#### A. Parameter Estimates

Table III reports the posterior means, posterior standard deviations, and the 1 and 99 posterior percentiles of the parameters in the various models. The parameters are quoted using a daily time interval following the convention in the time-series literature. Notice that the parameters need to be annualized to be comparable to typical results in the option pricing literature (e.g., Bates (2000), Pan (2002)).

Table III. ** Parameter Estimates** The table reports posterior means and standard deviations (in parenthesis), and 99 credibility intervals (in square brackets) for parameters in the jump diffusion models based on joint options and returns data. The parameters, η_{V} and η_{J}, denote the market premiums for volatility and jump risk, respectively. Parameter estimates correspond to a unit of time defined to be one day, and returns data scaled by 100. | SV | SVJ | SVCJ | SVSCJ |
---|

θ | 1.933 | 1.652 | 1.353 | 0.943 |

| (0.048) | (0.053) | (0.067) | (0.065) |

| [1.843, 2.064] | [1.523, 1.777] | [1.243, 1.560] | [0.767, 1.080] |

κ | 0.019 | 0.019 | 0.023 | 0.023 |

| (0.007) | (0.006) | (0.007) | (0.007) |

| [0.004, 0.035] | [0.005, 0.034] | [0.010, 0.038] | [0.010, 0.040] |

κ^{Q} | 0.009 | 0.011 | 0.011 | 0.006 |

| (0.000) | (0.000) | (0.000) | (0.000) |

| [0.009, 0.010] | [0.010, 0.011] | [0.010, 0.012] | [0.005, 0.008] |

η_{V} | 0.010 | 0.009 | 0.013 | 0.017 |

| (0.007) | (0.006) | (0.007) | (0.007) |

| [−0.005, 0.026] | [−0.006, 0.024] | [−0.001, 0.028] | [0.003, 0.033] |

ρ | −0.569 | −0.586 | −0.582 | −0.542 |

| (0.014) | (0.027) | (0.024) | (0.034) |

| [−0.601, −0.535] | [−0.652, −0.526] | [−0.646, −0.528] | [−0.620, −0.462] |

σ_{V} | 0.220 | 0.203 | 0.163 | 0.137 |

| (0.007) | (0.007) | (0.007) | (0.007) |

| [0.203, 0.240] | [0.187, 0.218] | [0.148, 0.181] | [0.122, 0.154] |

μ_{y} | | −0.388 | −6.062 | −1.535 |

| (3.456) | (2.274) | (0.184) |

| [−8.560, 7.631] | [−11.566, −0.881] | [−1.997, −1.120] |

μ^{Q}_{y} | | −2.002 | −7.508 | −7.902 |

| (1.867) | (0.932) | (0.843) |

| [−6.079, 2.022] | [−9.725, −5.527] | [−9.703, −5.984] |

η_{J} | | 1.613 | 1.446 | 6.367 |

| (3.789) | (2.474) | (0.842) |

| [−7.320, 10.478] | [−4.516, 6.939] | [4.472, 8.117] |

ρ_{J} | | −0.693 | −2.214 |

| (0.096) | (0.099) |

| [−0.856, −0.449] | [−2.435, −2.013] |

σ_{y} | | 6.634 | 3.630 | 2.072 |

| (1.081) | (1.106) | (0.302) |

| [4.972, 9.508] | [1.403, 6.261] | [1.439, 2.717] |

μ_{V} | | 1.638 | 1.503 |

| (0.790) | (0.279) |

| [0.833, 3.231] | [1.078, 2.065] |

λ_{0} | | 0.002 | 0.002 | 0.002 |

| (0.001) | (0.001) | (0.001) |

| [0.001, 0.003] | [0.001, 0.004] | [0.001, 0.004] |

λ_{1} | | 1.298 |

| (0.141) |

| [1.086, 1.606] |

There are several interesting features of the parameter estimates in Table III. We start with an examination of the estimates in the SV model. The long-term mean of the volatility process, θ is 1.93, which is relatively high. It corresponds to an annualized long-run volatility of 22%. The estimate is slightly higher than the unconditional sample variance of 1.832 (see Table II). Typically, estimates reported elsewhere of the unconditional variance of S&P 500 returns are somewhat below 1%, corresponding to an annualized value somewhat less than 15%. Our estimates of 1.933/1.832 are indicative of the relatively volatile sample period used. For the SVJ and SVCJ models, the estimates of θ are much smaller, suggesting that the jump components are explaining a significant portion of the unconditional return variance.

The κ parameter is low under both the objective and risk neutral measures. As a rule of thumb, estimates of κ can be interpreted as approximately one minus autocorrelation of volatility. Hence, estimates ranging from 0.019 to 0.025 imply volatility autocorrelations in the range 0.975–0.981 which is in line with the voluminous time-series literature on volatility models.

The speed of mean reversion parameter under the risk neutral measure, κ^{Q}, is a parameter of particular interest. The difference between these parameters across the two measures, η_{V}=κ−κ^{Q}, is the risk premium associated with volatility risk. These volatility risk premiums are estimated to be positive across all models. For the SVSCJ model, the estimate of this parameter is “significant”, in the sense that its 1 percentile is greater than zero. The volatility risk premium is similarly found to be “significant” at the 5% level for the SVCJ model, and insignificant for the SVJ and SV models. A positive value for this parameter implies that investors are averse to changes in volatility. For option prices, this implies that whenever volatility is high, options are more expensive than what is implied by the objective measure as the investor commands a higher premium. Conversely, in low-volatility periods, the options are less expensive, ceteris paribus. Figure 1 quantifies this effect in terms of annualized Black–Scholes volatility for an ATM option across maturities for different values of the initial volatility, *V*_{t}. As can be seen by the figure, whenever volatility is high, the risk premium is positive at all maturities and peaks at about 2 percentage points (annualized) at 130 days to maturity. Conversely, the low initial spot volatility gives a decreasing and then increasing premium. The limiting premium (as maturity increases) is zero since the limiting (unconditional) stock price distribution does not depend on the speed of mean reversion under either measure.

Next, we examine the volatility of volatility, σ_{V}, and correlation, ρ, between Brownian increments. Our estimates of σ_{V} are a little more than two times those obtained for the same models in Eraker et al. (2003) from time-series analysis on a longer sample of S&P 500 returns (see below). The correlation coefficient is also larger in magnitude. There is a certain disagreement in the literature as to the magnitude of both these parameters, as well as whether estimates obtained previously, are reasonable. We examine these issues in more detail below.

We now turn to discuss the jump size parameters and the related jump frequencies. First, we note that the jumps occur extremely rarely: The λ estimates in Table III indicate that one can expect about two to three jumps in a stretch of 1,000 trading days. The unconditional jump frequency is only marginally higher for the state dependent SVSCJ model. Whenever spot volatility is high, say a daily standard deviation of 3%, the estimate of λ_{1} is indicative of an instantaneous jump probability of about 0.003—a 50% increase over the constant arrival intensity specifications. According to SVJ estimates, observed jumps will be in the range [−13.4,12.6] with 95% probability when they occur.

A notable implication of the estimates of the mean jump sizes, μ_{y} and μ^{Q}_{y} in Table III, is that under both measures these parameters are difficult to identify. In particular, this parameter is difficult to estimate accurately under the objective risk probability measure. The reason is simple: Option prices do not depend on μ_{y}, so this parameter is only identified through the returns data. To illustrate how the difficulties arise, assume a hypothetical estimator based on knowledge of when the jumps occurred and by how much the process jumped. Then μ_{y} would be identified as the mean of these observations' jump sizes. But, since there are very few observations in the sample for which jumps are estimated to have occurred, this hypothetical estimator becomes very noisy. In reality, it is even more difficult to identify μ_{y} because jump times and sizes are not known. Not only does this introduce more noise than the hypothetical estimator, but it gives an improper posterior distribution if the investigator does not impose an informative (proper) prior.^{5}

There are equivalent problems in estimating the other parameters in the jump distribution for low jump-frequencies. This manifests itself by the fairly high posterior standard deviations in jump-size parameters, particularly for the SVJ model. This contrasts with the parameters which govern the dynamics in the volatility process, which, without exception, are extremely sharply estimated. The posterior standard deviations are somewhat smaller in the SVCJ model. The reason is that this model incorporates simultaneous jumps in volatility and returns, and since the latent volatility is very accurately estimated (see below), so are the jump times. Since jump times are accurately estimated, it is easier for the method to find the jump sizes, and hence jump size parameters under the objective measure. As a consequence, for the SVCJ model, we estimate μ_{y} more accurately than for the SVJ model.

The difference between the mean jump sizes under the respective probability measures, η_{J}, is the premium associated with jump risk, and is therefore of particular interest. The premium is estimated to be positive for all models, but with a very sizable credibility interval for both the SVJ and the SVCJ models. Hence, we cannot conclude that there is a significant market premium to jump risk for these models. For the SVSCJ model, the posterior 1 percentile for η_{J} is 4.47, indicating that for this model the jump risk premium is significant. The estimates reported here are much smaller in magnitude than corresponding estimates in Pan (2002).

The parameter estimates in Table III are interesting in light of estimates obtained elsewhere. The results in BCC are a particularly interesting reference because they are obtained purely by fitting the option prices, and are not conditioned on time-series properties of returns and volatility. The BCC estimates are indeed very similar to the ones obtained here. Notably, BCC estimate the jump frequency for the SVJ model is 0.59/252 = 0.0023 daily jump probability, and the jump-size parameters μ^{Q}_{y} and σ_{y} are −5 and −7%, respectively. Hence, using only options data, BCC obtain quite similar estimates from those reported here for the jump component of the model. Their volatility process parameters are also close to the ones reported here: Their estimate of κ and σ_{V} are 0.008 and 0.15, respectively, when converted into daily frequencies.^{6}

Pan (2002) and Bates' (2000) model specifications differ from the simple price-jump model (SVJ) considered here, in that the jump frequency, λ, depends on the spot volatility, and hence the jump size and jump frequency parameters are not really comparable to those reported in BCC and here. Interpreting the differences with this in mind, the average jump intensity point estimates in Pan are in the range [0.0007, 0.003] across different model specifications. Hence, her estimates are in the same ballpark as those reported in BCC and here. Interestingly, Bates obtains quite different results, with an average jump intensity of 0.005.^{7} Bates also reports a jump size mean ranging from −5.4 to −9.5% and standard deviations of about 10–11%. Hence, Bates' estimates imply more frequent and more severe crashes than the parameter estimates reported in BCC and in this paper. The practical implication of the difference is that his estimates will generate more skewness and kurtosis in the conditional returns distributions, and consequently also steeper Black–Scholes implied volatility smiles across all maturity option contracts. In particular, since the jump intensity in Bates depends on volatility, his model will generate particularly steep implied volatility curves whenever the spot volatility, *V*_{t}, is high.

Finally, to put the estimates in perspective, Table IV shows parameter estimates obtained by using returns data only. Estimates are shown for the SV, SVJ, and SVCJ models only. The estimates were obtained using returns collected over the period January 1970 to December 1990, so comparisons with the estimates in Table III should be done with this in mind. Not too surprisingly, the estimates in Table IV are close to those reported in EJP although some sensitivity to the sampling period in use is inevitable: the two papers use about 20 years of data, with the 1980s being common. So while the sample used in EJP includes the −7% returns on October 27, 1997 and August 31, 1998, comparably large moves did not occur in the 1970s. This leads to a different estimate of jump frequency in this paper (λ= 0.004 versus 0.0055, respectively). The other parameters are in the same ballpark as those reported in EJP. Notice again how the volatility-of-volatility parameter, σ_{V}, is noticeably smaller than in Table III.

Table IV. ** Parameter Estimates from Returns Data** The table reports posterior means and standard deviations (in parenthesis) and 99% credibility intervals (in square brackets) for parameters in the jump diffusion models based on returns data only. Parameter estimates were obtained using the estimation procedure in Eraker et al. (2003) using 5,307 time-series observations of the S&P 500 index from January 1970 to December 1990. | SV | SVYJ | SVCJ |
---|

*a* | 0.026 | 0.026 | 0.030 |

| (0.011) | (0.011) | (0.011) |

| [0.000, 0.051] | [0.000, 0.050] | [0.004, 0.056] |

θ | 0.881 | 0.834 | 0.573 |

| (0.098) | (0.122) | (0.078) |

| [0.692, 1.163] | [0.590, 1.221] | [0.397, 0.750] |

κ | 0.017 | 0.012 | 0.016 |

| (0.005) | (0.006) | (0.003) |

| [0.008, 0.030] | [0.004, 0.022] | [0.009, 0.023] |

ρ | −0.373 | −0.468 | −0.461 |

| (0.056) | (0.065) | (0.073) |

| [−0.500, −0.242] | [−0.601, −0.295] | [−0.616, −0.237] |

σ_{V} | 0.108 | 0.079 | 0.058 |

| (0.011) | (0.011) | (0.012) |

| [0.082, 0.137] | [0.061, 0.104] | [0.030, 0.078] |

μ_{y} | | −3.661 | −3.225 |

| (2.486) | (2.523) |

| [−10.752, 1.281] | [−10.086, 2.436] |

ρ_{J} | | 0.312 |

| | (1.459) |

| | [−3.580, 3.833] |

σ_{y} | | 6.628 | 4.918 |

| (1.697) | (1.272) |

| [3.714, 11.742] | [2.880, 9.295] |

μ_{V} | | 1.250 |

| | (0.381) |

| | [0.681, 2.523] |

λ | | 0.003 | 0.004 |

| (0.001) | (0.001) |

| [0.001, 0.006] | [0.001, 0.007] |

#### B. Option Price Fit

In this section, we discuss the empirical performance of the various models in fitting the historical option prices. Table V reports the posterior means of absolute option pricing errors for the different models, conditional upon moneyness and maturity.

Table V. ** Absolute Pricing Errors** The table reports mean absolute pricing errors for different option models conditional on time to maturity and moneyness. The errors are not corrected for serial correlation. All pricing errors in dollars. Maturity | | Moneyness (Strike/Spot) |
---|

<0.93 | 0.93–0.97 | 0.97–1.0 | 1.0–1.03 | 1.03–1.07 | >1.07 | All |
---|

| # | 13 | 86 | 289 | 272 | 71 | 9 | 740 |

| SV | 0.67 | 0.35 | 0.35 | 0.48 | 0.53 | 0.47 | 0.42 |

<1 m | SVJ | 0.61 | 0.35 | 0.36 | 0.46 | 0.51 | 0.55 | 0.42 |

| SVCJ | 0.69 | 0.35 | 0.32 | 0.43 | 0.48 | 0.51 | 0.39 |

| SVSCJ | 0.76 | 0.35 | 0.29 | 0.40 | 0.43 | 0.38 | 0.36 |

| # | 13 | 105 | 248 | 312 | 257 | 76 | 1011 |

| SV | 0.33 | 0.42 | 0.32 | 0.43 | 0.54 | 1.14 | 0.48 |

1–2 m | SVJ | 0.32 | 0.41 | 0.34 | 0.44 | 0.54 | 1.21 | 0.49 |

| SVCJ | 0.35 | 0.45 | 0.33 | 0.44 | 0.52 | 1.19 | 0.49 |

| SVSCJ | 0.30 | 0.49 | 0.38 | 0.46 | 0.54 | 1.20 | 0.51 |

| # | 10 | 74 | 125 | 188 | 127 | 79 | 603 |

| SV | 0.37 | 0.39 | 0.47 | 0.43 | 0.48 | 0.58 | 0.46 |

2–3 m | SVJ | 0.37 | 0.40 | 0.50 | 0.45 | 0.51 | 0.63 | 0.49 |

| SVCJ | 0.36 | 0.40 | 0.48 | 0.39 | 0.48 | 0.55 | 0.45 |

| SVSCJ | 0.36 | 0.36 | 0.41 | 0.39 | 0.51 | 0.49 | 0.43 |

| # | 22 | 68 | 110 | 194 | 140 | 176 | 710 |

| SV | 1.09 | 0.38 | 0.38 | 0.57 | 0.50 | 0.39 | 0.48 |

3–6 m | SVJ | 0.98 | 0.37 | 0.39 | 0.61 | 0.51 | 0.39 | 0.49 |

| SVCJ | 0.95 | 0.35 | 0.36 | 0.57 | 0.50 | 0.42 | 0.48 |

| SVSCJ | 1.14 | 0.46 | 0.38 | 0.49 | 0.50 | 0.45 | 0.48 |

| # | 5 | 14 | 23 | 42 | 40 | 82 | 206 |

| SV | 0.74 | 0.64 | 0.39 | 0.63 | 0.74 | 0.45 | 0.56 |

>6 m | SVJ | 0.69 | 0.68 | 0.43 | 0.67 | 0.78 | 0.46 | 0.58 |

| SVCJ | 0.73 | 0.70 | 0.42 | 0.65 | 0.72 | 0.41 | 0.55 |

| SVSCJ | 0.84 | 0.71 | 0.42 | 0.60 | 0.64 | 0.39 | 0.52 |

| # | 63 | 347 | 795 | 1008 | 635 | 422 | 3270 |

| SV | 0.71 | 0.40 | 0.37 | 0.48 | 0.53 | 0.57 | 0.47 |

All | SVJ | 0.65 | 0.40 | 0.38 | 0.49 | 0.54 | 0.60 | 0.47 |

| SVCJ | 0.66 | 0.41 | 0.36 | 0.46 | 0.52 | 0.58 | 0.46 |

| SVSCJ | 0.74 | 0.43 | 0.35 | 0.44 | 0.52 | 0.58 | 0.46 |

The results in Table V may seem surprising at first. The pricing errors for all four models are about 47 cents on average. Information about bid/ask spreads is missing from this database. However, similar data were used by BCC who reported that the spread ranged from 6 to 50 cents. Consequently, the pricing errors reported here are likely to be somewhat larger than the average bid/ask spread.

At first glance, the fact that the simple SV model fits the options data as well as the SVSCJ model seems implausible. After all, the SVSCJ model incorporates six more parameters, so how is it possible that it does not improve on the simple SV model? The answer is a combination of explanations: First, our MCMC approach does not minimize pricing errors. Simply, there are no objective functions that are optimized, but rather the results in Table V are mean errors over a large range of plausible (in the sense that they have positive posterior probability mass), parameter values. This way of conducting Bayesian model comparisons differs fundamentally from classical methods. An analogy is the computation of Bayes factors which are the ratio of likelihood functions that are averaged over the posterior distributions. This will sometimes give (marginalized) likelihoods which favor the most parsimonious model. Moreover, the jump models do not really increase the degrees of freedom by a notable amount. The number of parameters increases by six in the SVSCJ model, but there are 3,270 sample option prices, so the increase in degrees of freedom is marginal and practically negligible. This is an important difference between this study and the one by Bakshi et al. (1997). In their paper, since the model parameters are recalibrated every day, adding one parameter increases the degrees of freedom by the number of time-series observations. Notice that the improvement in fit for the SVJ model over the SV model reported in BCC, is in the order of a few cents only. By contrast, BCC report that the SV model improves more than 90 cents on the Black–Scholes model. BCC concludes that “once stochastic volatility is modeled, adding other features will usually lead to second-order pricing improvements.”

The lack of improvements in pricing results in Table V is somewhat at odds with the results in Bates (2000) and Pan (2002) who both conclude that jumps (in prices) are important in capturing systematic variations in Black–Scholes volatilities. Figure 2 presents further evidence on the fit of the various models. The figure plots the model and market implied Black–Scholes volatilities (IV) for high, medium, and low initial spot-volatility, and for different maturity classes. Notice that the IVs are very similar for all three models, and on average, they tend to fit the observed data reasonably well. There are a few exceptions:

- •.
For the short and medium maturity contracts during days of high or average spot volatility, the models do not generate sufficiently steep IVs.

- •.
For long maturity contracts trading on days with low spot volatility, all models except the SVSCJ model overprice.

The first observation motivated the generalization into state-dependent jump intensity incorporated by the SVSCJ model. Models with such state dependency will potentially generate steeper volatility smiles in high-volatility environments because the chance of observing large, negative jumps increases. However, the estimate of λ_{1} is not large enough to yield a significantly steeper volatility smile for this model relative to the simpler ones.

It is important to recognize that the results in Table V could be sensitive to the dollar/cent denomination specification for the pricing errors. For example, if the errors were measured in percent of the option price, the results would place heavier weight on short-term OTM or ITM contracts and thus, put relatively more emphasis on the tails of the return distributions at short horizons. The same effect is likely to occur if the errors are measured in terms of Black–Scholes implied volatilities. This is not only true for the pricing errors in Table V, but also for the parameter estimates in Table III. In particular, it is true that the values of the joint posterior distribution of the SVSCJ model are not too different for different parameter constellations for which the importance of jumps increases relative to that of the volatility component. In particular, parameter constellations involving more frequent jumps tend to improve the option price fit at short horizons, but deteriorate the fit for long-term contracts. Hence, judging by measures that would lend more weight to shorter contracts, such parameter constellations would potentially be found important. It may be that a different benchmark measure of performance could alter our conclusions. To examine the reasonableness of this conjecture, it is important to realize that the distribution of the pricing errors should be the guideline for the specification. To see this, assume that there were no pricing errors. We should then be able to identify all model parameters as well as latent volatility with arbitrarily good precision *regardless of whether the likelihood function is defined over dollar values, implied volatilities, or relative pricing errors*. Since this assumption is obviously violated, the all-important issue becomes the distribution of pricing errors. For example, assume that we modeled the errors in terms of implied Black–Scholes volatilities. We know that implied volatilities become increasingly variable for contracts in or out of the money (see Jackwerth and Rubenstein (1996)). Hence, if we were to specify the likelihood function over implied Black–Scholes volatility, we would have to incorporate this heteroskedastic feature of the data into the specification of the likelihood function. In doing so, we would in fact specify a likelihood function which would lend equal weight to all observations across different moneyness categories. Hence, it is not clear that short ITM/OTM contracts are given greater weight in a correctly specified likelihood function defined over relative errors or implied Black–Scholes volatilities.

Our choice of error distribution based on dollars and cents can be motivated from two observations. First, errors caused by discreteness of quotes should be uniformly distributed in dollars across moneyness and maturity. Second, and perhaps more important, the mean absolute errors in Table V are reasonably similar across moneyness and maturities. Thus, there do not seem to be systematic patterns of heteroscasticity across different option classes.

A final note of caution on the performance of the models: The lack of improvement for the jump models does not indicate that these models are incorrect. Indeed, jumps may very well be warranted to model the time-series behavior of the returns. I elaborate on this below.

#### C. Out-of-Sample Performance

The results discussed so far reflect in-sample fit obtained over the period January 1987 to December 1990. This section presents results of out-of-sample fit for the period January 1991 to March 1996. This leaves a total of 35,890 observations to be used in the out-of-sample performance study.

The procedure used to construct the out-of-sample errors is as follows. Given parameter estimates in Table III, a so-called particle filtering approach is used to estimate volatility *V*_{t} for each day *t* using information available at *t*− 1. This approach has similarities with the popular method of inserting yesterday's Black–Scholes implied volatility into the Black–Scholes model to obtain today's prices. Pricing errors resulting from this approach will generally be “close” to the ones obtained by actual minimization (with respect to *V*_{t} only). The reader should bear in mind that since the parameters are fixed, there is no way in which we can calibrate the shape of the term structure/volatility smile as market conditions change. This is therefore a very restrictive exercise, and results should be interpreted with this in mind.

Table VI presents the pricing errors broken down in maturity and moneyness categories. The average pricing errors are not dauntingly large and vary between 0.51 cents (SVSCJ) and 0.67 cents (SVCJ) on average. Hence, pricing errors are larger out-of than in-sample. No uniform relationship between the model complexity and the out-of-sample performance is evident. The SVSCJ model produces prices which on average differ from market prices by 4 cents less than the SV model. The SVJ and SVCJ models perform the worst.

Table VI. ** Out-of-Sample Absolute Pricing Errors** The table reports mean absolute pricing errors for different option models conditional on time to maturity and moneyness. Results are based on parameters estimated in Table II for the subsequent period January 1991 to March 1996. For each day, the model re-estimates spot volatility estimates using a particle filtering method. All pricing errors in dollars. Maturity | | Moneyness (Strike/Spot) |
---|

<0.93 | 0.93–0.97 | 0.97–1.0 | 1.0–1.03 | 1.03–1.07 | >1.07 | All |
---|

| # | 1213 | 2623 | 3306 | 3159 | 1398 | 161 | 11860 |

| SV | 0.19 | 0.29 | 0.40 | 0.40 | 0.17 | 0.07 | 0.32 |

<1 m | SVJ | 0.26 | 0.26 | 0.44 | 0.45 | 0.16 | 0.07 | 0.34 |

| SVCJ | 0.10 | 0.16 | 0.43 | 0.54 | 0.16 | 0.08 | 0.33 |

| SVSCJ | 0.76 | 0.38 | 0.46 | 0.36 | 0.43 | 0.50 | 0.45 |

| # | 964 | 1621 | 2830 | 3159 | 1675 | 244 | 10493 |

| SV | 0.37 | 0.50 | 0.52 | 0.52 | 0.33 | 0.17 | 0.46 |

1–2 m | SVJ | 0.37 | 0.48 | 0.51 | 0.51 | 0.32 | 0.18 | 0.45 |

| SVCJ | 0.21 | 0.32 | 0.46 | 0.58 | 0.35 | 0.25 | 0.43 |

| SVSCJ | 0.35 | 0.36 | 0.55 | 0.41 | 1.14 | 0.45 | 0.55 |

| # | 552 | 668 | 1446 | 2027 | 998 | 303 | 5994 |

| SV | 0.36 | 0.47 | 0.42 | 0.41 | 0.38 | 0.29 | 0.40 |

2–3 m | SVJ | 0.39 | 0.45 | 0.39 | 0.41 | 0.39 | 0.30 | 0.40 |

| SVCJ | 0.31 | 0.39 | 0.45 | 0.48 | 0.42 | 0.41 | 0.43 |

| SVSCJ | 0.29 | 0.30 | 1.20 | 0.39 | 0.46 | 0.48 | 0.58 |

| # | 635 | 482 | 861 | 1430 | 905 | 474 | 4787 |

| SV | 0.35 | 0.55 | 0.64 | 0.74 | 0.82 | 0.74 | 0.67 |

3–6 m | SVJ | 0.37 | 0.62 | 0.81 | 0.97 | 1.09 | 0.88 | 0.84 |

| SVCJ | 0.57 | 0.89 | 1.09 | 1.20 | 1.25 | 1.04 | 1.06 |

| SVSCJ | 0.40 | 0.49 | 0.52 | 0.51 | 0.38 | 0.84 | 0.50 |

| # | 354 | 289 | 390 | 598 | 516 | 609 | 2756 |

| SV | 0.93 | 1.59 | 1.66 | 2.37 | 2.53 | 2.66 | 2.10 |

>6 m | SVJ | 0.91 | 1.66 | 1.82 | 2.70 | 2.93 | 3.05 | 2.36 |

| SVCJ | 1.35 | 2.20 | 2.33 | 3.23 | 3.40 | 3.46 | 2.84 |

| SVSCJ | 0.44 | 0.38 | 0.36 | 0.49 | 0.49 | 0.71 | 0.50 |

| # | 3718 | 5683 | 8833 | 10373 | 5492 | 1791 | 35890 |

| SV | 0.36 | 0.46 | 0.52 | 0.60 | 0.59 | 1.18 | 0.56 |

All | SVJ | 0.39 | 0.44 | 0.55 | 0.66 | 0.66 | 1.35 | 0.61 |

| SVCJ | 0.36 | 0.40 | 0.59 | 0.79 | 0.75 | 1.56 | 0.67 |

| SVSCJ | 0.49 | 0.37 | 0.61 | 0.41 | 0.65 | 0.65 | 0.51 |

Figure 3 plots the pricing errors for different maturity contracts for the different models. As can be seen from the figure, the pricing errors are moderately small in the first year or two following the estimation period. They then become progressively larger. This is particularly obvious for the first three models.

The mechanical explanation for the above findings is as follows. The volatility of the out-of-sample period is significantly below that of the estimation period. During the out-of-sample period, markets experienced unusually low volatility and spot volatility (as well as implied Black–Scholes volatility) can be shown to move as low as 5 to 7%. This change of market conditions is hard to predict from the volatile estimation sample which implicitly incorporates a long-term, unconditional volatility which of course matches that of the sample average over the estimation period.

There is a simple mechanical explanation for the above findings: The first period used for parameter estimation included the crash of 1987, and the subsequent high-volatility. This is reflected, for instance, in the relatively high parameter estimates for θ. For the SV model, the estimate of θ implies an average volatility of 22%. Comparably, the annualized volatility during the out-of-sample period is 15.3% which is very close to the long-term sample standard deviation of S&P 500 returns. Since the average volatility will determine the relative expensiveness of the longer term contracts, a too high value of θ will lead to a systematic overpricing of long-term contracts. Conversely, short-term contracts depend less heavily on the value of θ.

The premiums on long-dated contracts become increasingly sensitive to the initial volatility as the value of κ^{Q} decreases. Smaller values of κ^{Q} imply that the term structure of option premiums will shift parallel in response to volatility changes. In the out-of-sample period, both short- and long-dated contracts become much less expensive than in the estimation period, so the out-of-sample data indicate a parallel shift consistent with a small value of κ^{Q}. Since κ^{Q} was estimated to be much lower for the SVSCJ model, this again explains why this model outperforms the other models out-of-sample.

To further shed light on how the high values of θ affect the pricing errors, I conducted another out-of-sample exercise where the values of θ were adjusted to match the historical average volatility of 15%. Not surprisingly, the models do a lot better after this adjustment, and the average pricing error for the SVCJ model is only 36 cents, a mere half of the previous error. The SVCJ model is particularly accurate in predicting the prices of the short-term contracts and the average error in the less than one month category is only 22 cents, almost half that of the SV model. Hence, these numbers support the notion that jump diffusion models give better descriptions of the relative pricing of short-term option contracts.^{8}

The out-of-sample results described in this section nevertheless illustrate one important shortcoming of stochastic volatility/jump models: The term structure of implied volatility cannot be matched with constant parameter values throughout the nine year history considered here. In essence, the expensiveness of options in the first period implies a value of θ that is too high to be consistent with the relatively cheaper prices of long-term options found in the latter period.

The above evidence is consistent with various types of model misspecification including structural shifts or other parameterization errors. It seems most likely that a model which allows for more complex volatility dynamics could capture the pricing errors. For instance, in a previous section it was argued that the option pricing errors were consistent with a higher speed of mean reversion in high-volatility states, and vice versa. This could potentially also explain the failure of the parameter estimates to capture the term structure of implied volatility in the out-of-sample period because if mean reversion were systematically lower in low volatility states, the term structure of implied volatility would be flatter in those states. Hence, the impact of the too high values of θ would diminish.

The evidence is also consistent with two-factor volatility models where the second factor determines the long-run volatility. Such models have been introduced to deal with the overly simple interest rate models which run into similar problems as the volatility models considered here (see Dai and Singleton (2000) and Andersen and Lund (1997) for examples). Two factor volatility specifications have been considered by Bates (2000) and Chernov et al. (1999) in the context of pure time-series analysis. Both papers consider models where the (stock) diffusion term take on the form . Given a sufficiently generous correlation structure, factor rotations can make such specifications observationally equivalent to the stochastic mean volatility specification suggested above.

#### E. Time-Series Fit

In what follows, we examine an essential question: Do option prices imply stock price dynamics consistent with time-series data? The first topic of interest is a re-examination of the evidence in BCC, Bates (2000) and Pan (2002) suggesting that the volatility of volatility, σ_{V} implied by option prices, cannot be reconciled with time-series estimates. In particular, BCC and Bates show that their estimated volatility paths are too smooth to be consistent with the relatively high σ_{V} estimated from option prices.

Table VIII reports estimates of σ_{V} from the simulated values of the historical volatilities, *V*^{g}_{t}, *t*= 1,…, *T*, *g*= 1, …, *G* analogous to those in BCC and Bates. The numbers match almost exactly those reported in Table III. The point estimates in Table VIII are slightly smaller than the posterior means reported in Table III, however the posterior credibility intervals in Table III do indeed overlap with the point estimates in Table VIII. Hence, the mismatch between the option implied volatility of volatility and the variability in the estimated volatility series reported elsewhere, cannot be replicated here.

Table VIII. ** Estimates of σ**_{V} from Filtered Volatility Series The table reports posterior means and standard deviations (in parenthesis) of estimates of σ_{V} from posterior simulations of the latent spot volatility, *V*_{t}. Estimates are obtained as the mean square errors ofacross posterior simulations, *g*= 1, …, *G*. Parameter estimates correspond to a unit of time defined to be one day, and returns data scaled by 100. | SV | SVJ | SVCJ | SVSCJ |
---|

Mean | 0.202 | 0.198 | 0.151 | 0.134 |

*SD* | 0.005 | 0.004 | 0.005 | 0.007 |

Some reflections on this result are in order. First, the MCMC estimator explicitly imposes the time-series constraint of the volatility dynamics through the likelihood function of the volatility path. This is also true for the full likelihood estimates in Bates (2000), however, Bates uses about 10 times as many option prices as here and about 1/3 as many time-series observations. Hence, in his analysis, the likelihood function is much more heavily influenced by option prices than the time-series dynamics. The converse is true here so the restrictions are much more likely to hold true. Second, Jacquier et al. (1994) show that their MCMC method provides much more erratically behaving volatility paths than other methods based on Kalman filtering, and quasi-maximum likelihood methods. Still, this is likely to play less of a role in explaining the difference between the results reported here and those in Bates (2000) because the volatility paths are estimated much more precisely than what is typically the case from returns data only.

The MCMC estimator employed here enables the investigator to obtain historical estimates of the Brownian increments ▵*W*^{S}_{t}=*W*^{i}_{t}−*W*^{i}_{t−1} for *i*={*S*, *V*}, the continuously arriving shocks to prices and volatility. These can be interpreted as the model standardized “residuals.”Figure 4 plots these residual for returns. It is evident from these plots that the jump models fare far better in explaining large stock price movements than do the simple SV model. For example, the −22% crash of October 19, 1987 and the 6% drop on October 13, 1989 produce too large return residuals relative to the prevailing market volatility at the times, to be consistent with the SV model. The SVJ model also has problems explaining the large market movements, and produces surprisingly large residuals for the same dates. In a nutshell, the reason for the large residuals is that the SVJ model partly fails to identify the large negative returns on these dates as jumps. This is again related to the fact that the unconditional jump probability, λ, is estimated to be so low. For the SVCJ and SVSCJ models, this changes because the large simultaneous moves in prices and volatility on these dates makes the algorithm identify these dates as “jump dates” in spite of the low unconditional jump probability. Hence, the return residuals shown for the SVCJ model in Figure 4 are not too different from what can be expected under the assumed *N*(0, 1) distribution.

Figure 5 plots the estimated (Brownian) shocks to the volatility process. As is well known, the infamous crash of 1987 had a huge impact on option prices across all maturities, especially in the days following the crash. The effect on the estimated, latent spot volatility is a huge jump in volatility. Regardless of whether such jumps are specified as part of the model, or not, the estimated spot volatility increases dramatically on the day of the crash. The SVSCJ and SVCJ models attribute this large increase to jumps in volatility and consequently produce plausible residual values. The SV and SVJ models produce residuals which are about 10 standard deviations from zero. Hence, these models require implausibly large Brownian volatility increments to deal with the market data from October 1987.

Interestingly, all four models generate too large *negative* movements in the days following the 1987 crash. Indeed, allowing for jumps to volatility in the SVCJ model does not explain the fall in spot volatility following the huge increase on the day of the crash. The reason is that the volatility jumps are restricted to be positive.

Finally, Table IX quantifies the magnitudes of the model violations discussed above through the posterior distributions of sample skewness and sample kurtosis in the residual series. The numbers confirm our conclusions from studying the residual plots: the residuals do not conform to the assumed normal distribution, although the magnitude of the violations is less for the jump models and least for the SVCJ model. Notice that the SVCJ model does markedly better in capturing the tails of both returns and volatility, and produces a residual kurtosis with a lower first percentile of 3.56 and 4.33, respectively.

Table IX. ** Residual Skewness and Kurtosis** The table reports skewness and kurtosis for estimated residuals (i.e., daily Brownian increments) for returns and volatility, respectively. The table contains posterior means, standard deviations (in parenthesis), and 99% credibility intervals (in square brackets). | SV | SVJ | SVCJ | SVSCJ |
---|

| Return Residuals |

Skewness | −1.891 | −0.692 | −0.129 | 0.138 |

| (0.088) | (0.294) | (0.060) | (0.707) |

| [−2.078, −1.709] | [−1.633, −0.364] | [−0.240, −0.016] | [−0.068, 0.532] |

Kurtosis | 20.352 | 7.223 | 3.959 | 3.757 |

| (1.121) | (3.282) | (0.267) | (1.321) |

| [18.083, 22.615] | [4.525, 18.630] | [3.563, 4.620] | [3.244, 5.798] |

| Volatility Residuals |

Skewness | 2.487 | 2.045 | −0.300 | −0.446 |

| (0.290) | (0.249) | (0.134) | (0.220) |

| [1.940, 3.194] | [1.475, 2.638] | [−0.564, −0.068] | [−0.782, −0.058] |

Kurtosis | 27.758 | 19.256 | 6.017 | 5.616 |

| (4.515) | (3.076) | (0.693) | (1.596) |

| [19.876, 38.507] | [13.687, 28.204] | [4.337, 7.700] | [4.190, 8.036] |