Distributions of Historic Market Data -- Implied and Realized Volatility
aa r X i v : . [ q -f i n . M F ] A p r Distributions of Historic Market Data – Implied and Realized Volatility
M. Dashti Moghaddam a , Zhiyuan Liu a , R. A. Serota a,1 a Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011
Abstract
We undertake a systematic comparison between implied volatility, as represented by VIX (new methodology)and VXO (old methodology), and realized volatility. We compare visually and statistically distributions ofrealized and implied variance (volatility squared) and study the distribution of their ratio. We find thatthe ratio is best fitted by heavy-tailed – lognormal and fat-tailed (power-law) – distributions, depending onwhether preceding or concurrent month of realized variance is used. We do not find substantial differencein accuracy between VIX and VXO. Additionally, we study the variance of theoretical realized variance forHeston and multiplicative models of stochastic volatility and compare those with realized variance obtainedfrom historic market data.
Keywords:
Volatility, Implied, Realized, VIX, Fat Tails
1. Introduction
The implied volatility index VIX was created in order to estimate, looking forward, the expected realizedvolatility. CBOE introduced the original VIX (now VXO) in 1986. It was based on an inverted Black-Scholes formula, where S&P 100 near-term, at-the-money options were used to calculate a weighted averageof volatilities. However, the Black-Scholes formula assumes that the volatility in the stock returns equation iseither a constant, or at least does not have a stochastic component, while in reality it was already understoodthat volatility itself is stochastic in nature. A number of well-studied models of stochastic volatility haveemerged, such as Heston (HM) [1, 2] and multiplicative (MM) [3, 4]. Consequently, a need arose for animplied volatility index, which would not only be based on stochastic volatility but would also be agnosticto a particular model of the latter [5, 6].CBOE introduced its current VIX methodology on September 22, 2003 [7] to fulfill the above requirementsand was based on [8, 9], where a closed-form formula for the expected value of realized volatility [10] wasderived using call and put prices. Notably, it utilized the S&P 500 index, which is far more representative ofthe total market, both near-term and next-term options and a broader range of strike prices. CBOE publisheshistoric data using both methodologies, VIX (new) and VXO (old) dating back to 1990 [11] (historic stockprices used in calculation of realized volatility can be found at [12]). Here we call 1990 through September19, 2003 VIX Archive and VXO Archive and from September 22, 2003 through December 30, 2016 VIXCurrent and VXO Current.Naturally, the question arises of whether VIX, designed to be a superior methodology, has a bettertrack record than VXO. The short answer is that it is unclear. All-in-all, VIX/VXO is still too young tohave accumulated sufficient amount of data and only time will tell how reliable it is in predicting realizedvolatility. Still, one of our notable observations discussed below is that the ratio of realized to impliedvariance (squared volatility) is best fitted with a fat-tailed (power-law) distribution, which clearly signalsoccasional large discrepancies between prediction and realization. This is not surprising, given that we aretrying to predict the future (by pricing options) based on what we know today and thus are unaware ofunexpected future events that can spike the volatility. [email protected] Preprint submitted to arXiv April 17, 2018 n the other hand, we also find that the distribution of the ratio of realized variance of the preceding month to the implied volatility, as well as of its inverse, is distributed with lognormal distribution. Whilethe latter is heavy-tailed, this nonetheless shows that VIX is better attuned to the known volatility. Wenote a recent surmise that VIX can be manipulated [13, 14] and that Nasdaq is working on its own volatilityindex [15]. Hopefully, this work will establish a proper framework for testing implied volatility indices.This paper is the second in a series devoted to analysis of historic market data, the other two discussing,respectively, stock returns [16] and relaxation and correlations [17]. It is organized as follows. In Section 2we give a detailed visual and statistical comparison between realized volatility ( RV ) and implied volatilityrepresented by VIX and VXO. More precisely, we compare distributions of realized variance RV with V IX and V XO and, in particular, we analyze KS statistics of fits of RV /V IX and RV /V XO by variousdistributions, from normal to fat-tailed. In Section 3 we compare the variance of the RV distributionagainst the analytical results obtained using Heston and multiplicative models respectively. We concludewith the discussion of open questions and future work.
2. Comparing distributions of RV and V IX Realized variance (index) is defined as follows RV = 100 × n n X i =1 r i (1)where r i = ln S i S i − (2)are daily returns and S i is the reference (closing) price on day i . Time-averaged realized variance can becalculated from stochastic volatility σ t [10], [16] as1 τ Z τ σ t d t (3)Evaluation of the implied volatility is based on the evaluation of the expectation value of (3) [8, 9]. VIXuses options prices to estimate this expectation value via the generalized formula [7] V IX = (100) × T X i ∆ K i K i e RT Q ( K i ) − T [ FK − ! (4)where T is the time to expiration; F is the forward index level desired from index option price; K is the firststrike below the forward index level, F ; K i is the strike price of i th out-of-money option: a call if K i > K ,a put if K i < K and both a put and a call if K i = K ; ∆ K i is the interval between strike prices, that is halfthe difference between the strike on either side of K i , ∆ K i = ( K i +1 − K i − ) / R is the risk-free interestrate to expiration and Q ( K i ) is the midpoint of bid-ask spread for each option with strike K i . This formulais then used for near- and next-term options [7] and the final expression for VIX is effectively an averagebetween the two so the latter and the sum in (4) are intended to approximate the time average in (3).VIX and VXO were designed to measure a 30-day expected volatility. However, in their final form V IX and V XO are annualized by the ratio of 365 / ≈
12 [7]. As is clear from (1), RV is also annualizedand for comparison with VIX/VXO, we should take n = 21, so that 252 /
21 = 12; unlike VIX/VXO, RV iscalculated based on the number of trading days. Accordingly, to compare the distributions of
V IX and V XO with RV , we must rescale one of them with the ratio of their mean values. Table 1 lists ratios ofthe mean of V IX and V XO over the mean of RV . In what follows, the distributions of RV are rescaledwith the respective ratios from Table 1. We also analyze data for VIX Current and VXO Current both inaggregate form and split nearly evenly for a period covering the financial crisis and after (see Appendix).2t should be emphasized that for n = 21 in (1) the distribution of RV should be approaching normal.Fig. 1 hints at that but with an extended tail. The tail may be exponential or power-law, depending on howsingle-day returns are distributed. While the longer-time returns are better described by the Heston modeland exponential tails [16], single-day returns is still an open question. As always, the tail behavior is hardto pinpoint, especially with smaller data sets. Fig. 2 confirms the RV distribution approach to modifiednormality. RV -4 P D F n = 1n = 2n = 3n = 4 RV -4 P D F n = 1n = 7n = 14n = 21 Figure 1: PDFs of n P ni =1 r i for n =1,2,3,4 (left) and n =1,7,14,21 (right). n KS s t a t i s t i c Ga PDExGaIGa PDN n-10.5-10-9.5-9 Log V a r i an c e VarianceFit
Figure 2:
Left:
Kolmogorov-Smirnov statistics for fitting n P ni =1 r i ; lower numbers indicate a better fit. N and ExGaare normal and exGaussian distributions respectively. Ga PD and IGa PD are product distributions of gamma and inversegamma distributions respectively and normal distribution – which describe the distributions of stock returns in the Heston andmultiplicative models [16] – modified by a change of variables to squared returns. Right:
Log-log plot of the variance of the RV distribution versus n . The slope of the straight-line fit is -0.9635. .2. Visual comparison of realized volatility and VIX/VXO As previously mentioned, realized variance RV is scaled by entries in Table 1. In Figs. 3 and 4 (whichis just exaggerated, squared version of 3) we show scaled RV and scaled RV vis-a-vis their volatility indicescounterparts. In Figs. 5-10 we show histograms and their contour plots for RV vis-a-vis V IX and V XO .KS statistics for comparing the latter two with the scaled RV is collected in Table 2 (lower numberscorrespond to a better match). Further split of the 2003-2016 data is summarized in the Appendix. Table 1: Ratio of mean
V IX Theory Ratio365 /
252 1.448430 /
21 1.4286Date Ratio1990-2016 1.49111990-2003 1.66912003-2016 1.34462003-2010 1.28612010-2016 1.4104
V XO Theory Ratio365 /
252 1.448430 /
21 1.4286Date Ratio1990-2016 1.52571990-2003 1.83722003-2016 1.29852003-2010 1.28502010-2016 1.3097
Table 2: KS test results
V IX Date KS statistic1990-2016 0.17231990-2003 0.14782003-2016 0.23942003-2010 0.22152010-2016 0.2734
V XO Date KS statistic1990-2016 0.15891990-2003 0.16322003-2016 0.21572003-2010 0.20342010-2016 0.23764
Year S c a l ed R V , V I X VIXScaled RV
90 92 95 97 00 02 05 07 10 12 15 17 20
Year S c a l ed R V , VX O VXOScaled RV
Figure 3: VIX (top) and VXO (bottom) with scaled RV, from Jan 2nd, 1990 to Dec 30th, 2016. Year S c a l ed R V , V I X VIX Scaled RV
90 92 95 97 00 02 05 07 10 12 15 17 20
Year S c a l ed R V , VX O VXO Scaled RV Figure 4:
V IX (top) and V XO (bottom) with Scaled RV , from Jan 2nd, 1990 to Dec 30th, 2016. Scaled RV and VIX P D F -3 Scaled RV VIX Scaled RV and VIX P D F -3 Scaled RV VIX Figure 5: PDFs of scaled RV and V IX from Jan 2nd, 1990 to Dec 30th, 2016. Scaled RV and VXO P D F -3 Scaled RV VXO Scaled RV and VXO P D F -3 Scaled RV VXO Figure 6: PDFs of scaled RV and V XO from Jan 2nd, 1990 to Dec 30th, 2016. Scaled RV and VIX Archive P D F -3 Scaled RV VIX Scaled RV and VIX Archive P D F -3 Scaled RV VIX Figure 7: PDFs of scaled RV and V IX from Jan 2nd, 1990 to Sep 19th, 2003. Scaled RV and VXO Archive P D F -3 Scaled RV VXO Scaled RV and VXO Archive P D F -3 Scaled RV VXO Figure 8: PDFs of scaled RV and V XO from Jan 2nd, 1990 to Sep 19th, 2003. Scaled RV and VIX Current P D F -3 Scaled RV VIX Scaled RV and VIX Current P D F -3 Scaled RV VIX Figure 9: PDFs of scaled RV and V IX from Sep 22nd, 2003 to Dec 30th, 2016. Scaled RV and VXO Current P D F -3 Scaled RV VXO Scaled RV and VXO Current P D F -3 Scaled RV VXO Figure 10: PDFs of scaled RV and V XO from Sep 22nd, 2003 to Dec 30th, 2016. .3. Ratio distribution To further compare the volatilities, we examined the ratios RV /V IX and RV /V XO . In plotsbelow we show their time series and distribution functions. The latter are fitted using maximum likelihoodestimation (MLE) and the parameters of the fits and KS statistics are collected in the tables. Six functions –normal, lognormal, inverse gamma, gamma, Weibull and inverse Gaussian were used but only three best fitsare shown with data PDFs. Clearly, barring VIX Archive, the fat-tailed IGa was the best fit. The hypothesisis that fat tails are due to sudden spikes in RV. The inverse distributions V IX /RV and V XO /RV aregiven to illustrate that there were no unexpected surges in VIX and illustrate consistency of MLE.
90 92 95 97 00 02 05 07 10 12 15 17 20
Year R V / V I X RV / VIX P D F dataLNIGaIG Figure 11: RV / VIX , from Jan 2nd, 1990 to Dec 30th, 2016.
90 92 95 97 00 02 05 07 10 12 15 17 20
Year V I X / R V VIX / RV P D F dataLNGaWbl Figure 12: VIX / RV , from Jan 2nd, 1990 to Dec 30th, 2016.Table 3: MLE results for “RV / VIX ” and “VIX / RV ” type parameters KS StatisticNormal N( 1.0000, 0.9067) 0.1940LogNormal LN( -0.2027, 0.5867) 0.0446IGa IGa( 3.3595, 2.3466) 0.0246Gamma Gamma( 2.6219, 0.3814) 0.0978Weibull Weibull( 1.1124, 1.4009) 0.1224IG IG( 1.0000, 2.3168) 0.0607 type parameters KS StatisticNormal N( 1.0000, 0.5626) 0.0972LogNormal LN( -0.1562, 0.5867) 0.0446IGa IGa( 2.6219, 1.8314) 0.0978Gamma Gamma( 3.3595, 0.2977) 0.0246Weibull Weibull( 1.1306, 1.8882) 0.0500IG IG( 1.0000, 2.3168) 0.073410 Year R V / VX O RV / VXO P D F dataLNIGaIG Figure 13: RV / VXO , Jan 2nd, 1990 to Dec 30th, 2016.
90 92 95 97 00 02 05 07 10 12 15 17 20
Year VX O / R V VXO / RV P D F dataLNGaWbl Figure 14: VXO / RV , from Jan 2nd, 1990 to Dec 30th, 2016.Table 4: MLE results for “RV / VXO ” and “VXO / RV ” type parameters KS StatisticNormal N( 1.0000, 0.8747) 0.1910LogNormal LN( -0.1973, 0.5795) 0.0449IGa IGa( 3.4629, 2.4438) 0.0224Gamma Gamma( 2.6897, 0.3718) 0.0971Weibull Weibull( 1.1150, 1.4256) 0.1230IG IG( 1.0000, 2.3981) 0.0611 type parameters KS StatisticNormal N( 1.0000, 0.5467) 0.0925LogNormal LN( -0.1513, 0.5795) 0.0449IGa IGa( 2.6897, 1.8982) 0.0971Gamma Gamma( 3.4629, 0.2888) 0.0224Weibull Weibull( 1.1308, 1.9374) 0.0499IG IG( 1.0000, 2.3981) 0.072911 Year R V / V I X A r c h i v e RV / VIX Archive P D F dataLNIGaIG Figure 15: RV / VIX , from Jan 2nd, 1990 to Sep 19th, 2003.
90 91 92 93 94 95 96 97 98 99 00 01 02 03 04
Year V I X A r c h i v e / R V VIX Archive / RV P D F dataLNGaWbl Figure 16: VIX / RV , from Jan 2nd, 1990 to Sep 19th, 2003.Table 5: MLE results for “RV / VIX ” and “VIX / RV ” type parameters KS StatisticNormal N( 1.0000, 0.6380) 0.1421LogNormal LN( -0.1570, 0.5460) 0.0275IGa IGa( 3.6963, 2.7429) 0.0358Gamma Gamma( 3.3420, 0.2992) 0.0640Weibull Weibull( 1.1310, 1.7263) 0.0913IG IG( 1.0000, 2.8769) 0.0344 type parameters KS StatisticNormal N( 1.0000, 0.5372) 0.1011LogNormal LN( -0.1413, 0.5460) 0.0275IGa IGa( 3.3420, 2.4800) 0.0640Gamma Gamma( 3.6963, 0.2705) 0.0358Weibull Weibull( 1.1328, 1.9825) 0.0578IG IG( 1.0000, 2.8768) 0.041912 Year R V / VX O A r c h i v e RV / VXO Archive P D F dataLNIGaIG Figure 17: RV / VXO , from Jan 2nd, 1990 to Sep 19th, 2003.
90 91 92 93 94 95 96 97 98 99 00 01 02 03 04
Year VX O A r c h i v e / R V VXO Archive / RV P D F dataLNGaWbl Figure 18: VXO / RV , from Jan 2nd, 1990 to Sep 19th, 2003.Table 6: MLE results for “RV / VXO ” and “VXO / RV ” type parameters KS StatisticNormal N( 1.0000, 0.6013) 0.1451LogNormal LN( -0.1404, 0.5155) 0.0349IGa IGa( 4.1290, 3.1633) 0.0269Gamma Gamma( 3.7185, 0.2689) 0.0723Weibull Weibull( 1.1328, 1.8135) 0.0979IG IG( 1.0000, 3.2759) 0.0405 type parameters KS StatisticNormal N( 1.0000, 0.5058) 0.0911LogNormal LN( -0.1260, 0.5155) 0.0349IGa IGa( 3.7185, 2.8489) 0.0723Gamma Gamma( 4.1290, 0.2422) 0.0269Weibull Weibull( 1.1326, 2.0978) 0.0536IG IG( 1.0000, 3.2759) 0.048113 Year R V / V I X C u rr en t RV / VIX Current P D F dataLNIGaIG Figure 19: RV / VIX , from Sep 22nd, 2003 to Dec 30th, 2016.
03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
Year V I X C u rr en t/ R V VIX Current / RV P D F dataLNGaWbl Figure 20: VIX / RV , from Sep 22nd, 2003 to Dec 30th, 2016.Table 7: MLE results for “RV / VIX ” and “VIX / RV ” type parameters KS StatisticNormal N( 1.0000, 1.0338) 0.2273LogNormal LN( -0.2391, 0.6201) 0.0564IGa IGa( 3.1107, 2.0677) 0.0360Gamma Gamma( 2.2436, 0.4457) 0.1178Weibull Weibull( 1.0981, 1.2920) 0.1344IG IG( 1.0000, 1.9826) 0.0814 type parameters KS StatisticNormal N( 1.0000, 0.5864) 0.0959LogNormal LN( -0.1693, 0.6201) 0.0564IGa IGa( 2.2436, 1.4914) 0.1178Gamma Gamma( 3.1107, 0.3215) 0.0360Weibull Weibull( 1.1282, 1.8115) 0.0564IG IG( 1.0000, 1.9827) 0.094814 Year R V / VX O C u rr en t RV / VXO Current P D F dataLNIGaIG Figure 21: RV / VXO , from Sep 22nd, 2003 to Dec 30th, 2016.
03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
Year VX O C u rr en t/ R V VXO Current / RV P D F dataLNGaWbl Figure 22: VXO / RV , from Sep 22nd, 2003 to Dec 30th, 2016.Table 8: MLE results for “RV / VXO ” and “VXO / RV ” type parameters KS StatisticNormal N( 1.0000, 0.9475) 0.2130LogNormal LN( -0.2263, 0.6125) 0.0595IGa IGa( 3.1750, 2.1454) 0.0282Gamma Gamma( 2.3626, 0.4233) 0.1169Weibull Weibull( 1.1059, 1.3451) 0.1270IG IG( 1.0000, 2.0837) 0.0778 type parameters KS StatisticNormal N( 1.0000, 0.5731) 0.0911LogNormal LN( -0.1657, 0.6125) 0.0595IGa IGa( 2.3626, 1.5965) 0.1169Gamma Gamma( 3.1750, 0.3150) 0.0282Weibull Weibull( 1.1289, 1.8480) 0.0486IG IG( 1.0000, 2.0837) 0.093015 .4. Ratio distribution revisited Here we repeat the calculation from the Sec. 2.3 except with the RV for the month preceding the monthfor which VIX/VXO is calculated. For instance, if on March 31 VIX/VXO predict RV for April, we comparethem to RV for March. This is to test the hypothesis that VIX/VXO are pretty much as good a predictoras the RV they are already aware of. Indeed, LN distribution fits best both RV /V IX and RV /V XO ,as well as their inverse, consistent with the fact that for a LN distribution the distribution of the inversevariable is also LN. In Sec. 2.3 we hypothesized large spikes of RV as the reason for fat tails. This iscongruent with uncertainty – even with the knowledge of preceding RV – reflected in heavy LN tails.
90 92 95 97 00 02 05 07 10 12 15 17 20
Year R V / V I X RV / VIX P D F dataLNGaIG Figure 23: RV / VIX , from Jan 2nd, 1990 to Dec 30th, 2016.
90 92 95 97 00 02 05 07 10 12 15 17 20
Year V I X / R V VIX / RV P D F dataLNIGaIG Figure 24: VIX / RV , from Jan 2nd, 1990 to Dec 30th, 2016.Table 9: MLE results for “RV / VIX ” and “VIX / RV ” type parameters KS StatisticNormal N( 1.0000, 0.4974) 0.0992LogNormal LN( -0.1099, 0.4689) 0.0147IGa IGa( 4.6889, 3.7619) 0.0431Gamma Gamma( 4.7110, 0.2123) 0.0381Weibull Weibull( 1.1325, 2.1250) 0.0672IG IG( 1.0000, 4.0580) 0.0215 type parameters KS StatisticNormal N( 1.0000, 0.4999) 0.1059LogNormal LN( -0.1104, 0.4689) 0.0147IGa IGa( 4.7110, 3.7796) 0.0381Gamma Gamma( 4.6889, 0.2133) 0.0431Weibull Weibull( 1.1329, 2.1186) 0.0751IG IG( 1.0000, 4.0580) 0.016316 Year R V / VX O RV / VXO P D F dataLNIGaIG Figure 25: RV / VXO , Jan 2nd, 1990 to Dec 30th, 2016.
90 92 95 97 00 02 05 07 10 12 15 17 20
Year VX O / R V VXO / RV P D F dataLNGaIG Figure 26: VXO / RV , from Jan 2nd, 1990 to Dec 30th, 2016.Table 10: MLE results for “RV / VXO ” and “VXO / RV ” type parameters KS StatisticNormal N( 1.0000, 0.4915) 0.1064LogNormal LN( -0.1041, 0.4539) 0.0150IGa IGa( 5.0351, 4.0948) 0.0331Gamma Gamma( 4.9618, 0.2015) 0.0454Weibull Weibull( 1.1316, 2.1383) 0.0730IG IG( 1.0000, 4.3548) 0.0203 type parameters KS StatisticNormal N( 1.0000, 0.4768) 0.0933LogNormal LN( -0.1026, 0.4539) 0.0150IGa IGa( 4.9618, 4.0352) 0.0454Gamma Gamma( 5.0351, 0.1986) 0.0331Weibull Weibull( 1.1319, 2.2099) 0.0689IG IG( 1.0000, 4.3548) 0.021217 . Variance of realized variance The goal of this section is to evaluate the expectation value of theoretical variance of realized variance E [( 1 T Z T v t d t − E [ 1 T Z T v t d t ]) ] = E [( 1 T Z T v t d t − θ ) ] (5)and compare it with the market data using historic squared stock returns. Here v t = σ t is the stochasticvariance and θ is the mean (expectation) value of v t in the mean-reverting models, E [ v t ] = θ (6)Below we discuss two such models – Heston and multiplicative [16]. In the Heston model the equation for stochastic variance is given byd v t = − γ ( v t − θ )d t + κ √ v t d W t (7)To evaluate (5), we need to know the correlation function [17] E [ v t v t + τ ] = θ + κ θ γ e − γτ (8)where κ θ γ = E [ v t ] − ( E [ v t ]) (9)From (5) and (8) we find E [( 1 T Z T v t d t − θ ) ] = 1( γT ) ( κ θγ )( − e − γT + γT ) (10)with the following limits E [( 1 T Z T v t d t − θ ) ] ≈ ( κ θ γ , γT ≪ κ θγ ( γT ) − , γT ≫ In the multiplicative model the equation for stochastic variance is given byd v t = − γ ( v t − θ )d t + κv t d W t (12)In this case [17] E [ v t v t + τ ] = θ + κ θ γ − κ e − γτ (13)where κ θ γ − κ = E [ v t ] − ( E [ v t ]) (14)From (5) and (13) we find E [( 1 T Z T v t d t − θ ) ] = 1( γT ) ( 2 κ θ γ − κ )( − e − γT + γT ) (15)18ith the following limits E [( 1 T Z T v t d t − θ ) ] ≈ ( κ θ γ − κ , γT ≪ κ θ γ − κ ( γT ) − , γT ≫ Clearly, for both models the following holds E [( T R T v t d t − θ ) ] E [ v t ] − ( E [ v t ]) = 2( γT ) ( − e − γT + γT ) (17)with the following limits E [( T R T v t d t − θ ) ] E [ v t ] − ( E [ v t ]) ≈ ( γT ≪ γT ) − , γT ≫ T and is consistent with central limit theorem otherwise.In Fig. 27 we compare the historic market data with theoretical predictions (10) and (15), including thelimiting behaviors (11) and (16). To do so, we need to identify the values of parameters θ , κ and γ . one wayto accomplish this is by fitting historic data with (6), (8) and (13) and the results are summarized in Table11. Thus found values of parameters are close to those obtained from leverage [17], [18] and by averagingthe values of parameters obtained from fitting multi-day stock returns [16], [17]. Alternatively, in Table 12we use parameters obtained from single-day returns ( τ = 1 values in [16], with the same γ as in Table 11).While multiplicative model does better, a continuous model may not be appropriate for single-day returns. Table 11: Parameters for Heston and multiplicative models for S&P 500
Heston model θ γ κ . × − . × − Multiplicative model θ γ κ . × − Table 12: Parameters for Heston and multiplicative models for S&P 500 from single-day returns
Heston model θ γ κ . × − . × − Multiplicative model θ γ κ . × − γ , which is wellestablished correlation and relaxation scale for mean reverting models [16]-[19].19
50 100 150 200 250T01234567 V a r o f V a r -8 DataIGaGa (T)-9-8.8-8.6-8.4-8.2-8-7.8-7.6-7.4-7.2 Log ( V a r o f V a r) DataIGaGa V a r o f V a r -8 DataIGaGa (T)-8.8-8.6-8.4-8.2-8-7.8-7.6-7.4-7.2 Log ( V a r o f V a r) DataIGaGa
Figure 27: Historic data vis-a-vis (10) and (15). Top row: with parameters from Table11. Bottom row: with parameters fromTable12. Straight lines on the right are best data fits with slopes -0.0113 and -0.992 to compare with the limiting behaviors(10), (15) and (18). V a r o f V a r DataIGaGa (T)-1-0.8-0.6-0.4-0.200.20.4 Log ( V a r o f V a r) DataIGaGa
Figure 28: Historic data vis-a-vis (17). Straight lines on the right have the same slopes as in Fig. 27. . Conclusions Previously [16] we showed that the multi-day – typically longer than several weeks, to account forrelaxation processes – stock returns are better described by the Heston model. Realized volatility, conversely,is calculated from single-day returns, which are presumably on the borderline between intra-day jumpprocesses and continuous processes, such as Heston and multiplicative. Furthermore, realized volatility issquare root of the realized variance, which is a sum of roughly 21 realized daily variances representing thetrading month. As such, it is clear that the realized variance should be approaching normal distribution.Figs. 1-2 reflect this conjecture, with the caveat that the distribution maintains a tail similar to that of asingle-day variance distribution.Regarding the latter, we showed that an exGaussian distribution provides an excellent fit. This, howeverdoes not necessarily indicate that the tail is in fact exponential. An exGaussian is a sum of normal andexponential distributions and is an artificial construct, where all parameters are shape parameters and thedistribution rescales only when all parameters rescale simultaneously. On the other hand, exGaussian has theright properties for the realized variance distribution and can be described analytically. It is quite possiblethat a distribution exists, which combines normality with a heavy, perhaps fat, tail that gives comparableKolmogorov-Smirnov statistics to exGaussian. This is something we intend to study in the future.With respect to comparison of the realized volatility to predictions of volatility indices, we found nodiscernible advantages between VIX and VXO. We also found that the ratio of the realized variance tosquared VIX and VXO is best fitted by a fat-tailed (power-law) distribution – in this case Inverse Gamma.This most likely reflects large unexpected spikes of realized volatility not foreseen by volatility indices. Theinverse distribution, which is the distribution of the inverse variable, is best fitted with an exponentiallydecaying distribution – in this case Gamma. This reflects no unexpected surges in volatility indices relativeto realized volatility. When we use realized volatility of the preceding month, however, the ratio distributionsare all best fitted with lognormal distribution. This most likely reflects the fact that while we predict futurevolatility based on what we presently know (past realized volatility and current information [20]), largespikes in volatility still result in heavy lognormal tails of the ratios due to uncertainty associated with suchspikes. We will discuss relationship between volatility spikes and tails in a future work.Finally, evaluation of the theoretical dependence of variance of realized variance and its limiting behaviorscompares well vis-a-vis historic data, with the prediction of the multiplicative model having an edge over theHeston model. The latter may be due that single-day returns, on which realized variance is based, are betterdescribed by the Student’s distribution [16], [21]. Interestingly, 21 days, over which the realized variance iscalculated, is also roughly the relaxation time of stochastic volatility, γ − . Additionally, the mean value ofthe volatility in the mean-reverting models may be stochastic itself [18] – with similar time scales – whichmay as well affect the comparison. Ideally, one would also want to study the higher moments of the realizedvariance distribution. The difficulty, however, is that decoupling the higher-order correlation functions ofstochastic variance into pair-wise correlation functions leads trivially to a normal distribution, which is notthe case per above. This is a challenging problem that also needs to be addressed.21 ppendix A. VIX Current and VXO Current Here we split 2003-2016 data in two roughly equal time periods, before and after the financial crisis.
Appendix A.1. Visual Comparison
Scaled RV and VIX Current P D F -3 Scaled RV VIX Scaled RV and VIX Current P D F -3 Scaled RV VIX Figure A.29: PDFs of scaled RV and V IX from Sep 22nd, 2003 to Aug 30th, 2010. Scaled RV and VXO Current P D F -3 Scaled RV VXO Scaled RV and VXO Current P D F -3 Scaled RV VXO Figure A.30: PDFs of scaled RV and V XO from Sep 22nd, 2003 to Aug 30th, 2010. Scaled RV and VIX Current P D F -3 Scaled RV VIX Scaled RV and VIX Current P D F -3 Scaled RV VIX Figure A.31: PDFs of scaled RV and V IX from Aug 31st, 2010 to Dec 30th, 2016. Scaled RV and VXO Current P D F -3 Scaled RV VXO Scaled RV and VXO Current P D F -3 Scaled RV VXO Figure A.32: PDFs of scaled RV and V XO from Aug 31st, 2010 to Dec 30th, 2016. ppendix A.2. Ratio Distribution
03 04 05 06 07 08 09 10 11
Year R V / V I X C u rr en t RV / VIX Current P D F dataLNIGaIG Figure A.33: RV / VIX , from Sep 22nd, 2003 to Aug 30th, 2010.
03 04 05 06 07 08 09 10 11
Year V I X C u rr en t/ R V VIX Current / RV P D F dataLNGaWbl Figure A.34: VIX / RV , from Sep 22nd, 2003 to Aug 30th, 2010.Table A.13: MLE results for “RV / VIX ” and “VIX / RV ” type parameters KS StatisticNormal N( 1.0000, 1.0244) 0.2338LogNormal LN( -0.2280, 0.5943) 0.0763IGa IGa( 3.5292, 2.4224) 0.0383Gamma Gamma( 2.3463, 0.4262) 0.1297Weibull Weibull( 1.1013, 1.3060) 0.1588IG IG( 1.0000, 2.1887) 0.0969 type parameters KS StatisticNormal N( 1.0000, 0.5124) 0.0613LogNormal LN( -0.1483, 0.5943) 0.0763IGa IGa( 2.3463, 1.6105) 0.1297Gamma Gamma( 3.5292, 0.2833) 0.0383Weibull Weibull( 1.1290, 2.0447) 0.0392IG IG( 1.0000, 2.1887) 0.112824 Year R V / VX O C u rr en t RV / VXO Current P D F dataLNIGaIG Figure A.35: RV / VXO , from Sep 22nd, 2003 to Aug 30th, 2010.
03 04 05 06 07 08 09 10 11
Year VX O C u rr en t/ R V VXO Current / RV P D F dataLNGaWbl Figure A.36: VXO / RV , from Sep 22nd, 2003 to Aug 30th, 2010.Table A.14: MLE results for “RV / VXO ” and “VXO / RV ” type parameters KS StatisticNormal N( 1.0000, 0.9130) 0.2237LogNormal LN( -0.2069, 0.5763) 0.0715IGa IGa( 3.6954, 2.6086) 0.0358Gamma Gamma( 2.5708, 0.3890) 0.1274Weibull Weibull( 1.1119, 1.3839) 0.1435IG IG( 1.0000, 2.4002) 0.0922 type parameters KS StatisticNormal N( 1.0000, 0.5009) 0.0635LogNormal LN( -0.1414, 0.5763) 0.0715IGa IGa( 2.5708, 1.8147) 0.1274Gamma Gamma( 3.6954, 0.2706) 0.0358Weibull Weibull( 1.1296, 2.0966) 0.0375IG IG( 1.0000, 2.4002) 0.105525 Year R V / V I X C u rr en t RV / VIX Current P D F dataLNIGaIG Figure A.37: RV / VIX , from Aug 31st, 2010 to Dec 30th, 2016.
10 11 12 13 14 15 16 17
Year V I X C u rr en t/ R V VIX Current / RV P D F dataLNGaWbl Figure A.38: VIX / RV , from Aug 31st, 2010 to Dec 30th, 2016.Table A.15: MLE results for “RV / VIX ” and “VIX / RV ” type parameters KS StatisticNormal N( 1.0000, 1.0445) 0.2245LogNormal LN( -0.2513, 0.6450) 0.0510IGa IGa( 2.8188, 1.8171) 0.0420Gamma Gamma( 2.1414, 0.4670) 0.1082Weibull Weibull( 1.0947, 1.2762) 0.1251IG IG( 1.0000, 1.8141) 0.0745 type parameters KS StatisticNormal N( 1.0000, 0.6306) 0.1114LogNormal LN( -0.1877, 0.6450) 0.0510IGa IGa( 2.1414, 1.3804) 0.1082Gamma Gamma( 2.8188, 0.3548) 0.0420Weibull Weibull( 1.1263, 1.7028) 0.0654IG IG( 1.0000, 1.8141) 0.086826 Year R V / VX O C u rr en t RV / VXO Current P D F dataLNIGaIG Figure A.39: RV / VXO , from Aug 31st, 2010 to Dec 30th, 2016.
10 11 12 13 14 15 16 17
Year VX O C u rr en t / R V VXO Current / RV P D F dataLNGaWbl Figure A.40: VXO / RV , from Aug 31st, 2010 to Dec 30th, 2016.Table A.16: MLE results for “RV / VXO ” and “VXO / RV ” type parameters KS StatisticNormal N( 1.0000, 0.9878) 0.2060LogNormal LN( -0.2494, 0.6511) 0.0504IGa IGa( 2.7773, 1.7884) 0.0301Gamma Gamma( 2.1563, 0.4638) 0.1090Weibull Weibull( 1.0984, 1.3018) 0.1153IG IG( 1.0000, 1.8085) 0.0670 type parameters KS StatisticNormal N( 1.0000, 0.6258) 0.1065LogNormal LN( -0.1907, 0.6511) 0.0504IGa IGa( 2.1563, 1.3885) 0.1090Gamma Gamma( 2.7773, 0.3601) 0.0301Weibull Weibull( 1.1262, 1.7084) 0.0516IG IG( 1.0000, 1.8085) 0.084327 eferences