Internet Appendix for

"Lévy jump risk: Evidence from options and returns✩"

Chayawat OrnthanalaiRotman School of Management, University of Toronto

In this note, we provide additional results that are left out of the paper due to spaceconsiderations. Section 1 discusses the difference and similarity between GARCH and SVapproaches. Section 2 shows that the affi ne GARCH model that we use is not void of thevolatility risk premium effect. Section 3 provides further results on the models’ in-sampleoption pricing performance. Finally, Section 4 provides results from re-estimating the modelsusing daily S&P 500 returns from January 1980 to December 2009. We show that our con-clusions regarding the models’performance in Figure 7 of the main text are robust to the useof a longer return period that includes the 1987’s crash as well as the 2008 crisis.

1. Comparison between GARCH and SV approaches

We assume that the dynamic of the variance of the normal innovation, hz,t+1, and thetime-homogeneous parameter of the jump innovation, hy,t+1, are

hz,t+1 = wz + bzhz,t +azhz,t

(zt − czhz,t)2 (1)

hy,t+1 = wy + byhy,t +ayhz,t

(zt − cyhz,t)2 .

The added filtering step required to implement (1) means that the process can be viewed asa hybrid between the GARCH and the stochastic volatilities (SV) approach. It differs fromGARCH in that the next period’s conditional volatility is not a function of the total returnresidual. On the other hand, it differs from SV because the next period’s conditional volatilityis known ex-post of the return today. Although the implementation of the model in (1) ismore involved than a standard GARCH, it is simpler to implement than the SV approaches.For clarity, we distinguish the mechanic of (1) from the SV approaches by comparing it tothe affi ne stochastic volatilities with stochastic jump intensities model (SVSJI). We choosethe SVSJI model for the comparison which our Lévy GARCH model converges to in thecontinuous-time limit. For ease of comparison, we cast the SVSJI model into a discrete-timesetting by means of the Euler discretization. Its dynamic is then given by

Rt = r + µt +√vtz1,t + yt (2)

vt+1 = av + bvvt + σv√vt

(ρvz1,t+1 +

√1− ρ2vz2,t+1

)(3)

λt+1 = aλ + bλλt + σλ√λt

(ρλz1,t+1 +

√1− ρ2λz3,t+1

), (4)

Preprint submitted to Journal of Financial Economics August 9, 2013

where yt represents the Lévy jump innovation with time-varying intensity λt, and z1,t+1, z2,t+1,and z3,t+1 are three i.i.d. normal innovations at time t+1. The correlation between return andvariance is captured by Corrt (Rt+1, vt+1) = ρv. Similarly, the correlation between return andjump intensity is captured by Corrt (Rt+1, λt+1) = ρλ. In the above dynamics (2)-(4), there area total of four independent shocks. Given that today’s return Rt is known, we can decomposethe diffusive part

√vtz1,t from the jump part yt in (2) using a filtering density similar to our

approach in this paper. However, unlike the affi ne GARCH model in (1), we cannot determinethe next period’s conditional volatility and jump intensity immediately because vt+1 and λt+1also depend on the Brownian shocks z2,t+1 and z3,t+1 respectively, which cannot be inferredfrom today’s return. In most cases, the filtration of the future period’s volatility and jumpintensity in the SVSJI model requires additional steps. This involves simulating two setsof i.i.d normal shocks z2,t+1 and z3,t+1. After, vt+1 and λt+1 can be inferred based on theirlikelihood of fitting the next period’s return Rt+1.

2. GARCH and the volatility risk premium

By the nature of GARCH, normal shocks in returns and volatilities are perfectly correlated.The "normal" risk premium values that we report in our empirics represent the combinationof the normal and the volatility risk. Although we cannot separately identify the volatilityrisk premium, the Lévy GARCH models are not void of the volatility risk premium effect.Specifically, we show that the normal risk premium causes the volatility dynamics under thephysical and risk-neutral measures to differ in two important ways that are consistent withthe continuous-time stochastic volatility models. First, the normal risk premium intensifiesthe asymmetric volatility feedback effect (i.e., the leverage effect). Second, it raises the un-conditional mean of the return variance.We illustrate the effect of the market price of normal risk (λz) on the asymmetric volatility

feedback by computing conditional covariance between Rt+1 and σ2t+2

CovQt (Rt+1, σ2t+2) = EQt

[(Rt+1 − EQt [Rt+1]

) (σ2t+2 − EQt

[σ2t+2

])](5)

= −2(az (cz + λz) + ξ

∗′′

y (0) ay (cy + λz))hz,t+1,

where the superscript Q denotes that the expectation is taken under the risk-neutral measure.We also note that ξ

∗′′

y (0) is the second derivative of the risk-neutral coeffi cient in the cumulantexponent of the Lévy jump innovation evaluated at 0. Equation (5) shows that a larger value ofλz will drive return Rt+1 to co-vary more negatively with σ2t+2 under the risk-neutral measure.This feature is known as the asymmetric volatility feedback effect. Therefore, a larger normalrisk premium implies a larger negative correlation between returns and conditional volatilities.The normal risk premium also affects the unconditional variance of Rt+1 by increasing

its long-run mean. The long-run risk-neutral level of σ2t+1 can be derived by taking theunconditional expectation of the GARCH dynamic (see Proposition 2 in the main paper)

EQ [V art (Rt+1)] ≡ EQ[σ2t+1

]= EQ [hz,t+1] + EQ [hy,t+1] ξ

∗′′

y (0)

=wz + az

1− bz − az (cz + λz)2 +

ay + wy + ay (cy + λz)2 σ2z

1− byξ∗′′

y (0)

2

The above equation shows the long-run risk-neutral return variance increases with the marketprice of normal risk λz.

3. Further analysis of option pricing performance

This sections further analyzes the models’in-sample performance by looking at their abilityto fit index option prices. Table IA.1 reports in-sample option pricing performance for themodels across moneyness and maturity. We compute the mean of RIVRMSE for variousmoneyness (Panel A) and maturity bins (Panel B). The raw RIVRMSE values are reported forthe GARCH model. For the other four models, we report their RIVRMSE ratios with respectto the GARCH model to facilitate the comparisons. RIVRMSE ratios below one indicatethat the model has a lower pricing error than the GARCH model. Using all observations, wefind that the NIG model has the lowest RIVRMSE, followed by the CGMY and VG models.These results confirm the previous findings in Table 4 of the paper. Panels B-C show thatthe performance of the NIG model is very robust across moneyness and maturity. The NIGmodel performs remarkably well at pricing deep out-of-the-money puts and calls. The LS,VG and CGMY, on the other hand, do not perform better than the GARCH model for themoneyness bin F/K > 1.10.Before going further, we note that the in-sample option pricing performance in Table 4

of the main text and those in Table IA.1 cannot be compared to studies that calibrate theparameters as well as the latent variables directly to option prices. For instance, Bakshi, Caoand Chen (1997) find that adding Merton jumps to the squared root SV model of Heston(1993) improves the sum of option squared pricing errors by 40 percent. Similarly, Broadie,Chernov and Johannes (2007) find that jump models outperform the SV model in-sample byabout 50 percent. Clearly, our results based on the joint MLE approach show a more modestevidence that adding jumps to returns improve in-sample option pricing performance. Eraker(2004) arrives at a similar finding to ours by jointly estimating S&P 500 index options andreturns from January 1987 to December 1990 using the MCMC technique. Our results, aswell as those in Eraker (2004), reflect the models’performance at reconciling the dynamics ofoptions and returns, and hence cannot be compared to studies that directly fit their modelsto options data.Fig. IA.1 plots the implied volatility smirks across moneyness for four different maturity

buckets: 15-45, 120-180, and 250-365 days to maturity. Table 4 in the main text show that theGARCH model has the largest in-sample option pricing errors. Therefore, to avoid clutter, wedo not plot the implied volatility smirks for the GARCH model. We present the analysis forthree different volatility periods. The first sample period is from 2004/09/01 to 2004/12/31.The average value of the VIX over this period is 13.75%, and we refer to it as a low volatilityperiod. The second sample period is from 1997/02/01 to 1997/05/31. This is when the averageVIX value is about 19.96%, and we deem it a medium volatility period. The third sampleperiod is from 2008/09/01 to 2008/12/31, when the average VIX level is 51.59%, which is ahigh volatility period. The main conclusion from Fig. IA.1 is that all the jump models cangenerate the steep implied-volatility smirks at short maturities. The NIG model, however, isslightly better at matching the implied-volatility slope observed in the data during in low andmedium volatility periods. For longer maturities, Fig. IA.1 shows that all models performequally well.

3

Finally, we assess the models’performance by looking at the option fitting residuals basedon the loss function

εj =σBSj − IV (Oj)

σBSj,

Fig. IA.2 shows the conventional QQ-plots of the standardized relative implied volatilityresiduals from the joint MLE. These plots show that the option fitting residuals for all modelsdeviate from normality. Nevertheless, Fig. IA.2 illustrates that the violation of the non-normality assumption is the most severe for the GARCH model which confirms our previousresults.

4. Daily return MLE: 1980-2009

We show that our conclusions regarding the return MLE performance of our jump modelsare robust to the use of longer return time series. We re-estimate the models using daily S&P500 returns from January 1980 to December 2009. This return period includes the extremelylarge crash of 1987 as well as the 2008 credit crisis where we large positive and negative jumpsarrive consecutively. We estimate the following return dynamic

Rt+1 = rt+1 + µzhz,t+1 + µyhz,t+1 + zt+1 + yt+1,

where zt+1 is the normal innovation and yt+1 is the Lévy jump innovation. The parametersµz and µy are the conditional in-mean parameters. The GARCH dynamics for the variance ofthe normal return innovation, hz,t+1, and the time-homogeneous jump parameter, hy,t+1, are

hz,t+1 = wz + bzhz,t +azhz,t

(zt − czhz,t)2 (A2)

hy,t+1 = wy + byhy,t +ayhz,t

(zt − cyhz,t)2 . (A6)

We report the estimation results in Table IA.2.Panel B of Table IA.2 shows that most of he parameters describing the volatility and jump

dynamics are well estimated and statistically significant. An exception is the parameter cy inthe MJ model which controls the leverage effect in the jump intensity. This finding differs fromour MLE results using daily returns from 1995-2009. The statistically insignificance estimateon cy suggests that jump arrival rates in the MJ model does not asymmetrically responseto negative and positive normal shocks. However, volatility of the normal component stillexhibits strong leverage effect, i.e. cz is positive and significant. This finding is consistentwith Christoffersen, Jacobs and Ornthanalai (2012) who estimate a complex jump intensityspecification that nests our MJ model using S&P 500 returns from 1962-2009. They find thatnormal return innovation does not induce asymmetry in the news impact curve of the jumpintensity.Looking at the log likelihood values, we find the CGMY model significantly outperforms

the other jump models. Clearly, the CGMY model is the best performing one followed by theVG model. Similar to the results in Table 5 of the main paper, Table IA.2 shows that the loglikelihood values of the MJ and NIG models are almost identical. Overall, we find that ourperformance ranking based on the log likelihood values is robust to different sample periods.

4

Using the MLE estimates on 1980-2009 returns in Table IA.2, we plot the annualizedconditional daily return volatility (solid lines) for each model. Fig. IA.3 reports the results.The annualized conditional volatility is calculated as

√V art (Rt+1)× 252, where V art (Rt+1)

is given byV art(Rt+1) = hz,t+1 + hy,t+1ξ

′′y (0) .

which is identical to equation (5) in the main text. In order to see the contribution of jumpsto daily return volatilities, we plot the annualized conditional jump volatility calculated as√V art (yt+1)× 252 where V art (yt+1) = hy,t+1ξ

′′y (0) . Fig. IA.3 shows the conditional volatility

of the GARCH model increases excessively high during the 1987 crash, while for the jumpmodels, the increase in volatility is more moderate. We find that jumps from the finite-activity MJ model contribute the least to the total return volatility while infinite activityjumps contribute to approximately half or more of the daily return volatility. The time-seriesaverage of the ratio of jump volatility to total return volatility for the MJ model is 41% whilefor the infinite-activity VG, NIG, and CGMY, they are 48%, 61% and 58% respectively. Thisfinding is consistent with our results in Fig. 2 in the main text. Overall, our conclusionthat infinite-activity jumps explain a significantly larger share of uncertainties in returns thanfinite-activity jumps remains intact.The rows labeled "Average skewness" and "Average kurtosis" in Table IA.2 summarize

the time-series mean of daily conditional skewness and kurtosis. Consistent with Tables 3 and5 in the main text, conditional return distribution implied by the NIG model has the mostnegative skewness and fat-tailed characteristics. The MJ model, on the other hand, exhibitsthe least fat-tailed distribution. The sample skewness and excess kurtosis calculated usingdaily returns for 1980-2009 are −1.29 and 28.8. MJ model therefore severely underestimatesthe conditionally heavy left-tailed distribution of daily returns which is consistent with ourprevious findings.Using estimates in Table IA.2, we apply the particle filtering algorithm to back out the

jump component in daily returns. Fig. IA.4 plots the results (solid lines). For a quickcomparison with daily returns, the grey market ’.’ in each panel plots the time-series of S&P500 returns. We find that the filtered jump components in Fig. IA.4 behave similarly to theircounterparts in Figs. 3 and 5, in the main text, confirming that our jump parameters arewell estimated and robust to different sample periods. We find the MJ model has the smallestjump magnitude during the 1987 crash as well as during the 2008. This finding confirms ourresults in the main text and is consistent with the fact that the MJ model exhibits the leastconditional non-normality among the jump models.

References

[1] Bakshi, G., Cao, C., Chen, Z., 1997. Empirical performance of alternative option pricingmodels. Journal of Finance 52, 2003-2049.

[2] Broadie, M., Chernov, M., Johannes, M., 2007. Model specification and risk premia: theevidence from the futures options. Journal of Finance 62. 1453-1490.

[3] Christoffersen, P., Jacobs, K., Ornthanalai, C., 2012. Dynamic jump intensities and riskpremiums: evidence from S&P 500 returns and options. Journal of Financial Economics106, 447—472.

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[4] Eraker, B., 2004. Do stock prices and volatility jump? Reconciling evidence from spot andoption prices. Journal of Finance 59, 1367-1403.

[5] Heston, S., 1993. A closed-form solution for options with stochastic volatility with appli-cations to bond and currency options. Review of Financial Studies 6, 327-343.

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0.95 1 1.05 1.10.1

0.12

0.14

0.16

0.18

0.215

< D

TM <

 45

Low

Data Merton VG NIG CGMY

0.95 1 1.05 1.10.1

0.12

0.14

0.16

0.18

0.2

120 

< D

TM <

 180

0.95 1 1.05 1.10.1

0.12

0.14

0.16

0.18

0.2

250 

< D

TM <

 365

0.95 1 1.05 1.1

0.15

0.2

0.25

Medium

0.95 1 1.05 1.1

0.15

0.2

0.25

0.95 1 1.05 1.1

0.15

0.2

0.25

0.95 1 1.05 1.10.3

0.4

0.5

0.6

High

0.95 1 1.05 1.10.3

0.4

0.5

0.6

0.95 1 1.05 1.10.3

0.4

0.5

0.6

0.7

Fig. IA.1. Implied volatility smirks at different maturities for selected models. We use Black-Scholes implied volatilities computed from the joint MLE and plot implied volatility smirksof the data and the models for three different periods: a low volatility period, a mediumvolatility period and a high volatility period. The low volatility period is between 2004/09/01to 2004/12/31, when the average VIX level is 13.75. The medium volatility period is between1997/02/01 to 1997/05/31, when the average VIX level is 19.96. The high volatility periodis between 2008/09/01 to 2008/12/31, when the average VIX level is 51.59. The moneynessis on the horizontal axis and each row of panels corresponds to a different maturity. We plotimplied volatility smirks for the data and four jump models: MJ, VG, NIG and CGMY. Toavoid clutter, we do not plot the implied volatility smirks for the GARCH model which hasrelatively larger option pricing errors.

7

­4 ­2 0 2 4

­4

­2

0

2

4Q

uant

iles 

of in

put s

ampl

e

GARCH

­4 ­2 0 2 4

­4

­2

0

2

4

MJ

­4 ­2 0 2 4

­4

­2

0

2

4

Qua

ntile

s of

 inpu

t sam

ple

VG

­4 ­2 0 2 4

­4

­2

0

2

4

Standard Normal quantiles

NIG

­4 ­2 0 2 4

­4

­2

0

2

4

Standard Normal quantiles

Qua

ntile

s of

 inpu

t sam

ple

CGMY

Fig. IA.2. QQ-plots of option fitting residuals from the joint MLE. We plot the quantilesof option fitting residuals from the joint MLE for each model against the standard normalquantiles (QQ-plot). The quantiles of standard normal distribution are represented on thex-axis, and the quantiles of the option fitting residuals are represented on the y-axis. The topleft panel shows the QQ-plot for the option fitting residuals of the Heston-Nandi GARCH(1,1)model, which has no jump component. The other panels show the QQ-plots for the optionfitting residuals of the five jump models that we consider. The dash-dot line in each panelrepresents the case when the sample input is a normal density.

8

1980 1985 1990 1995 2000 2005 20100

0.5

GARCH

1980 1985 1990 1995 2000 2005 20100

0.2

0.4

MJ

1980 1985 1990 1995 2000 2005 20100

0.2

0.4

VG

1980 1985 1990 1995 2000 2005 20100

0.5

NIG

1980 1985 1990 1995 2000 2005 20100

0.5

CGMY

Total return volatilty Jump volatility

Fig. IA.3. Conditional return and jump volatilities: returns-based MLE 1980-2009. Usingthe estimates in Table IA.2, we plot the conditional return volatility,

√V art (Rt+1), for the

GARCH model and the four jump models. All values are expressed in annualized terms.For the four jump models, we also plot the conditional volatility that is due to the jumpcomponent,

√V art (yt+1).

9

1980 1985 1990 1995 2000 2005 2010­0.25

­0.2

­0.15

­0.1

­0.05

0

0.05

0.1

MJ

1980 1985 1990 1995 2000 2005 2010­0.25

­0.2

­0.15

­0.1

­0.05

0

0.05

0.1

VG

1980 1985 1990 1995 2000 2005 2010­0.25

­0.2

­0.15

­0.1

­0.05

0

0.05

0.1

NIG

1980 1985 1990 1995 2000 2005 2010­0.25

­0.2

­0.15

­0.1

­0.05

0

0.05

0.1

CGMY

Fig. IA.4. Filtered jump components: returns-based MLE from 1980-2009. We use the dailyreturns MLE estimates in Table IA.2 and apply a particle filtering (PF) algorithm with 10,000particles to estimate the jump component of daily returns from January 1995 to December2009. The solid line plots the time series of jumps in returns while the grey marker ’.’plots thedaily S&P 500 returns. Jumps in the MJ model follow the finite-activity Compound Poissonprocess. Jumps in the VG, NIG and CGMY models follow the infinite-activity VarianceGamma, Normal Inverse Gaussian, and CGMY processes, respectively.

10

Table IA.1RIVRMSE and RIVRMSE ratiosWe use joint MLE estimates from Table 3 in the main paper to compute the mean of

Relative Implied Volatility Root-Mean-Squared Error (RIVRMSE) for various moneyness,and maturity bins. GARCH refers to the Heston-Nandi GARCH(1,1) model which has nojump component. MJ refers to the model that relies on the finite-activity Merton jumpprocess. VG, NIG and CGMY refer to the models that rely on the infinite-activity VarianceGamma, Normal Inverse Gaussian, and CGMY jump processes, respectively. The RIVRMSEis reported in percentages for the GARCH model. For the jump models, we report theirRIVRMSE ratios with respect to the GARCH model.

RIVRMSE

GARCH Merton VG NIG CGMY

F/K<0.95 17.43% 0.888 0.893 0.876 0.9220.95<F/K<0.975 16.86% 0.918 0.913 0.915 0.8820.975<F/K<1.00 15.96% 0.962 0.938 0.948 0.8751.00<F/K<1.025 14.57% 0.930 0.926 0.929 0.9121.025<F/K<1.05 14.54% 0.956 0.942 0.873 0.9551.05<F/K<1.10 15.40% 0.985 0.953 0.796 0.974

1.10<F/K 18.27% 1.020 1.001 0.675 0.979All 16.26% 0.945 0.934 0.866 0.926

RIVRMSEGARCH Merton VG NIG CGMY

DTM<30 17.10% 0.962 0.970 0.883 0.93430<DTM<80 17.15% 0.940 0.931 0.855 0.892

80<DTM<180 15.82% 0.967 0.940 0.853 0.918180<DTM<250 15.28% 0.959 0.942 0.881 0.949

250<DTM 15.99% 0.904 0.912 0.888 0.978All 16.26% 0.945 0.934 0.866 0.926

RIVRMSE ratio with respect to GARCH

Panel A: Sorted by moneyness

Panel B: Sorted by maturity

RIVRMSE ratio with respect to GARCH

11

Table IA.2Daily return MLE estimates: 1980-2009We apply MLE to daily returns on the S&P500 index from January 1980 to December

2009. We report the results for the GARCH model which has no jump component and thefour jump models that we consider: MJ, VG, NIG, CGMY. Reported in brackets underneatheach estimate is the standard error computed using the outer product of the gradients. PanelA reports the estimates of conditional mean parameters. Panel B reports the estimates for theGARCH parameters governing the volatility and jump dynamic. Panel C reports estimatesfor the jump parameters. We report the time-series means of the conditional return momentsimplied by the parameter estimates. "Average volatility" reports the mean of daily returnvolatilities calculating and expressed in annualized terms. "Average skewness" and "Averagekurtosis" report the means of daily skewness and excess kurtosis, respectively. "Averageleverage" reports the sample mean of the daily correlations between returns and variance.The last row reports the log-likelihood values from the MLE.

12

Table IA.2 (continued...)Daily return MLE estimates: 1980-2009

GARCH MJ VG NIG CGMY

Panel A: Conditional mean paramterμz 1.49E+00 ­2.60E+00 1.40E­01 1.45E+00 4.90E­05

(1.04E+00) (1.37E+00) (1.90E­02) (2.45E­01) (2.21E­08)μy 2.03E­02 1.00E­02 3.89E­01 (4.16E+01)

(1.33E­02) (1.11E­02) (5.29E­02) (6.06E­03)Panel B: Volatility and jump intensity parameters

wz ­7.47E­13 ­1.57E­06 ­1.42E­06 ­1.36E­06 ­1.69E­06(1.36E­07) (1.03E­07) (1.26E­09) (1.85E­09) (3.08E­10)

bz 8.82E­01 9.45E­01 9.33E­01 9.32E­01 9.35E­01(7.29E­03) (2.25E­03) (5.29E­05) (7.07E­05) (8.00E­06)

az 4.21E­06 2.60E­06 2.60E­06 2.55E­06 2.56E­06(2.19E­07) (8.19E­08) (1.94E­08) (2.21E­08) (2.73E­09)

cz 1.38E+02 1.27E+02 1.45E+02 1.49E+02 1.45E+02(9.98E+00) (2.74E­05) (4.19E­08) (9.09E­01) (1.54E­01)

wy ­6.48E­04 ­2.98E­04 ­4.04E­06 ­8.36E­08(2.60E­11) (4.37E­07) (7.15E­09) (1.19E­11)

by 9.85E­01 9.29E­01 9.33E­01 9.38E­01(5.22E­10) (1.22E­04) (8.77E­05) (1.21E­05)

ay 9.33E­04 5.41E­04 8.99E­06 1.70E­07(2.02E­10) (7.55E­06) (1.66E­07) (4.53E­10)

cy 1.82E+01 1.34E+02 1.59E+02 1.54E+02(1.44E+01) (5.54E­01) (1.07E+00) (1.17E­01)

Panel C: Jump parameters

θ ­3.12E­03 ­7.19E+00(1.91E­03) (1.12E­02)

δ 2.80E­02(1.38E­03)

α 1.48E+01(6.54E­02)

β 6.31E+02 ­9.27E+00(6.84E+00) (1.64E­01)

G 7.89E+00(2.80E­03)

M 1.08E+01(4.04E­03)

Y 1.80E+00(3.42E­05)

Average volatility 0.165 0.169 0.174 0.194 0.169Average skewness 0.00 ­0.19 ­0.82 ­7.38 ­0.88Average excess kurtosis 0.00 5.84 17.07 354.23 15.83Average  leverage 0.98 ­0.73 ­0.83 ­0.78 ­0.75Log Likelihood 24482 24732 24742 24732 24770

13