β =.8, Figure 6, we see that the shape of the price surface is rather different for each of the three assumptions about the correlation ρ. In contrast, when β =.5 the 13

14 shape of the price surface changes very little with the correlation. For instance, with β =.5 the price of the straddle option is always increasing in the mean-reversion speed κ v of the variance rate. In contrast, when β = 1 or β =.8 we see that the relation between the price of the straddle option and the mean-reversion speed κ v clearly depends on the level of the correlation ρ. Again, notice that the assumption β =.5 is rejected for all five stock market indices. Finally, if we restrict ourselves to the possibilities β > and ρ, we see that going from ρ = to ρ < has a different impact when β = 1 compared with when β =.8. In the first case increasing the mean-reversion speed κ v almost always leads to lower prices for the straddle options. In contrast, for β =.8 this effect is only found when ρ =. When ρ =.7 we see that the relation between the price of the straddle option and the mean-reversion speed κ v is not monotonic. For β =.8 and ρ =.7 we see that for all but the lowest starting value for the variance rate, increasing κ v initially reduces the price of the straddle option. However, for the different starting values the relation between the option price and the mean-reversion speed becomes increasing somewhere in the range 1 to 1.5. Observe that a value for κ v larger than 1 corresponds to overshooting of the mean-reversion level. When β = 1 and ρ =.7 overshooting has no impact on the relation between the price of a straddle option and the mean-reversion speed, as this relation is decreasing for all values for κ v. In contrast, when β =.8 and ρ =.7 we see that this relation always becomes increasing when overshooting occurs. 6.3 ATMF Straddles and Volatility in Different Stock Markets Here we calculate 3 calendar day ATMF straddle prices with the CEV model using Monte Carlo simulation 5 from 4th January 2 to 23rd July 28. On each day, we assume that the instantaneous variance is equal to the square of implied volatility index divided by 1. The upper part of Figure 8 shows the ATMF straddle price in FTSE 1 stock market from 4th January 2 to 23rd July 28, and lower part shows FTSE 1 index with its implied 5 The number of observation sample paths used here is 5, and the number of totally time steps is 3. It can be noticed that we use calendar days here for real market data in consistent with our ML estimation and CBOE model free volatility index calculation, while we used trading days for previous numerical analysis of ATMF. 14

15 volatility index VFTSE (The flat time series is index level.). This figure shows that the ATMF straddle series has a pattern very similar to that of the VFTSE index. When the FTSE 1 index and the VFTSE index both have high values, e.g. begin of 2 and around 28, ATMF straddle prices are especially high. When the VFTSE is high but the FTSE 1 has low values, e.g. the end of 22 and 23, these two facts have an opposite impact on the ATMF straddle prices. Under the BS assumptions we can derive the implied volatility from ATMF straddle from equation (16) as: σ AT MF ST AT M = t 2π. (17) 2S t T t This measure for volatility uses the same methodology as the CBOE BS at-the-money volatility index. So we can expect σ AT MF to have a behavior similar to at-the-money volatility index. 6 Figure 9 shows the series for σ AT MF calculated with equation (17) together with the VFTSE index, as well as the difference(σ AT MF ST V F T SE). We see that σ AT MF and the VFTSE index indeed show a very similar pattern. Their differences are not very large, most of them are between.2 and.2. Interestingly, the difference(σ AT MF ST V F T SE) tends to be negative when VFTSE is positive and vice versa. Which suggest that the VFTSE index is more volatile than σ AT MF ST. Figure 1 shows the differences series of σ AT MF ST V olind for the other 4 stock markets. We see that the differences series are smaller for FTSE 1 and S&P 5, and bigger for BEL 2, which suggests that our model works better for FTSE 1 and S&P 5 stock market, but is not very suitable to BEL 2 stock index. This result is consistent with our previous ML estimation results for the CEV model. We found in Section 5 that the log likelihood function for the the CEV model is the highest for the FTSE 1 and S&P 5 stock indices, and the lowest for the BEL 2 index. 6 We use model free volatility index to estimated CEV SV model parameters and also use it as initial variance to estimate ATMF straddle values. So it can be understand that we found σ AT MF which estimated here is more close to model free volatility index series. 15

16 6.4 ATMF Straddle Option under CEV Model in Stock Markets Figure 11 gives the ATMF straddle option for the FTSE 1 market index on 12rd March 23 and 31st October The volatility index for 12rd March 23 is.485 and for 31st October 27 it is.212. Prices were calculated with the CEV model. The various parameter values are taken from Table 7. The number of sample paths in the Monte Carlo simulation is 5,, and the number of time steps is always 365 (independent of the actual time to maturity). The option values are calculated for different strike prices and times to maturity (t t ). The strike prices range from to 2 percent of the current index level. The time to maturity varies from (1 day) to.5 (half a year). The value of the straddle option decreases with maturity on both days. But the values drop more steeply on 12rd March 23 as volatility is higher on that day. Figures 12 to 15 give the ATMF straddle option prices calculated with the CEV model for the S&P 5, AEX, BEL 2 and CAC 4 indices. We find that the results are very similar across the different markets. 6.5 Pricing ATMF Straddle Options on the FTSE 1 and the S&P 5 Figure 16 compares the prices for straddle options on the FTSE 1 index obtained with the GARCH model with the prices implied by the CEV model, and Figure 17 compares the prices implied by the Heston model with those of the CEV model. Again, all parameter values are taken from Section 5. Comparing these two figures, we see that the Heston model almost always overshoots the option price on both days, with the difference going well above 8% on March 12 for some of the options. In contrast, the differences between the prices of the GARCH model and those of the CEV model are far smaller, and they can be of either sign. This result confirms the outcome of the hypothesis testing in Section 5, where the Heston model was clearly rejected for the FTSE 1 index, but the GARCH model was not rejected. 7 On 12rd Marh 23 FTSE 1 index reaches its lowest value of and on 31st October 27 FTSE 1 reaches its highest value of from 4th January 2 to 23rd July

17 In Figure 18 and Figure 19, we make the same comparison between these three different models for the S&P 5 index and find a different result. As with the FTSE 1 index, the Heston model again systematically overshoots the price compared with the CEV model. However, the GARCH model now systematically leads to the prices below those of the CEV model, and the difference between these two models is far larger than in the case of the FTSE 1 index. Again this confirms the results in Section 5, where both the Heston model and the GARCH model were rejected in S&P 5 stock market. 7 Conclusion In this paper, we calibrate three stochastic models to five different stock indices using the ML method of Ait-Sahalia and Kimmel (27). For the S&P5 index we find, like Sahalia (27), that both the Heston model and the GARCH model are rejected. However, this result is not obtained for any of the other 4 stock indices. For the AEX and CAC 4 we find that again the Heston model and the GARCH model are rejected by the data in favor of the CEV model. But the GARCH model is not rejected for the FTSE 1 and the BEL 2 indices. Moreover, for the BEL 2 index we only find very little evidence of the leverage effect. We also tested whether the difference in statistical fit of the three models translates in a significant difference in prices for at-the-money straddle options. We generate prices for the straddle option for both the FTSE 1 and the S&P 5 indices based on the parameter values of the maximum-likelihood estimation. The results confirm that for the S&P5 the GARCH and Heston models give significantly different prices than the CEV model, whereas for the FTSE 1 index the prices implied by the GARCH model are quite close to those of the CEV model. That is, the difference in statistical fit across the different stock indices translates in non-trivial differences in option prices. 17

18 References Ait-Sahalia, Y. (21): Closed-form likelihood expansions for multivariate diffusions, Working Paper, Princeton University. Ait-Sahalia, Y., and R. Kimmel (27): Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics, 83, An, Y., A. Assaf, and J. Yang (26): Hedging Volatility Risk: The Effectiveness of Volatility Options, International Journal of Theoretical and Applied Finance, 1, Black, F., and M. Scholes (1973): The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, Bollerslev, T. (1986): Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, pp Brenner, M., and D. Galai (1989): New financial instruments for hedging changes in volatility, Financial Analysts Journal, 45(4), Brenner, M., E. Y. Ou, and J. E. Zhang (26): Hedging volatility risk, Journal of Banking and Finance, 3, Brenner, M., and M. Subrahmanyam (1988): A simple formula to compute the implied standard deviation, Financial Analysts Journal, 44(5), Carr, P., and D. Madan (1998): Towards a theory of volatility trading, Risk Book on Volatility Risk, New York. (1999): Option valuation using the Fast Fourier Transform, Journal of Computational Finance, 3, Chacko, G., and L. M. Viceira (1999): Spectral GMM Estimation of Continuous-Time Processes, working paper. Chan, K. C., G. A. Karolyi, F. A. Longstaff, and A. B. Sanders (1992): An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, Journal of Finance, 47,

19 Cox, J., J. Ingersoll, and S. Ross (1985): A theory of the term structure of interest rates, Economitrica, 53, Demeter, K., E. Derman, M. Kamal, and J. Zou (1999): A guide to volatility and variance swaps, Journal of Derivatives, 6, Gastineau, G. (1977): An index of listed option premiums, Financial Analysts Journal, 3, Grunbichler, A., and F. A. Longstaff (1996): Valuing Futures and Options on Volatility, Journal of Banking & Finance, 2, Heston, S. (1993): A Closed-form Solution For Options With Stochastic Volatility with Application to Bond and Currency Options, Review of Financial Studies, 6, Hull, J., and A. White (1987): The pricing of options on assets with stochastic volatilities, The Journal of Finance, Vol. 42, No. 2, pp Jones, C. S. (23): The dynamics of stochastic volatility: evidence from underlying and options markets, Journal of Econometrics, 116, Lewis, A. L. (2): Option Valuation Under Stochastic Volatility, Finance Press,Newport Beach, CA. Meddahi, N. (21): An eigenfunction approach for volatility modeling, Technical Report,Universite de Montreal. Nelson, D. (199): ARCH Models as Diffusion Approximations, Journal of Econometrics, 45, Poteshman, A. (2): Forecasting future volatility from option prices,. Whaley, R. E. (1993): Derivatives on Market Volatility: Hedging Tools Long Overdue, Journal of Derivatives, 1, (2): The Investor Fear Gauge: Explication of CBOE VIX, The Journal of Portfolio Management, 26,

20 Figure 1: Time series of index level on 5 financial markets (AEX, BEL 2, CAC 4, FTSE1 and S&P 5) from 4th January 2 to 23rd July 28. AEX BEL2 CAC4 Index level of 5 financial markets Year Year Year Year Year FTSE1 S&P5 2

21 Figure 2: Time series of 5 volatility indices divided by 1 (VAEX, VBEL, VCAC, VFTSE and VIX) from 4th January 2 to 23rd July 28. VAEX/ Volatility index/1 of 5 financial markets Year VIX/1 VFTSE/1 VCAC/1 VBEL/ Year Year Year Year Table 1: Summary statistics of 5 volatility indices divided by 1 (VAEX, VBEL, VCAC, VFTSE and VIX) from 4th January 2 to 23rd July 28. VAEX VBEL VCAC VFTSE VIX Min Max Median Mean STD Skew Kurt

22 Figure 3: Time series of realized volatility (RV) on 5 financial markets (AEX, BEL 2, CAC 4, FTSE 1 and S&P 5) from 4th January 2 to 23rd July 28. RV AEX Realized volatility of 5 financial markets Year RV S&P5 RV FTSE1 RV CAC4 RV BEL Year Year Year Year Table 2: Summary statistics of realized volatility (RV) of 5 financial markets (AEX, BEL 2, CAC 4, FTSE 1 and S&P 5) from 4th January 2 to 23rd July 28. AEX BEL 2 CAC 4 FTSE 1 S&P 5 Min Max Median Mean STD Skew Kurt Table 3: Correlations between returns on five stock market indices (AEX, BEL 2, CAC 4, FTSE 1 and S&P 5) from 4th January 2 to 23rd July 28. AEX BEL 2 CAC 4 FTSE 1 S&P 5 AEX BEL CAC FTSE S&P

23 Table 4: Correlations between the five implied volatility indices (VAEX, VBEL, VCAC, VFTSE and VIX) from 4th January 2 to 23rd July 28. VAEX VBEL VCAC VFTSE VIX VAEX VBEL VCAC VFTSE VIX Table 5: ML Estimation of Heston model, ds t = [ (r t d t ) + ( λ ( 1 ρ 2) ] 2) 1 vt dt + vt dwtp s, dv t = δ v (v t v)dt + k v vt dwtp v, E tdwtp s dw tp v = ρdt AEX BEL 2 CAC 4 FTSE 1 S&P 5 Est. s.e. Est. s.e. Est. s.e. Est. s.e. Est. s.e. δ v v k v ρ λ ) L (ˆθ Table 6: ML Estimation of GARCH model ds t = [ (r t d t ) + ( λ ( 1 ρ 2) ] 2) 1 vt dt + vt dwtp s, dv t = δ v (v t v)dt + k v v t dwtp v, E tdwtp s dw tp v = ρdt AEX BEL 2 CAC 4 FTSE 1 S&P 5 Est. s.e. Est. s.e. Est. s.e. Est. s.e. Est. s.e. δ v v k v ρ λ ) L (ˆθ Table 7: ML Estimation of CEV model ds t = [ (r t d t ) + ( λ ( 1 ρ 2) ] 2) 1 vt dt + vt dwtp s, dv t = δ v (v t v)dt + k v v β t dwtp v, E tdwtp s dw tp v = ρdt AEX BEL 2 CAC 4 FTSE 1 S&P 5 Est. s.e. Est. s.e. Est. s.e. Est. s.e. Est. s.e. δ v v k v β ρ λ ) L (ˆθ

24 Table 8: Heston Hypothesis Statistic AEX BEL 2 CAC 4 FTSE 1 S&P Likelihood Ratio (.) (.) (.) (.) (.) Wald (.) (.) (.) (.) (.) Table 9: GARCH Hypothesis Statistic AEX BEL 2 CAC 4 FTSE 1 S&P Likelihood Ratio (.) - - (.) Wald (.7537) (.) (.278) (.22) (.) Figure 4: A straddle with a long position Put only Call only 4 2 Straddle profit or loss 2 Strike Price stock price 24

25 Figure 5: The value of straddle option under CEV SV model (β = 1., GARCH SV model) with different initial variance var and κ v. ρ=.6, β=1., K sto =11.5 ρ=, β=1., K sto =11.5 ρ=.7, β=1., K sto = var var var κ v κ v κ v 7 ρ=.6, β=1., K sto = ρ=, β=1., K sto = ρ=.7, β=1., K sto = var var 1 var κ v κ κ v v 25

26 Figure 6: The value of straddle option under CEV SV model (β =.8) with different initial variance var and κ v. ρ=.6, β=.8, K sto =11.5 ρ=, β=.8, K sto =11.5 ρ=.7, β=.8, K sto = var var var κ v κ v κ v ρ=.6, β=.8, K =11.5 sto ρ=, β=.8, K sto =11.5 ρ=.7, β=.8, K =11.5 sto var 6 var 1 var κ v κ v κ v 26

27 Figure 7: The value of straddle option under CEV SV model (β =.5, Heston SV model) with different initial variance var and κ v. ρ=.6, β=.5, K sto =11.5 ρ=, β=.5, K sto =11.5 ρ=.7, β=.5, K sto = var var var κ v κ v κ v 5 ρ=.6, β=.5, K sto = ρ=, β=.5, K sto = ρ=.7, β=.5, K sto = var var var κ v κ v κ v 27

28 Figure 8: Time series of ATMF Straddle in FTSE 1 stock market with FTSE 1 index and VFTSE from 4th January 2 to 23rd July ATMF straddle price of FTSE1 25 ATMF Straddle Year FTSE1 index level with its volatility index VFTSE FTSE Year

29 Figure 9: Volatility series from ATMF straddle in FTSE 1, VFTSE index series and their difference (σ AT MF ST V F T SE) from 4th January 2 to 23rd July 28. σ ATMFST VFTSE σ ATMFST VFTSE Volatility from ATMF straddle in FTSE year Volatility Index of FTSE 1, VFTSE year Difference of volatility from ATMF straddle and VFTSE year 29

30 Figure 1: Differences of volatility from ATMF straddle and its corresponding volatility index (σ AT MF ST volatility index) from 4th January 2 to 23rd July 28 in 5 financial markets (AEX, BEL 2, CAC 4, FTSE 1, and S&P 5). AEX BEL2 CAC4 FTSE1 SP5 Difference of volatility from ATMF straddle and volatility index year year year year year Figure 11: ATMF straddle option under CEV model in FTSE 1 stock market Straddle Option on 12rd Mar 23 Straddle Option on 31st Oct (Pound) 1 5 (Pound) (Pound) t (Pound) t 3

31 Figure 12: ATMF straddle option under CEV model in S&P 5 stock market on 12rd Mar 23 in S&P 5 (CEV); on 31st Oct 27 in S&P 5 (CEV) t t Figure 13: ATMF straddle option under CEV model in AEX stock market on 12rd Mar 23 in AEX (CEV); on 31st Oct 27 in AEX (CEV) t t 31

32 Figure 14: ATMF straddle option under CEV model in BEL 2 stock market on 12rd Mar 23 in BEL 2 (CEV); on 31st Oct 27 in BEL 2 (CEV) t t Figure 15: ATMF straddle option under CEV model in CAC 4 stock market on 12rd Mar 23 in CAC 4 (CEV); on 31st Oct 27 in CAC 4 (CEV) t t 32

33 Figure 16: ATMF straddle option under Heston SV model and the difference of ATMF straddle option between CEV and Heston SV model in FTSE 1 stock market on 12rd Mar 23 with Heston Model on 31st Oct 27 with Heston Model 15 2 (Pound) 1 5 (Pound) (Pound) t (Pound) t CEV Heston on 12rd Mar 23 CEV Heston on 31st Oct 27 Difference between CEV and Heston(Pound) Difference between CEV and Heston(Pound) (Pound) t (Pound) t 33

34 Figure 17: ATMF straddle option under GARCH SV model and the difference of ATMF straddle option between CEV and GARCH SV model in FTSE 1 stock market on 12rd Mar 23 with GARCH Model on 31st Oct 27 with GARCH Model 15 2 (Pound) 1 5 (Pound) (Pound) t (Pound) t CEV GARCH on 12rd Mar 23 CEV GARCH on 31st Oct 27 Difference between CEV and GARCH(Pound) Difference between CEV and GARCH(Pound) (Pound) t (Pound) t 34

35 Figure 18: ATMF straddle option under Heston SV model and the difference of ATMF straddle option between CEV and Heston SV model in S&P 5 stock market on 12rd Mar 23 in S&P 5 (Heston); on 31st Oct 27 in S&P 5 (Heston) t t CEV Heston on 12rd Mar 23 (S&P 5) CEV Heston on 31st Oct 27 (S&P 5) Difference between CEV and Heston Difference between CEV and Heston t.1 t 35

36 Figure 19: ATMF straddle option under GARCH SV model and the difference of ATMF straddle option between CEV and GARCH SV model in S&P 5 stock market on 12rd Mar 23 in S&P 5 (GARCH); on 31st Oct 27 in S&P 5 (GARCH) t t CEV GARCH on 12rd Mar 23 (S&P 5) CEV GARCH on 31st Oct 27 (S&P 5) Difference between CEV and GARCH Difference between CEV and GARCH t.1 t 36