#### A. Time-Series Specification Tests

To implement the time-series-based tests, we use a representative option price to compute a model-based estimate of *V*_{t}. We select a representative daily option price that (1) is close to maturity (to minimize the American feature), (2) is at the money, (3) is not subject to liquidity concerns, (4) is an actual transaction (not recorded at the open or the close of the market), (5) has a recorded futures transaction occurring at the same time, and (6) is a call option (to minimize the impact of the American early exercise feature). Appendix B describes the procedure in greater detail.

We also report the summary statistics using the VIX index and ATM implied volatilities extracted from daily transactions using our interpolation scheme (see Whaley (2000) for a description of the VIX index). Although not reported, we also compute all of the statistics using a sample of put options, and none of our conclusions change. We adjust the options for the American feature, as described in Appendix A. We use interpolated Treasury bill yields as a proxy for the risk-free rate.

Table II summarizes the implied volatilities and scaled returns for the different data sets and models. In the first panel, the first two rows labeled “VIX” provide summary statistics for the VIX index (including and excluding the crash of 1987), the rows labeled “Calls” provide statistics for our representative call option data set, and the rows labeled “Interpolated” use the ATM implied volatility interpolated from all of the daily transactions.^{12} In this first panel, the implied volatility is based on the Black–Scholes model (BSIV) and the subsequent panels report model-based (as opposed to Black–Scholes) implied variances.

Table II. ** Volatility and Return Summary Statistics** The first three rows provide summary statistics for variance increments and standardized returns using the VIX index, a time series of call option implied volatility (see Appendix B), and the ATM interpolated implied volatility (see Appendix B). In these three cases, the variance used is that from the Black–Scholes model. The second, third, and fourth panels contain model implied variances for the SV, SVJ, and SVCJ models assuming options are priced based on the objective measure. We also include risk premiums (RP) and document the effect of increasing σ_{v} in the SV model. | Period | *V*_{kurt} | *V*_{skew} | *R*_{kurt} | *R*_{skew} |
---|

VIX (BSIV) | 1987 to 2003 | 2996.58 | 50.41 | 13.72 | −1.02 |

1988 to 2003 | 20.93 | 1.74 | 5.69 | −0.43 |

Calls (BSIV) | 1987 to 2003 | 1677.16 | 32.78 | 22.99 | −1.46 |

1988 to 2003 | 15.17 | 1.25 | 5.64 | −0.40 |

Interpolated (BSIV) | 1987 to 2003 | 2076.58 | 38.21 | 21.04 | −1.38 |

1988 to 2003 | 25.17 | 1.79 | 5.82 | −0.43 |

SV Model | 1987 to 2003 | 1035.71 | 23.85 | 17.39 | −1.18 |

1988 to 2003 | 14.33 | 1.29 | 6.04 | −0.41 |

SV Model (RP) | 1987 to 2003 | 907.57 | 21.85 | 17.87 | −1.20 |

1988 to 2003 | 13.44 | 1.25 | 5.74 | −0.40 |

SV Model (σ_{v}= 0.2) | 1987 to 2003 | 1039.98 | 23.91 | 17.97 | −1.22 |

1988 to 2003 | 14.41 | 1.29 | 6.10 | −0.43 |

SVJ Model | 1987 to 2003 | 850.41 | 21.16 | 17.75 | −1.20 |

1988 to 2003 | 15.66 | 1.37 | 6.10 | −0.42 |

SVJ Model (RP) | 1987 to 2003 | 1048.51 | 24.21 | 15.91 | −1.06 |

1988 to 2003 | 16.01 | 1.40 | 7.09 | −0.42 |

SVCJ Model | 1987 to 2003 | 1015.13 | 23.62 | 16.62 | −1.12 |

1988 to 2003 | 15.16 | 1.34 | 6.31 | −0.40 |

SVCJ Model (RP) | 1987 to 2003 | 546.52 | 16.08 | 15.96 | −1.02 |

1988 to 2003 | 13.44 | 1.38 | 6.77 | −0.35 |

Although there are some *quantitative* differences across data sets, the *qualitative* nature of the results is unchanged: We observe large positive skewness and excess kurtosis in the variance increments and negative skewness and positive excess kurtosis in standardized returns. The minor variations across the data sets are due to differences in underlying indices (the VIX is based on the S&P 100 index) and in the timing and nature of the price quotes (the VIX is based on close prices, the calls are actual transactions in the morning, and the interpolated set averages all transactions in a given day).

For the formal tests, we use the call option data set. Our conclusions are the same using the other data sets, although the call option data have fewer issues (interpolation, averaging effects, stale quotes, etc.). The bottom three panels in Table II report statistics using model-based implied variances for the SV, SVJ, and SVCJ models using three sets of parameters. The first set is from EJP who, as we mention earlier, report higher σ_{v} estimates than other papers. Because the parameter σ_{v} primarily controls the kurtosis of the volatility process, this set of parameters gives the SV and SVJ models the best chance of success. For robustness, we include statistics using two additional parameter sets. The results in the rows labeled “RP” incorporate risk premia in order to gauge their impact on implied variances.^{13} The third set of results uses the SV model and σ_{v}= 0.20, which is roughly five standard deviations away from the point estimate in EJP in the SV model.

Table III provides quantiles of the finite sample distribution for each of the statistics using the Monte Carlo procedure described in the previous section and for each of the model-parameter configurations. Note that these results are simulated under the -measure. Thus, there are no separate entries for the cases incorporating risk premia, as the -measure behavior does not change.

Table III. ** Statistics' Finite Sample Distribution** For each model and set of parameters, we report the appropriate quantiles from the statistics' finite sample distribution. The base parameters are taken from Eraker, Johannes, and Polson (2003) as reported in Table I. | Quantile | *V*_{kurt} | *V*_{skew} | *R*_{kurt} | *R*_{skew} |
---|

SV model | 0.50 | 3.27 | 0.34 | 3.02 | −0.05 |

0.95 | 3.51 | 0.41 | 3.14 | −0.10 |

0.99 | 3.67 | 0.43 | 3.19 | −0.12 |

SV model | 0.50 | 3.55 | 0.48 | 3.05 | −0.06 |

σ_{v}= 0.2 | 0.95 | 3.96 | 0.55 | 3.16 | −0.12 |

0.99 | 4.26 | 0.60 | 3.23 | −0.14 |

0.50 | 3.05 | 0.15 | 22.05 | −1.48 |

SVJ model | 0.95 | 3.23 | 0.22 | 106.05 | −5.07 |

0.99 | 3.34 | 0.26 | 226.77 | −8.66 |

0.01 | 261.02 | 9.94 | 7.73 | −0.63 |

0.05 | 320.24 | 13.18 | 10.72 | −0.92 |

SVCJ model | 0.50 | 615.40 | 21.04 | 24.77 | −1.91 |

0.95 | 1649.03 | 34.87 | 78.90 | −4.15 |

0.99 | 2500.51 | 43.73 | 175.67 | −6.66 |

0.01 | 3.28 | 0.20 | 3.02 | 0.06 |

SVCJ model | 0.05 | 13.21 | 1.01 | 3.21 | −0.02 |

μ_{v}= 0.85 | 0.50 | 217.70 | 8.98 | 7.13 | −0.46 |

λ= 0.0026 | 0.95 | 1150.76 | 27.53 | 37.92 | −2.20 |

0.99 | 2012.62 | 39.34 | 94.16 | −4.11 |

The SV model cannot generate enough positive skewness or excess kurtosis to be consistent with the observed data. For example, the model generates *V*_{kurt}= 3.67 at the 99th quantile, which is orders of magnitude lower than the value observed in the data (around 1,000). Similarly, the SV model cannot generate the large positive skewness observed in the data. We also note in passing that, not surprisingly, the SV model cannot come close to generating the observed nonnormalities in returns either.

Before concluding that the SV model is incapable of capturing the behavior of option implied volatility, it is important to document that our results are robust. To do so, we show that the results are unchanged even if we ignore the crash of 1987, we account for volatility risk premia, or we increase σ_{v}. The rows labeled “1988 to 2003” in Table II provide the statistics from 1988 to 2003, a period excluding the crash of 1987.^{14} Based on the post-1987 sample, the SV model is still incapable of generating these observed statistics, even though the parameters are estimated including the crash. If the SV model were estimated using post-1987 data, it is very likely that θ_{v}, κ_{v}, and σ_{v} would be lower, which implies that the model generates even less nonnormalities. The conclusion is unchanged even if we increase σ_{v} to 0.20.

Finally, the row labeled “SV model (RP)” in Table II indicates that the results are robust to realistic risk premia. Diffusive volatility and volatility jump risk premia change the level and speed of mean reversion in volatility, which can have a significant impact on implied variance in periods of very high volatility (e.g., October 1987). Risk premia, however, cannot explain the nonnormalities in the observed data. This is most clear in the post-crash subsample, in which risk premia have a minor impact. Thus, we can safely conclude that the SV model is incapable of capturing the observed behavior of option prices.

In the SV model, volatility increments over short time intervals,

- (9)

are approximately conditionally normal (see also Table III). The data, however, are extremely nonnormal, and thus the square root diffusion specification has no chance to fit the observed data.

The following example provides the intuition by way of specific magnitudes. Consider the mini-crash in 1997: On October 27th the S&P 500 fell about 8% with Black–Scholes implied volatility increasing from 26% to 40%. In terms of variance increments, daily variance increased from 2.69 to 6.33, which translates into a standardized increment, of 2.22. To gauge the size of this move, it can be compared to the volatility of standardized increments over the previous 3 months, which was 0.151 (remarkably close to the value σ_{v}= 0.14 used above). Thus, the SV would require a roughly 16-standard deviation shock to generate this move. This example shows the fundamental incompatibility of the square root specification with the observed data. What we have here is not an issue of finding the right parameter values; rather, the model is fundamentally incapable of explaining the observed data. Whaley (2000) provides other examples of volatility “spikes.”

The SV and SVJ models share the same square root volatility process, suggesting the SVJ model is also incapable of fitting the observed data. The third panel in Table III indicates that it does generate different implied variances (due to the different volatility parameters and jumps), but it cannot generate the observed skewness or kurtosis. Subsamples or risk premia do not change the conclusion. Since the SV and SVJ models share the same volatility process, the conclusions are unchanged with σ_{v}= 0.20. The SVJ model can generate realistic amounts of skewness and kurtosis in returns. This should not be surprising as the jumps generate the rare, large negative returns observed in prices.

The lower panels in Tables II and III demonstrate that the SVCJ model is capable of capturing both the behavior of implied variances and standardized returns for the full sample. In Table III, the panel reports the 1st, 5th, 50th, 95th, and 99th quantiles of each of the statistics. For example, *V*_{kurt} based on option prices is about 1,000, and the sample statistics fall somewhere between the 50th (about 600) and 95th quantile (1,600). The skewness generated by the model is almost identical to the value observed in the data; with the 50th quantile equal to 21.04 in simulations, compared to 21.16 in the data. The infrequent, exponentially distributed jumps in volatility naturally generate the combination of high kurtosis and positive skewness observed in the data. The model can also capture the conditional distribution of returns. The final rows in Table II indicate that the conclusions are unchanged if we include diffusive volatility and jump risk premia. The kurtosis in the SVCJ model falls in the full sample with risk premia because the jump premia (volatility and jumps in prices) alter the model-implied variances, especially during the crash of 1987.

A comparison of the subsamples in Table II with the quantiles generated by the SVCJ models using the base parameters indicates that over the post-1987 period, the SVCJ model with parameters from EJP generates too much kurtosis and skewness. This can be seen by comparing, for example, *V*_{kurt} from 1988 to 2003 in Table II with the 1st quantile for the SVCJ model in Table III. This result is not at all surprising since the base case parameters in EJP are estimated using data that include the crash of 1987: Because jumps are rare events that generate conditional nonnormalities, if one removes these outliers the observed data (variance increments or standardized returns) become more normal, by construction. Of course, this does not indicate that the SVCJ model is misspecified, but rather when using the parameters estimated using the full sample, that it generates too much excess kurtosis and skewness. Naturally, if the SVCJ model were estimated omitting the crash of 1987, it is likely that the parameters governing the jump sizes would change.

To document that this is not a generic problem with the SVCJ model, the final panel in Table III shows that if one reduces the jump intensity to λ= 0.0026 and μ_{v}= 0.85 (about two standard errors below the point estimates in EJP), the model fits both periods within a 5% to 95% confidence band.^{15} Note that we do not suggest using these ad hoc parameters; we simply use them to illustrate the flexibility of the SVCJ model. The key point is to contrast this result with those that obtain using the SV and SVJ models: These models, due to the diffusion specification, could not fit the data over *either* of the samples, even using a value of σ_{v} that is implausibly high.

We conclude that the SV and SVJ models are incapable of capturing the time-series behavior of option implied variances, while the SVCJ model can easily capture the observed behavior. Since a primary goal of this paper is to estimate risk premia, it is important to have a well-specified model as model misspecification can easily distort risk premia estimates. Finally, our results are related to Jones (2003), who shows that the square root and constant elasticity of variance models cannot explain the dynamics of implied volatility. We confirm Jones's (2003) findings, and, in addition, we provide a model with jumps in volatility that is capable of capturing the observed dynamics.

#### B. Model Specification and Risk Premium Estimates

Table IV provides risk premium estimates and overall option fit for each model, and Table V evaluates the significance of the model fits. We estimate the risk-neutral parameters and compute total RMSE using the procedure in Section II.C based on the cross section of option prices. While the time-series results above indicate that the SV and SVJ models are misspecified, we still report cross-sectional results for these models in order to quantify the nature of the pricing improvement in the SVCJ model and to analyze the sensitivity of the factor risk premia to model misspecification.

Table IV. ** Risk-Neutral Parameter Estimates** For each parameter and model, the table gives the point estimate, computed as the average parameter value across 50 bootstrapped samples, and the bootstrapped standard error. For the SVJ and SVCJ models, an entry of σ_{s} in the column indicates that we impose the constraint . | η_{v} | (%) | (%) | | RMSE (%) |
---|

SV | 0.005 (0.07) | — | — | — | 7.18 |

SVJ | 0.010 (0.03) | −9.97 (0.51) | σ_{s} | — | 4.08 |

SVJ | 0.006 (0.02) | −4.91 (0.36) | 9.94 (0.41) | — | 3.48 |

SVJ | 0 | −9.69 (0.58) | σ_{s} | — | 4.09 |

SVJ | 0 | −4.82 (0.33) | 9.81 (0.58) | — | 3.50 |

SVCJ | 0.030 (0.21) | −6.58 (0.53) | σ_{s} | 10.81 (0.45) | 3.36 |

SVCJ | 0.031 (0.18) | −5.39 (0.40) | 5.78 (0.70) | 8.78 (0.42) | 3.31 |

SVCJ | 0 | −7.25 (0.50) | σ_{s} | 5.29 (0.18) | 3.58 |

SVCJ | 0 | −5.01 (0.38) | 7.51 (0.83) | 3.71 (0.22) | 3.39 |

Table V. ** Model Comparison Results** Comparison of the RMSEs across models. The table reads as follows: The probability that model [name in a row] is better than model [name in a column] by [number in a row]% is equal to [number in the intersection of the respective row and column]. The numbers in parentheses are out-of-sample values. For example, the probability that the RMSE of the SVCJ model is smaller than the RMSE of the SVJ model by 10% is 0.76. | SV | SVJ | SVJ | SVCJ |
---|

| | |
---|

| 5% | | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) |

SV | 10% | | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) |

15% | | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) |

40% | | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) |

5% | 1.00 (1.00) | | 1.00 (1.00) | 0.26 (0.66) |

SVJ | | 10% | 1.00 (1.00) | | 0.98 (0.98) | 0.04 (0.36) |

15% | 1.00 (1.00) | | 0.42 (0.66) | 0.00 (0.16) |

40% | 1.00 (1.00) | | 0.00 (0.02) | 0.00 (0.02) |

5% | 1.00 (1.00) | 0.00 (0.00) | | 0.00 (0.02) |

SVJ | | 10% | 1.00 (1.00) | 0.00 (0.00) | | 0.00 (0.00) |

15% | 1.00 (1.00) | 0.00 (0.00) | | 0.00 (0.00) |

40% | 0.70 (0.70) | 0.00 (0.00) | | 0.00 (0.00) |

5% | 1.00 (1.00) | 0.06 (0.00) | 0.96 (0.80) | |

SVCJ | | 10% | 1.00 (1.00) | 0.00 (0.00) | 0.76 (0.36) | |

15% | 1.00 (1.00) | 0.00 (0.00) | 0.26 (0.00) | |

40% | 0.96 (0.96) | 0.00 (0.00) | 0.00 (0.00) | |

We discuss model specification and risk premia in turn. Our procedure generates the following intuitive metric for comparing models. We compute the number of bootstrapped samples for which the overall pricing error (measured by the relative difference between BSIVs) for one model is lower than another by 5%, 10%, 15%, or 40%.

Because it is difficult to statistically identify η_{v}, we report the results for versions of the models with or without a diffusive volatility risk premium. Throughout this section, we constrain to be equal to λ and use the value from EJP. In general, it is only possible to estimate the compensator, , and not the individual components separately. Pan (2002) and Eraker (2004) imposed the same constraint. The estimate in EJP implies approximately 1.5 jumps per year, which is higher than most other estimates and will result in conservative price jump and volatility jump risk premia estimates.

Finally, for the jump models, we consider cases that depend on whether or not σ_{s} is equal to . This is the first paper that allows these parameters to differ. We believe it is important to document the size of this risk premium and how it affects the estimates of the other premia.

##### B.1. Model Specification: Pricing Errors

A number of points emerge from Tables IV and V. Regardless of the assumptions on the risk premium parameters, the SVJ and SVCJ models provide significant pricing improvement over the SV model. This is true in a point-wise sense in Table IV, as the RMSEs of the pricing errors are reduced by almost 50%, and is also true based on the bootstrapped samples. Under any of the risk premia assumptions and in all 50 bootstrapped samples, the SVJ and SVCJ models provide at least a 15% pricing error improvement over the SV model, and in most samples, more than a 40% improvement. These results are consistent with BCC (who find a 40% improvement) and in contrast to Bates (2000), Eraker (2004), and Pan (2002). The reasons for our clear results are twofold: We impose time-series consistency, and we use option prices that span a long time period.

Next, consider a comparison of the overall pricing errors in the SVJ and SVCJ models in Table IV. The SVJ model, with no constraints on risk premia, has an overall pricing error of 3.48%. By comparison, the unconstrained SVCJ model has pricing errors of 3.31%, which is an improvement of 5%. If we impose in the SVJ model, the SVCJ model generates a larger improvement of 18% (4.08% vs. 3.36%). If increases drastically, allowing the SVJ model to generate more conditional kurtosis in returns, a role very similar to that played by jumps in volatility. In fact, a comparison of the unconstrained SVJ model and the SVCJ model with the same number of parameters ( in the SVCJ model) shows that the pricing errors are quite similar: 3.48% and 3.36%, respectively. This result is consistent with objective measure time-series results in Table III, which show that the SVJ model can generate realistic amounts of nonnormalities in returns through jumps in prices.

In Table V, we test for differences for the most important variations of the models. We compare the SV model, the SVJ model with and without constraints on , and the SVCJ model assuming . The constrained SVCJ model is of interest for model fit because the unconstrained SVJ and constrained SVCJ models have the same number of parameters. The SVCJ model has pricing errors that are at least 5% lower than those of the SVJ model in 96% of the bootstrapped replications. Thus, the SVCJ model provides a statistically significant improvement in overall option fit. However, a comparison of the unconstrained SVJ model with the constrained SVCJ model indicates that the SVCJ model outperforms by more than 5% in only 6% of the trials, an indication that SVCJ adds little to the cross section of returns when the risk premia are unconstrained in SVJ. In order to ensure the robustness of our findings, we also evaluate “out of sample” RMSEs by using the parameter values reported in Table IV on 50 randomly selected days in our sample that were not used for risk premia estimation. For each of the 50 days, we estimate the spot variance and then computed RMSEs, holding the risk premia constant. The results are qualitatively the same, with a slight improvement in the unconstrained SVJ model relative to the SVCJ model.^{16}

We sort option pricing errors by maturity, strike, and volatility. The only major pattern that emerges is that pricing errors tend to be higher for all models in periods of high volatility. Given that our objective function focuses on absolute differences in volatility (as opposed to percentage differences), this is not a surprise. Pan (2002) finds a similar pattern. The SVJ and SVCJ models outperform the SV model in all categories, however, there is little systematic difference between the SVJ and SVCJ models.

We conclude that there is some in-sample pricing improvement by including jumps in volatility, but the effect is modest. However, as our tests in the previous section indicate, the SVJ model cannot capture the dynamics of *V*_{t}. Thus, the SVCJ is our preferred model as it is consistent simultaneously with the time series and the cross section.

##### B.2. Estimates of Risk Premia

Table IV summarizes the -measure parameter estimates. For each parameter and specification, the table provides estimates and bootstrapped standard errors based on 50 replications. A number of interesting findings are apparent.

The diffusive volatility risk premium η_{v} is insignificant in every model. As mentioned earlier, there are several reasons to believe that this parameter is difficult to identify, and our finite-sample procedure generates quite large standard errors relative to the point estimate. This does not necessarily mean that η_{v}= 0; instead it suggests only that we cannot accurately estimate the parameter. When we constrain η_{v}= 0, there is virtually no change in the RMSE, which indicates that its impact on the cross section is likely to be minor.

Why is it so hard to estimate η_{v}, even with a long data set? The main reason is that, as shown in Table VI, the implied volatility term structure is flat, at least over the option maturities that we observe. Focussing on the whole sample, the difference between short- (less than one month) and longer-dated (3 to 6 months) implied volatility is only about 0.1%.^{17} In the context of our models, it is important to understand what could generate such a flat term structure. Since jumps in prices contribute a constant amount to expected average variance over different horizons, any variation in the term structure shape will arise from the stochastic volatility component.

Table VI. ** The Average Term Structure of S+P 500 Index Implied Volatility** For each year, we compute average at-the-money implied volatility from our Black–Scholes implied volatility curves as computed in Appendix B and bin the results into three categories: options that mature in under 1 month, those that mature in 1 to 2 months, and those that mature in 3 to 6 months. Year | 1 Month | 2 Month | 3–6 Months |
---|

1987 | 28.01 | 28.96 | 21.62 |

1988 | 21.67 | 22.30 | 22.39 |

1989 | 14.79 | 15.35 | 15.90 |

1990 | 19.50 | 19.99 | 20.22 |

1991 | 16.12 | 16.55 | 17.02 |

1992 | 13.14 | 13.67 | 14.62 |

1993 | 10.81 | 10.99 | 12.06 |

1994 | 11.58 | 11.84 | 12.98 |

1995 | 10.44 | 10.62 | 11.55 |

1996 | 14.34 | 14.37 | 14.64 |

1997 | 20.06 | 20.24 | 20.16 |

1998 | 21.20 | 22.33 | 23.57 |

1999 | 21.33 | 22.18 | 24.02 |

2000 | 20.63 | 20.32 | 21.36 |

2001 | 23.69 | 23.43 | 21.82 |

2002 | 25.20 | 24.60 | 23.25 |

2003 | 28.72 | 28.09 | 26.27 |

Mean | 18.90 | 19.17 | 19.03 |

The stochastic volatility model could generate a flat average term structure via two conduits. First, η_{v} could be small, implying that the term structure is flat on average. Second, η_{v} could be large (of either sign), but the term structure could still be flat over short horizons if risk-neutral volatility is very persistent. Given the near unit root behavior of volatility under , volatility will also be very persistent under for a wide range of plausible η_{v}s, generating a flat term structure of implied volatility. This implies that we can only distinguish between these two competing explanations if we have long-dated options.

It is also the case that using a more efficient estimation procedure, such as one including returns and option prices, would improve the accuracy of the parameter and risk premia estimates. However, joint estimation must still confront the fact that the term structure is flat, which implies that merely using a different estimation procedure is not likely to alleviate the problem that η_{v} is insignificant, unless the procedure incorporates long-dated options. In large part, this explains why the existing literature using options and returns finds unstable, insignificant, or economically small estimates.

In both the SVJ and SVCJ models, there is strong evidence that , an effect that has not previously been documented. Moreover, as we note in the previous paragraph, this has important implications for the magnitudes of the premium attached to the mean price jump size. Estimates of σ_{s} in the SVJ model are around 4%, while estimates of are more than 9%. However, it appears that this premium is largely driven by specification. As mentioned earlier, the time-series tests indicate the presence of jumps in volatility. These jumps generate large amounts of excess kurtosis in the distribution of returns. Since the SVJ model does not allow volatility to jump, it can only fit observed option prices by drastically increasing to create a large amount of risk-neutral kurtosis. When jumps in volatility are allowed in the SVCJ model, estimates of fall to about 6% (with a standard error of 0.7%) in the unconstrained SVCJ model and to around 7.5% (with a standard error of 0.83%) when η_{v} is constrained to be zero. Unlike the significant estimates of and the insignificant estimates of η_{v} in all models, the very large risk premium attached to σ_{s} in the SVJ model appears to be driven by model specification, although, even with jumps in volatility, there is evidence for a modest premium.

This discussion illustrates that, given sufficiently flexible risk premia, one cannot distinguish among different models based on options' cross section only. However, a good option pricing model must be able to fit both cross-sectional and time-series properties. In our case, this means that SVCJ is the only model capable of successfully addressing all aspects of the data.

##### B.3. Interpreting the Risk Premia

In this section, we examine the degree to which the risk premia are reasonable, and we assess their economic significance. To do so, we examine the mean price jump size premia in the context of simple equilibrium models, the contribution of jump risk to the overall equity premium, and the impact of price and volatility jump risk premia on option returns.

Next, consider the jump risk contribution to the equity premium. Using -measure parameter estimates from EJP and the decomposition of the equity risk premium from Section I.A, the contribution of the price jump risk premia is 2.7% and 2.9% per annum in the SVJ and SVCJ models (with η_{v}= 0), respectively. Over our sample, the equity premium is about 8%, implying that jumps generate roughly one third of the total premium. As a benchmark, time-series studies find that jumps in prices explain about 10% to 15% of overall equity volatility (EJP or Huang and Tauchen (2005)). It appears that jumps generate a relatively larger share of the overall equity premium. While significant, it is difficult to argue that these premia are unreasonable.

Finally, we consider our risk premium estimates in the context of a rapidly growing literature that identifies a “put-pricing puzzle” (see e.g., Bondarenko (2003), Driessen and Maenhout (2004b), Jones (2006), or Santa-Clara and Saretto (2005)). Using data similar to ours (Bondarenko uses S&P 500 futures options from 1987 to 2000), these authors document that average monthly returns of ATM and OTM puts are approximately −40% to −95%, respectively, and have high Sharpe ratios. Naturally, average returns of this magnitude are difficult to explain using standard risk-based asset pricing models such as the CAPM or Fama–French three-factor model, and they are also puzzling from a portfolio perspective (Driessen and Maenhout (2004b)) or based on a nonlinear factor model (Jones (2006)).

Table VII. ** The Impact of Risk Premia on Option Returns in the SVCJ Model** We compare out-of-the-money (OTM) average option returns (measured in percent) and their bootstrapped percentiles reported by Bondarenko (2003) (Average, 5%, 95%) to population average options returns implied by the SVCJ model assuming zero risk premia (SVCJ-) and using the estimated risk premia (SVCJ-). The dagger (^{†}) denotes returns outside the confidence intervals. | Moneyness | 6% | 4% | 2% | 0% |
---|

Data | Average | −95.00 | −58.00 | −54.00 | −39.00 |

5% | −99.00 | −80.00 | −72.00 | −54.00 |

95% | −89.00 | −35.00 | −36.00 | −24.00 |

Model | SVCJ- | −20.70^{†} | −21.91^{†} | −21.78^{†} | −19.96^{†} |

SVCJ- | −68.46^{†} | −58.36 | −45.21 | −32.18 |

SVCJ- | −75.78^{†} | −64.67 | −50.01 | −35.27 |

Table VII provides a number of interesting implications. The SVCJ- results indicate that a model without any risk premia generates about −20% per month for average put returns. These large negative put returns are solely driven by the very high S&P 500 returns: A short put position has a high “beta” on the index (around −25 to −30 for ATM puts, see Coval and Shumway (2001)). Intuitively, if the equity premium is high, puts often end up OTM, with a 100% return to the writer. Thus, a large component of the put pricing puzzle is due to the high equity premium over the 1990s. The SVCJ- results indicate that adding a small mean price jump risk premium alone generates returns that are inside the confidence bands for most strikes. The final row indicates that adding the volatility jump risk premium generates option returns that are very close to the historical sample means. The only case that is not within the bounds is that of deep OTM puts; here, the returns are economically close.^{18}

We conclude that jump risk premia provide an attractive risk-based explanation for the put-pricing puzzle. Although the returns are extremely large, these option positions are highly levered. Since our intuition and models (CAPM) are often based on normal distributional assumptions, the high returns (and Sharpe ratios) seem puzzling. However, these models are not well suited for understanding nonnormal risks such as those embedded in jumps. In our model with jumps, once we allow for even modest jump risk premia, the returns on these option strategies are not necessarily puzzling.

Although this section argues that these jump premia are not unreasonably large and that they have important economic implications, we are agnostic about their exact sources. These jump risk premia may arise, for example, from standard utility functions; from asymmetric utility of gains and losses, which leads investors to care more about large negative returns; from the presence of heterogeneous investors with more risk-averse investors buying put options from the less risk-averse investors; from an inability to hedge jump risks; or from institutional explanations. Several papers explore some of these potential explanations. Liu, Pan, and Wang (2005) argue that an aversion to parameter uncertainty could generate the premia. Bollen and Whaley (2004) show that there are demand effects, in the sense that changes in implied volatility are related to the demand for options, and Garleanu, Pedersen, and Poteshman (2005) model demand effects with heterogenous investors. Disentangling the sources of the jump risk premia and characterizing their relationship to investor demand appears to be a fruitful avenue for future research.