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Keywords:

  • Generalized residuals;
  • Importance Sampling;
  • Jump-diffusion;
  • Option pricing;
  • Simulated Maximum Likelihood;
  • Stochastic volatility

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

Summary  In this paper, we consider joint estimation of objective and risk-neutral parameters for stochastic volatility option pricing models using both stock and option prices. A common strategy simplifies the task by limiting the analysis to just one option per date. We first discuss its drawbacks on the basis of model interpretation, estimation results and pricing exercises. We then turn the attention to a more flexible approach, that successfully exploits the wealth of information contained in large heterogeneous panels of options, and we apply it to actual S&P 500 index and index call options data. Our approach breaks the stochastic singularity between contemporaneous option prices by assuming that every observation is affected by measurement error, essentially recasting the problem as a non-linear filtering one. The resulting likelihood function is evaluated using a Monte Carlo Importance Sampling (MC-IS) strategy, combined with a Particle Filter algorithm. The results provide useful intuitions on the directions that should be followed to extend the model, in particular by allowing jumps or regime switching in the volatility process.


1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

In this paper, we consider joint estimation of objective and risk-neutral parameters for non-affine jump-diffusion stochastic volatility (SV) option pricing models using both stock and option prices. This problem has been the subject of much work in recent empirical financial econometrics.

A common strategy simplifies the task by limiting the analysis to just one option per date, instead of the full cross-section, and assuming that its price is observed without measurement error. In this set up, there exists a one-to-one relationship between the observed variables and the state variables, and this makes the latter effectively observable. As a consequence, the loglikelihood can be evaluated using the Jacobian formula. The same result can be obtained using, instead of a single option price, some proxy of the latent volatility state, that can e.g. be derived using the VIX index as a proxy of the risk-neutral expectation of the integrated variance, and neglecting any noise it may contain. The simplicity of this approach explains its widespread adoption in the literature; see e.g. Aït-Sahalia and Kimmel (2007, 2010) for an application to SV models and to term structure models, respectively.

It should be noted, however, that the assumption about which specific option is exempt from observation noise is essentially arbitrary, and that in principle many alternative and equally reasonable decisions would be possible. Some recent papers (e.g. Jiang and Tian, 2007), moreover, point to some systematic biases in the VIX. In either case, the estimates of the model parameters, and the filtered state variables and pricing errors, will in general depend on the assumptions made to recover the latent variables by inverting the model-implied expressions of the observable variables. This approach can also be problematic to implement in models in which the latent state variables are restricted to belong to a subset of the real line, a constraint that is not automatically satisfied by the inversion technique. Finally, it does not allow to price an option conditioning on more than just one option observed at the same date.

In this paper we develop an alternative inference strategy that does not need assumptions of this kind. Such a procedure has already been advocated by Tauchen (2002) and Bates (2003). Its features can be summarized as follows. We break the stochastic singularity between contemporaneous option prices by assuming that every observation is affected by measurement error. We deem this assumption more appealing than the above one. The price to pay for this increased flexibility is that the evaluation of the likelihood function poses some non-trivial numerical challenges, but we overcome them using a MC-IS strategy, combined with a Particle Filter algorithm along the lines suggested by Durham and Gallant (2002) and Durham (2007). We approximate the theoretical model-implied option prices using a highly flexible parametric model, which allows us to compute quickly and accurately a huge number of prices.

For readability, the discussion is based on a set of simplifying assumptions, but it is important to remark that many of these could be relaxed without difficulties, as they are not essential for the implementation of our approach. In particular, different assumptions concerning risk premia structures or jump intensities and size distributions could be handled fairly easily, and maximum likelihood (ML) inference would still be feasible. Notice, however, that a few existing contributions already considered some of these extensions with mixed results. Bates (2000) considers a specification in which the jump intensity can be an affine function of the volatility state, but fails to reject the null of a zero slope. A constant jump intensities is also supported by Chernov et al. (2003) and Andersen et al. (2001).

The paper is structured as follows. The following section outlines the option pricing model, and provides details on the specification that we actually consider in the empirical analysis. It also discusses some problems that are commonly encountered in the literature of empirical option pricing, such as the need to approximate the transition density of the state variables and the model-implied theoretical option pricing formula. Section 3 discusses the simpler strategy based on the assumption that one option at each date is devoid of measurement error, and on the Jacobian formula to compute the loglikelihood. We highlight its drawbacks and provide an empirical illustration. Section 4 outlines the alternative approach which assumes measurement errors on each option. We first describe the strategy we use to approximate the loglikelihood, and show that with some minor modifications of the same techniques we can also easily compute filtered values of the state variables and of functions thereof that can be extremely useful in testing the specification of the model and in using it for pricing purposes. Section 5 illustrates an application of this approach to a sample of call options on the S&P 500 equity index. Finally, Section 6 concludes. Details about the sample and the methodology employed in the paper are provided in the appendices.

2. OPTION PRICING UNDER JUMP-DIFFUSION SV

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

2.1. The model

In a jump-diffusion SV model, the dynamics under the risk-neutral probability measure inline image of the price St of the underlying asset and the associated volatility state Vt is described by

  • image((2.1))
  • image((2.2))

where rt and dt are the instantaneous riskless interest rate and dividend rate, respectively, and, under inline image, inline image and inline image are standard Brownian motions with instantaneous correlation inline image, inline image is a Poisson process with intensity inline image, and inline image is the jump size. We assume that the jump intensity and the jump size are independent from each other and from every other variable in the model, and that inline image. Finally, we denote inline image.

To derive the dynamics of the state variables under the objective measure inline image, we need some assumptions about the structure of the risk premia. In this paper, we assume that the return risk premium on the Brownian shocks is given by inline image, where inline image is the diffusion coefficient in the price process—see (2.3) below—and inline image is a constant parameter. The volatility and jumps-related risk premia could also be specified explicitly; however, following Broadie et al. (2007), we simply specify a different dynamics of V, as well as different jump intensity and jump size distribution, and we interpret the difference between the inline image and inline image parameters as risk premia.

We do not impose a priori any constraint on the volatility risk premium; the specific functional forms of inline image and inline image adopted in the empirical applications below allow to keep some simplicity in the model and are coherent with previous work in the field.

Under the previous assumptions, the dynamics of the state variables under inline image is given by

  • image((2.3))
  • image((2.4))

where under inline image, WSt and WVt are standard Brownian motions with instantaneous correlation inline image, Nt is a Poisson process with intensity inline image, and inline image. For estimation purposes, it is convenient to work with the log price Pt, whose dynamics can be easily derived from (2.3)

  • image((2.5))

Notice that inline image. For further reference, let us denote with inline image the drift coefficient in (2.5).

Previous work in this area focussed mainly on models in the affine class due to tractability considerations and to the existence of quasi-closed form expressions for option prices. Several works, however, emphasized the conclusion that affine models can be frequently badly misspecified; see, among others, Christoffersen et al. (2010). In this paper we consider non-affine models that seem to provide a better fit to the data, either thanks to a different assumption on the volatility process (the log volatility model), or to increased flexibility (the CEV model). More precisely, the models we consider can be obtained from the general specification above if we impose the following constraints on the unspecified drift and diffusion coefficients:

  • • 
    Log volatility (LOG-J) model:
    • image
  • • 
    Constant elasticity of variance (CEV-J) model:
    • image

In both models the volatility drift is linear under either inline image and inline image. Many studies based on this model assumed a single free parameter in the volatility risk premium, which implies that both drift parameters change between inline image and inline image, but not independently. On the contrary, we adopt a more flexible specification with two free parameters in the volatility risk premium. This allows inline image and inline image to vary independently across probability measures.

The CEV-J model collapses to an affine specification under the constraint inline image. Affine models have attracted a huge amount of attention in the literature, and we also considered this specification in the analysis. Given that this specification is overwhelmingly rejected by the data, and to save space, we do not report the corresponding results, and we limit our discussion to the LOG-J and CEV-J models. Finally, it should be noted that the LOG-J and CEV-J specifications can not be embedded into the affine class through the use of an augmented state, as it is the case, e.g. for the Linear Quadratic Jump Diffusion models examined in Cheng and Scaillet (2007).

These specifications are more general than those that have been considered in the literature so far; Broadie et al. (2007) fit on a large cross-section of S&P futures option prices from 1987 to 2003 an affine model in which the jump risk premia are similar to ours, but with a constrained risk premium specification on the volatility process; Durham (2010) considers the same non-affine models we do, but constrains the jump related parameters to be the same across the two probability measures. In the empirical implementations we will also sometimes consider constrained specifications in which some of the jump parameters coincide under the two measures. The analysis in Durham (2010) is also based on the time series of S&P 500 index returns and the VIX index, and neglects the cross-sectional dimension in options data.

For the non-affine processes we consider the transition density f(Pt, Vt|Pt−1, Vt−1) is unknown, and must be approximated. Several strategies have been advanced to solve this problem, the most successful two being the closed-form Hermite polynomials expansion of Aït-Sahalia (2008) or the IS strategy developed by Durham and Gallant (2002). This paper is based on the latter because of its greater flexibility, which will be particularly convenient in the approach illustrated in Section 4. Details on the IS approach we use are provided in Appendix B.

2.2. Numerical evaluation of theoretical option prices

Since volatility is not observable, we use option prices to extract information on the latent state. Our sample is a highly unbalanced panel of prices of European call options that for each observation date differ by strike price and/or time to maturity. Following Bates (2000), we focus on option prices normalized by the underlying asset price discounted at the dividend rate. Consider the generic ith option observed at date t, and let Cit, Kit and inline image denote its price, strike and time to maturity, respectively. The option’s normalized price (NP) is then defined as

  • image

We collect in inline image the inline image NPs at date t. Define inline image as the log discounted moneyness of the option, and let inline image be vector of the option’s characteristics. We denote with inline image the model implied theoretical NP. Notice that to simplify the notation we omit the occurrence of inline image in h.

Appendix C illustrates a numerical technique that can be used to evaluate option prices in non-affine jump diffusion SV models. Even if relatively fast, this approach is still too slow for our sample size. For this reason we approximate inline image using a polynomial interpolation scheme. We first construct a fixed three-dimensional grid inline image combining three univariate grids, spanning the range of variation of the corresponding variable. Given a value for inline image, we evaluate the theoretical option price at each point on the grid, Hg, using the approach described in Appendix C. We then use the set inline image to construct an interpolation scheme that approximates log  Hg with a polynomial in inline image for several reasons. First, its coefficients can be computed very quickly and accurately by OLS. Second, given the estimates, it is immediate to compute an approximation of inline image and of its derivative w.r.t. Vt, which is needed in the empirical applications below. We use a polynomial of order four with 35 parameters. To estimate them, we consider equally spaced univariate grids with 10 points for log  V, and 6 points for inline image and X. The three-dimensional grid contains G=360 points. The R2 coefficient of the interpolating regression is always larger than 0.998.

A natural alternative to numerical schemes would be to use one of the recently advanced analytical approximations of the theoretical NPs in jump diffusion SV models (see e.g. Lewis, 2000, Sircar and Papanicolau, 1999, Lee, 2001, and Medvedev and Scaillet, 2007), which provide extremely fast tools to evaluate theoretical NPs. In this paper, however, we prefer the latter because some preliminary Monte Carlo experiments highlighted that the quality of approximation characterizing the analytical expansions is lower than that of the numerical scheme.

2.3. Measurement errors

For any candidate inline image, the pricing model states that the NP of any option is a function of inline image, which in turn implies the existence of a set of exact relations between the NPs of different options at the same date. This conclusion is rejected in any data set. To overcome this issue we could consider just one option per date, but this amounts to neglect a huge amount of information on the latent state. Moreover, the choice of the specific single option to be considered at each date would be, to a large extent, essentially arbitrary.

Alternatively, the stochastic singularity can be broken by introducing additional sources of statistical uncertainty. Increasing the dimension of the state vector would be theoretically sound but extremely complicated. For this reason, the solution usually adopted is to assume that option prices are observed with an error that can be due to microstructure effects (e.g. bid-ask spreads and tick-by-tick price variations) and data issues (e.g. non-synchronous or only approximate observation of the relevant variables). Measurement errors in option prices can be assumed implicitly, e.g. when parameters are estimated through least squares techniques, or explicitly, as a component of the estimation strategy.

ML inference requires an assumption about the stochastic structure of the observation errors. In this paper, we assume additive measurement errors in log NPs, defined by

  • image

distributed independently through time and across options according to a Gaussian distribution with mean zero. We also allow for some heteroscedasticity by maintaining that

  • image

We merge in inline image the parameters appearing in the measurement errors distribution. The assumption of independence across dates and options is not essential, but we think that it is reasonable: (i) it limits the number of nuisance parameters and (ii) we believe that any correlation between options should be accounted for by the pricing model, and not by the measurement errors. Our assumptions are also largely confirmed by the empirical results below. Finally, the techniques we analyse could be extended to handle different definitions of the errors or of their distribution, including alternative forms of heteroscedasticity depending on Vt.

3. VOLATILITY FILTERING BY OPTION PRICES INVERSION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

3.1. Loglikelihood derivation using the Jacobian formula

A common approach assumes that at each date exactly one option is observed without error, whereas the remaining Nt−1 are affected by measurement noise. This choice equalizes the dimensions of the augmented vector of latent variables (volatility and measurement errors) and of the observed option prices, allowing to derive the likelihood contribution with an application of the Jacobian formula.

To illustrate this strategy, let us partition the observed NPs as in inline image, where, without loss of generality, H1t is assumed to be noise free, whereas H2t are Nt−1 NPs affected by error. Furthermore, let us denote with fPV(Pt, Vt|Pt−1, Vt−1) the transition density of the log price and its volatility derived from (2.4)-(2.5). The Jacobian formula then states that the transition density of the observables is given by

  • image

which is the date t likelihood contribution of Pt and H1t. The exact expression of the transition pdf is generally unknown, and in this paper we approximate it using the IS approach in Appendix B. To derive the likelihood contribution of the remaining NPs, we observe that the measurement error on the options 2 to Nt is given by

  • image

Given our assumption of independent inline image measurement errors, the density of H2t conditional on Vt, or equivalently on H1t, is given by inline image. The sample loglikelihood is then given by

  • image((3.1))

3.2. Issues

The simplicity of this approach explains its widespread adoption; see e.g. Aït-Sahalia and Kimmel (2007, 2010) for an application to SV models and to term structure models. Notice, however, that the choice of the option exempt from observation noise is arbitrary. In principle, many alternative and equally reasonable decisions would be possible. Moreover, the estimates of the parameters will in general depend on this choice; see the next section for an illustration in option pricing models. Finally, this approach can be problematic to implement in models in which the latent state variables are restricted to belong to a subset of the real line.

To elaborate on the latter point, notice that in a SV model the NP of an option can not be smaller than some lower bound. In the Black and Scholes (1973) model this lower bound is max [0, 1− exp (−X)], and it is attained for a zero diffusion coefficient. When volatility is stochastic the bound is still attained for a zero Vt, but it is higher, and it depends on inline image. For an option with characteristics inline image, we denote the lower bound for the NP with inline image. The crucial step in the previous approach, which effectively makes V observable, is to compute the solution of the T+1 non-linear equations

  • image

However, for these equations to admit a solution it is necessary that the following non-linear inequality constraints be satisfied:

  • image((3.2))

Hence the ML estimation problem must be formulated as an optimization under T+1 non-linear inequality constraints

  • image

The presence of a T+1 non-linear constraints on the parameters greatly complicates inference; see Duffee (2002) for a discussion in affine term structure models. In practice, it is impossible to solve the estimation problem using standard techniques of maximization under constraints. The only feasible strategy consists in imposing a huge penalty to the loglikelihood whenever inline image does not satisfy some of the inequality constraints. In turn, this introduces large discontinuities in the objective function, which essentially prevent the use of derivative-based optimization algorithms. Even algorithms that do not require derivatives (such as the Simplex method we use) almost always get stuck on the boundary of the parameter space generated by one of the constraints; as a consequence, the end result is usually a boundary local maximum.

We now provide an empirical illustration of the above discussion. For simplicity, we focus on the effect of the choice of the option observed without error, and we neglect the numerical issues posed by the presence of the huge number of non-linear constraints (3.2).

3.3. An application to S&P 500 options

In this section we apply the NP inversion approach to a sample of options on the S&P500 stock index. We defer to Appendix A for a description of the data set. As option pricing models, we consider the LOG-J and the CEV-J jump diffusion SV models discussed in Section 2.1.

Our purpose is to illustrate the impact on the parameters estimates and derived quantities of five possible criteria used to identify the noise free contract. For each criterion, we first select the options with time to maturity closest to some target value inline image, and then pick among them the one with discounted moneyness closest to some value inline image. Different criteria correspond to different choices about the target inline image and inline image.

  • Criterion 1: inline image, inline image

  • Criterion 2: inline image, inline image

  • Criterion 3: inline image, inline image

  • Criterion 4: inline image, inline image

  • Criterion 5: inline image, inline image

The first criterion is a slight modification of the selection rule used by Pan (2002); its targets inline image and inline image identify at-the-money short-lived options. Criteria 2 and 3 allow to appreciate the effect of a different choice of inline image, while criteria 4 and 5 set the target time to maturity at 30 days, and consider slightly out-of-the-money and in-the-money options, respectively. Notice that the subsamples of options used to recover the volatility states by the five criteria partially overlap; in particular, criterion 2 selects just 99 (3.9%) different contracts w.r.t. criterion 1. The other criteria overlap to a lesser degree: the numbers (percentages) of different options for criteria 3, 4 and 5 are 1309 (52%), 2321 (92.2%) and 2286 (90.8%), respectively.

Tables 1 and 2 illustrate the parameter estimates for the LOG-J and CEV-J models. The tables contain five columns, one for each of the criteria used to identify the NPs to be inverted. For each parameter, we report the SML estimates and the corresponding asymptotic standard error (in parenthesis) derived from the outer product of gradients estimate of the asymptotic variance matrix. To save space, we omit the estimates of the heteroscedasticity inline image parameters, but they are available at request. The parameter estimates are generally quite accurate, and in line with results reported elsewhere. The R2 coefficients of the polynomial interpolation used to approximate option prices are always larger than 0.998.

Table 1.  Estimates of the parameters for the LOG-J option pricing model.
  inline image= 30 inline image= 15 inline image= 60 inline image= 30 inline image= 30
  inline image= 0% inline image= 0% inline image= 0% inline image=−1% inline image= 1%
  1. Notes: The estimates are based on the sample of options on the S&P 500 index observed on each day from 4 January 1996 to 30 December 2005. Each column contains the estimates obtained by selecting the NPs to be inverted following one of the five criteria detailed in Section 3.3. Asymptotic standard errors in parenthesis.

inline image 3.673.017.86−0.9011.32
 (2.55)(2.61)(2.71)(2.24)(3.47)
inline image −0.0211−0.0288−0.0103−0.0163−0.0895
 (0.00725)(0.00921)(0.00448)(0.00612)(0.0175)
inline image −0.0410−0.0466−0.0285−0.0268−0.0657
 (0.00824)(0.00863)(0.00575)(0.00781)(0.00942)
inline image 0.280.330.200.250.48
 (0.00190)(0.00235)(0.00104)(0.00167)(0.00299)
inline image −0.78−0.75−0.88−0.75−0.84
 (0.00399)(0.00378)(0.00373)(0.00412)(0.00275)
inline image −0.00856−0.01710.00351−0.01000.00677
 (0.000283)(0.000401)(0.0000961)(0.000256)(0.000708)
inline image −0.0129−0.0151−0.0164−0.00827−0.0320
 (0.0000435)(0.0000590)(0.0000431)(0.0000512)(0.0000648)
inline image 0.03510.04030.02820.01910.24
 (0.000419)(0.000441)(0.000341)(0.0000550)(0.000887)
inline image −0.54−0.54−0.32−1.520.14
 (0.00703)(0.00552)(0.00658)(0.00449)(0.00171)
inline image 0.390.360.930.181.85
 (0.0243)(0.0222)(0.0507)(0.0143)(0.0992)
inline image −0.0881−0.0838−0.100.0207−0.00914
 (0.12)(0.15)(0.0533)(0.35)(0.0348)
inline image 1.811.981.182.881.02
 (0.0203)(0.0194)(0.0118)(0.00556)(0.00234)
loglik.32040.331599.630839.532428.929473.5
Table 2.  Estimates of the parameters for the CEV-J option pricing model.
  inline image inline image inline image inline image inline image
  inline image inline image inline image inline image inline image
  1. Notes: The estimates are based on the sample of options on the S&P 500 index observed on each day from 4 January 1996 to 30 December 2005. Each column contains the estimates obtained by selecting the NPs to be inverted following one of the five criteria detailed in Section 3.3. Asymptotic standard errors in parenthesis.

inline image −0.98−1.34−1.16−2.01 2.33
 (2.52)(2.55)(2.50)(2.54)(2.32)
inline image 0.002570.001930.002270.005690.00259
 (0.000749)(0.000622)(0.00112)(0.00228)(0.000609)
inline image 0.02720.02800.02450.02120.0257
 (0.0103)(0.00711)(0.00994)(0.0176)(0.00638)
inline image 0.270.270.260.260.29
 (0.000841)(0.000809)(0.00103)(0.000569)(0.000575)
inline image 1.201.211.211.161.22
 (0.000649)(0.000617)(0.00103)(0.000561)(0.000541)
inline image −0.77−0.79−0.73−0.77−0.93
 (0.00179)(0.00216)(0.00276)(0.00144)(0.00141)
inline image −0.000921  −0.000981−0.000620−0.000442−0.000605
 (0.0000157)(0.0000215)(0.0000188)(0.0000232)(0.0000131)
inline image 0.04260.04260.04330.03160.0702
 (0.000168)(0.000168)(0.000198)(0.000151)(0.000178)
inline image 0.04220.04640.01590.05400.20
 (0.000390)(0.000444)(0.0000741)(0.000418)(0.00116)
inline image −0.74−0.69−1.48−0.700.0877
 (0.00577)(0.00599)(0.00959)(0.00305)(0.00168)
inline image 0.440.500.290.582.38
 (0.0254)(0.0273)(0.0182)(0.0420)(0.16)
inline image −0.0543  −0.0471−0.0400−0.03080.00395
 (0.11)(0.0987)(0.22)(0.0598)(0.0319)
inline image 1.721.692.560.990.91
 (0.0152)(0.0146)(0.00900)(0.00317)(0.00207)
loglik.31579.930600.726681.132691.928037.1

Given the amount of information contained in our sample of options, it is not surprising that the drift and jump parameters under inline image are estimated with much higher precision than the corresponding parameters under inline image. For the CEV-J model, the estimates of inline image are always significantly larger than 1, which is coherent with the results in Jones (2003), although our estimates are somewhat lower. As shown by Conley et al. (2010), a value of inline image higher than 1 implies that the stationarity of the Vt process is ‘volatility-induced’ irrespectively from the sign of inline image, provided that inline image is positive. The latter condition is always satisfied under inline image, but violated under inline image. Hence, according to the estimates in Table 2, the volatility process is non-stationary under the risk-neutral measure.

Inspection of Tables 1 and 2 highlights several discrepancies in parameter estimates across measurement errors structures. As expected, given the percentage of non-overlapping observations outlined above, the size of the discrepancies is minimum for the first two columns, it increases when one compares columns 1 and 3, and it is maximum when considering the last two columns. It is also not surprising that the parameters most affected by the assumption about the measurement errors structure are those characterizing the risk-neutral measure, i.e. the diffusion coefficient parameters (with the lone exception of inline image), the risk-neutral drift and jump process.

The impact of the measurement errors structure on the model’s implications can also be evaluated by looking at the filtered volatility trajectories and the option pricing residuals. For simplicity, we focus here on the LOG-J model (analogous results for the CEV-J model are available at request). Table 3 reports summary statistics computed on the filtered volatilities, along with their percentage differences w.r.t. the volatility filtered using the first criterion. It is apparent that the specific assumption about the measurement error structure can be quite relevant, and somewhat systematic. The volatilities derived from out-of-the-money (in-the-money) options tend to be systematically higher (lower) than those derived under the baseline criterion; nevertheless, some notable exceptions can be spotted at dates in which the volatility marks (according to some criterion) a sharp increase.

Table 3.  Summary statistics of the filtered volatilities.
 Volatilities (inline image)Percentage differences w.r.t. inline image, inline image
  1. Note: The summary statistics are based on the LOG-J option pricing model estimated on the sample of options on the S&P 500 index observed on each day from 4 January 1996 to 30 December 2005. q(p) denotes the pth percentile.

inline image 301560303015603030
inline image 0%0%0%−1%1%0%0%−1%1%
Avg.0.9410.9250.9780.9170.546−10.09.9−0.7−57.9
Std.Dev.0.9130.9890.9150.8510.82412.719.413.518.0
RMSEnanananana16.221.813.660.6
Min0.0270.0130.0470.0200.003−50.8−75.7−47.8−90.6
q(0.05)0.1020.0690.1290.0940.021−33.8−13.1−15.4−80.6
q(0.25)0.3180.2630.3540.3200.094−18.0−3.0−7.2−71.7
Median0.7020.6420.7370.7150.273−7.44.6−2.3−60.6
q(0.75)1.1731.1471.2191.1570.606−1.219.54.2−48.0
q(0.95)2.7222.8662.7982.6182.0655.948.317.8−21.8
Max6.7057.6386.8286.5888.247143.8152.7280.523.6

Table 4 reports the average option pricing residuals by discounted moneyness X and time to maturity inline image. Overall, the model tends to systematically underprice (overprice) short (long) maturity options, irrespectively from the assumption about inline image and inline image. However, it is apparent that the extent of the pricing errors depend on the characteristics of the contract used to infer the volatility status. The overpricing in medium-to-long maturity, in-the-money options almost vanishes when using criterion 5, replaced by a symmetric overpricing for out-of-the-money contracts. Using criterion 3 generates larger underpricing errors for short maturity options. In general, moving from one criterion to another tends to reduce the errors for contracts close to inline image and inline image, and to increase those for contracts with very different characteristics.

Table 4.  Average option pricing residuals.
Time to maturity, inline imageDiscounted moneyness, X
(−5%,−3%)(−3%,−1%)(−1%,1%)(1%,3%)(3%,5%)All
  1. Note: The average option pricing residuals (inline image) were computed by discounted moneyness and maturity, for the LOG-J model and under alternative assumptions about inline image and inline image.

inline image= 30, inline image= 0%
(15,24)4.70.60.23.04.72.4
(25,33)0.2−0.9 0.30.9−0.6 0.0
(34,42)0.3−0.7 −0.1 −2.5 −6.0 −1.3 
(43,51)−0.5 −1.5 −0.9 −4.4 −8.7 −2.3 
(52,60)−3.3 −3.2 −2.4 −8.0 −12.9  −4.6 
All1.0−0.8 −0.5 0.9−2.1 −0.5 
inline image= 15, inline image= 0%
(15,24)4.40.5−0.2 2.13.62.0
(25,33)−0.2 −1.1 0.00.4−1.1 −0.4 
(34,42)0.2−0.4 0.3−2.0 −5.5 −1.1 
(43,51)−0.3 −1.3 −0.6 −3.4 −7.4 −1.8 
(52,60)−3.2 −2.9 −1.8 −6.4 −10.6  −3.9 
All0.9−0.8 −0.5 −0.8 −2.1 −0.5 
inline image= 60, inline image= 0%
(15,24)4.21.22.86.58.03.9
(25,33)−0.2 −0.3 2.34.03.01.5
(34,42)1.00.92.41.2−1.0 1.1
(43,51)−0.3 −0.3 0.4−2.2 −5.0 −1.0 
(52,60)−3.7 −3.6 −4.0 −9.0 −12.5  −5.6 
All0.9−0.1 1.61.91.21.0
inline image= 30, inline image=−1%
(15,24)3.1−0.7 −2.3 −0.1 3.20.4
(25,33)−0.4 −1.1 −0.7 −0.3 0.0−0.6 
(34,42)0.90.70.2−1.6 −2.8 −0.2 
(43,51)−0.2 −0.5 −0.2 −2.8 −4.5 −1.1 
(52,60)−2.6 −1.6 −0.3 −5.2 −6.1 −2.4 
All0.6−0.7 −0.9 −1.4 −0.7 −0.6 
inline image= 30, inline image=+1%
(15,24)6.72.10.94.45.13.6
(25,33)0.5−1.8 −0.9 1.61.3−0.1 
(34,42)−0.7 −2.7 −1.9 −0.4 −1.2 −1.5 
(43,51)−2.1 −3.6 −1.5 −0.4 −0.2 −1.9 
(52,60)−3.6 −3.2 0.31.33.3−1.0 
All1.3−1.2 −0.4 1.92.20.4

Overall, these results suggest that the specific assumption about the measurement errors structure is critical both on parameter estimates, volatility filtering and option pricing. We now turn our attention to an alternative inference strategy.

4. ESTIMATION BY NON-LINEAR FILTERING

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

In this section, we illustrate an alternative approach which does not require to assume the existence of an option observed without measurement error. Assuming that each observed NP is affected by observation noise is less arbitrary, and seems more natural, but it complicates the evaluation of the sample loglikelihood, because when there are more sources of uncertainty than observed quantities, the likelihood can not be computed using the Jacobian formula, but requires the evaluation of a high-dimensional integral. In some special cases, e.g. affine term structure models, the problem can be simplified by casting it in a Gaussian state space model and exploiting the Kalman filter recursions, as in De Jong (2000), but in general its solution requires Importance Sampling (IS) techniques. This is the avenue followed e.g. by Brandt and He (2005) in their analysis of affine term structure models.

In this paper, we show how to evaluate the loglikelihood by combining an IS scheme and a Particle Filter algorithm, along the lines suggested by Durham and Gallant (2002, sect. 7); we present it in detail in Section 4.1. In particular, we highlight that including option prices in the observation sample significantly improves the performance of the SML estimator because of the huge amount of information they convey about the latent state variable, i.e. the volatility.

4.1. Likelihood evaluation

Let inline image be the filtration generated by the variables observed up to time t, i.e. inline image, and inline image. The likelihood function is given by

  • image((4.1))

The second equality derives from the Markov property of the diffusion and the independence of measurement errors. The initial condition P0 is known, and V0 will be integrated out (see below). Consider the time t contribution to inline image

  • image((4.2))

This two-dimensional integral can be interpreted as an expected value w.r.t. Vt, Vt−1 under the distribution implicitly defined by the integrand. Its value can be approximated using an IS scheme by specifying a sampling density for the integration variables. However, the transition pdf f(Pt, Vt|Pt−1, Vt−1) is unknown, and must be approximated using the MBB strategy outlined in Appendix B. Luckily, the two IS schemes can be merged into a single one. To see how, let inline image, and inline image. Following (B.1), the integral on the right-hand side (rhs) of (4.2) can be approximated with

  • image((4.3))

where inline image and inline image are defined in Appendix B. According to (4.3), the likelihood evaluation requires to numerically approximate T integrals whose dimension equals M+1. This can be done using a single IS scheme. Let inline image be a pdf on inline image, and rewrite (4.3) as follows:

  • image((4.4))

(4.4) highlights that inline image can be seen as the expected value w.r.t. inline image of the ratio in the integrand under the joint distribution defined by the product of densities inline image. Let inline image, be L independent draws from inline image. The IS estimate of (4.4) is given by

  • image((4.5))

To implement (4.5), we need to specify (i) which density to choose as inline image, and how to draw from it; and (ii) how to draw from inline image. The next sections consider these points in turn.

4.1.1. The auxiliary densityinline image

Let Vt−1 be a generic lagged value of the latent volatility drawn from inline image (we will show how to simulate from this distribution in Section 4.1.2). In this section, we propose a sampling density inline image for inline image which has a simple functional form, is easy to sample from, and provides accurate estimates of the likelihood.

To keep low the MC variance of (4.5), the sampling density inline image should be as much as possible proportional to inline image over the whole support of Vt and inline image. This product is informative about the uncertainty surrounding Vt and inline image in two ways: it reflects (i) the information about Vt in the observed cross-section of NPs Ht through the measurement errors density f(Ht|Pt, Vt) and (ii) the information about both Vt and inline image contained in inline image. In our framework, the second source of information is clearly dominated by the information in the option prices, and this remark suggests that, instead of the usual recursive factorization, it is more convenient to factorize the auxiliary sampling density as

  • image

Consider first q(Vt|Vt−1). Ideally, this density should equal fa(Vt|Ht, Pt, Pt−1, Vt−1), which is unavailable, but can be approximated by noting that

  • image((4.6))

where the two densities on the rhs correspond to the two sources of information about Vt discussed above.

In this paper, we use as q(Vt|Vt−1) the Laplace approximation to (4.6). The Laplace approximation is a powerful and accurate strategy widely used in mathematics and statistics to represent unknown densities; see Gelman et al. (1995) for a general presentation, and Durham (2006) and Huber et al. (2009) for two applications in financial econometrics. In a nutshell, it consists of a Gaussian pdf centred at the mode of the target density, with dispersion given by minus the inverse of the Hessian matrix of the log of target, evaluated at the mode. In practice, we proceed as follows. Let us approximate fa(Pt, Vt|Pt−1, Vt−1) with the Gaussian distribution derived from the Euler discretization over the whole interval (t−1, t)—i.e. ignoring the subintervals defined above. We first compute

  • image

using Newton’s method, and

  • image

The Laplace sampling density for Vt is then Gaussian, with mean inline image and variance inline image. Notice that both inline image and inline image depend on Vt−1. In practice, this implies that the Laplace approximation must be computed for each simulated value of the lagged volatility. While this might seem complicated, it should be noted that the whole procedure amounts to solve a large number of straightforward univariate maximization problems, given the availability of good initial points and of analytical expressions of the derivatives of the function to be maximized. Usually (see e.g. Durham, 2006, 2007) the Laplace approximation is computed w.r.t. to the whole trajectory of the volatility state because the likelihood is not sequentially factorized as in (4.1), but rather defined as a single integral w.r.t. the volatility trajectory, whose dimension is equal to T. In this paper we prefer to work with the factorized loglikelihood for several reasons. The whole-trajectory strategy is well-suited for discrete-time models, but becomes much more complicated in a continuous-time setting, in which there are multiple ‘intermediate’ volatility values to integrate out. Moreover, the sequential strategy naturally provides a way to compute the generalized residuals that we will use later to conduct a specification analysis.

A couple of remarks about this result are in order. First, the usefulness of our sampling density depends on the validity of two simplifying approximations: using the Euler discretization instead of the true transition density to derive inline image and inline image, assuming at most one jump between t−1 and t. These steps, however, can be easily checked ex post by examining the MC variance of (4.5), and checking that this estimate has finite variance. We show in Section 5 that this variance is actually very low in all our applications.

Second, a similar approach could be used also to approximate the pdf inline image which is needed in order to integrate out V0 in (4.4) for t=1. To this end, given the lack of a lagged volatility, we use a Gaussian density computed as the Laplace approximation above, but based on a target density that neglects the transition density, and focuses exclusively on the measurement errors density f(H0|P0, V0).

It remains to discuss our choice of inline image, which is a pdf for the ‘intermediate’ volatility states inline image given Vt−1 and Vt. The ideal pdf would be inline image, which is unknown. However, given the density used to draw Vt, we argue that no information about inline image is lost if we drop the conditioning on Ht, Pt and Pt−1. This allows to factorize inline image as

  • image

We set each pdf in the product of the rhs as a Gaussian density with moments computed in the same way as the MBB strategy discussed in Appendix B. Notice however that, unlike the ‘pure’ MBB approach, the simulated inline image trajectories do not start from the same volatility state Vt−1, and do not end up in the same volatility state Vt, as both these values are simulated by inline image and q(Vt|Vt−1), respectively.

4.1.2. Drawing frominline image

In this section we show how to randomly draw from inline image for inline image; the case t=0 was already discussed in Section 4.1.1. The approach we adopt is basically the application of the Particle Filter discussed by Durham and Gallant (2002, sect. 7).

Let inline image be L independent draws from inline image, and

  • image

be the L simulated values of the ratio under integration used in (4.5). Define the normalized weights

  • image

By construction, inline image and inline image. Furthermore, by Theorem 1 in Geweke (1989), the collection inline image can be seen as a discrete approximation of inline image, in the sense that, by a Law of Large Numbers:

  • image

for any function g for which the expectation on the rhs exists and is finite.

There are various different ways to exploit this result to draw Vt from inline image. The simplest one, advanced by Rubin (1988), consists in drawing with replacement from inline image, where inline image is the probability that each inline image is drawn. The resulting likelihood, however, would not be continuous in the parameters, causing difficulties to the numerical optimizer. Durham and Gallant (2002, sect. 7) prefer to use the collection inline image to build a Hermite approximation of inline image, and draw from it. In this paper we use the bootstrap procedure based on univariate linear spline advanced by Pitt (2002) to get an approximated likelihood which is smooth in the parameters. With a multivariate latent state some other kind of multivariate interpolation technique should be used.

4.2. Diagnostic testing and filtered (generalized) residuals

To assess the validity of the models specification, we use simulation based techniques to estimate sequences of filtered estimates of the latent volatility Vt and of functions of Vt.

We consider filtering w.r.t. four alternative kinds of information sets. The first one is inline image, and it contains, at date t, only the observations up to date t−1. Conditioning on this information set, the filtered estimate of a generic function inline image is defined by

  • image

To estimate these quantities, we first use the Particle Filter technique outlined in Section 4.1.2 to generate draws from inline image, and then simulate the couple inline image from inline image by drawing blindly from the Euler discretization of the Vt process. Finally, Pt is drawn from the conditional distribution inline image derived in Appendix B. We label ‘predicted’ values of inline image the results of this procedure. They can be useful in diagnostic checking, but they do not represent the best way to predict the price of an option, in particular when some information about the forward date is available.

To this end, we consider some alternative information sets inline image that contain inline image, Pt and a subset inline image of the whole set of options Ht observed at date t. In this case the filtered value of a generic function of the volatility Vt (notice that since Pt is known conditionally on inline image, we can omit it from the arguments of inline image) is defined by

  • image

These integrals can be evaluated following the procedure described in Section 4.1. Notice that the denominator is equivalent to (4.3) using only the subset inline image of options instead of the full vector Ht; if inline image does not contain options at date t, then the inline image factor disappears from both integrals. To evaluate the numerator, we use the same simulated values of the denominator integrand, multiplied by inline image evaluated at the simulated Vt value. Geweke (1989) shows that the simulation variance of the estimate of the ratio is reduced if the same draws are used in the numerator and the denominator.

We consider three kinds of augmented information sets: (a) inline image; (b) inline image, where inline image is the date t option selected according to criterion 1, as defined in Section 3.3; and finally, (c) inline image. Case (a) considers the predicted values of inline image given past observations and the current value of the stock index. Knowledge of the latter carries some information on the current Vt value, and should allow more accurate predictions. We label ‘updated given Pt’ the results of this procedure.

Case (b) further enlarges the information set by including a single option at each date. Since the latter is a monotone function of Vt, this inclusion significantly increases the available information on the latent variable, and generates dramatically improved predictions of inline image, that we label ‘updated given Pt and inline image’. This kind of conditioning is very interesting from an operational point of view, as it allows to price any date t option conditionally on the observed price inline image.

It should be noted that pricing some options relative to one observed option is also possible in the approach of Section 3, in which one option at each date was assumed to be free of measurement error. The two procedures, however, are fundamentally different. On one hand, volatility is filtered by inverting an observed option price; on the other, it is filtered by estimating a conditional expectation given an option which is observed with error. It is likely that, if such error actually exists, neglecting it might induce biased volatility and option prices estimates. Furthermore, in our approach, there is no need to condition on just one option at each date; we might as well condition on all but one of the observed options, and compute the predicted price of the contract left over. This should further enhance the accuracy of the predictions, and it is of course impossible to do under the assumptions of the approach of Section 3.

Case (c) considers the widest information set comprising the log stock price and all the options observed at each date. We label the values of inline image predicted in this way as ‘fully updated’. Notice that their computation can be done using the volatility trajectories used in the likelihood evaluation discussed in Section 4.1. In all cases, 100,000 trajectories were used to approximate the above expressions using Monte Carlo integration techniques.

In our set up, predicted values can be computed for the options NPs and the log stock index price. In the case of options, they allow to compute residuals that, according to our hypothesis about measurement errors, should conform to a Gaussian distribution independent across dates. This can be checked using standard test procedures, such as the Box–Pierce test, either applied to the residuals or to their squares, and the Jarque–Bera test.

In the case of the log stock prices, the assumed distribution is not Gaussian; rather, it is a mixture of conditionally heteroscedastic Gaussian densities. To perform diagnostic checking, we computed the associated generalized residuals as follows. Consider the first kind of filtering rule discussed above, based on the conditioning information set inline image, and the predicted values for inline image, the conditional cdf corresponding to the pdf derived in Appendix . If the model specification is correct, these predicted values should be i.i.d. uniformly distributed in [0,1]. If we further transform these uniform generalized residuals using the inverse standard Gaussian cdf, we obtain generalized residuals that, under the null hypothesis of correct specification, should be i.i.d. standard Gaussian. This can be tested using, as in the case of options, the Box–Pierce or the Jarque–Bera tests.

5. IMPLEMENTING THE SML ESTIMATOR

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

5.1. Parameter estimation

We estimated a few variants of the LOG-J and the CEV-J models, differing by the parameterization of the volatility and the jump risk premia, along with the corresponding affine specifications. We limit our discussion to the two versions of the LOG-J and the CEV-J models, which, according to information criteria, should be preferred. The ‘baseline’ LOG-J and CEV-J models coincide with the specification estimated in Section 3.3; the ‘augmented’ versions labelled LOG-Ja and CEV-Ja assume different jump size variances under the actual and the risk-neutral measures.

Table 5 reports the results. For each parameter, the table contains the ML estimate, the estimated asymptotic standard error (in parenthesis), and the numerical standard error (in brackets). The latter is also computed for the loglikelihood at the optimum, and it is based on 100 estimates using independent draws. The R2 coefficients of the polynomial interpolation that we use to approximate option prices are equal to 0.9998 for each model, and suggest that the approximation is reliable.

Table 5.  Estimates of parameters for the LOG-J and CEV-J option pricing models.
 CEV-JCEV-JaLOG-JLOG-Ja
  1. Notes: The estimates are based on the sample of options on the S&P 500 index observed on each day from 4 January 1996 to 30 December 2005. Asymptotic (statistical) standard errors in parenthesis; numerical (Monte Carlo) standard errors in brackets.

inline image −3.68 −2.29 1.194.86
 (2.15)(2.86)(2.47)(2.59)
 [0.0129][0.0121][0.00815][0.0340]
inline image 0.001430.000815−0.0245−0.0207
 (0.000545)(0.000459)(0.00833)(0.00730)
 [0.00000899][0.0000111][0.0000219][0.0000372]
inline image 0.03080.0327−0.0323−0.0274
 (0.0269)(0.0316)(0.00932)(0.00880)
 [0.0000398][0.0000295][0.0000488][0.0000497]
inline image 0.270.270.290.27
 (0.000864)(0.000955)(0.00224)(0.00173)
 [0.00000601][0.00000343][0.00000756][0.00000900]
inline image 1.081.09n.a.n.a.
 (0.000362)(0.000426)  
 [0.0000273][0.00000281]  
inline image −0.75−0.77−0.80−0.85
 (0.00206)(0.00228)(0.00429)(0.00391)
 [0.0000244][0.0000176][0.0000181][0.0000269]
inline image 0.171.340.281.68
 (0.0127)(0.16)(0.0168)(0.14)
 [0.000136][0.000296][0.0000855][0.000461]
inline image −0.23−0.0110−0.0175−0.0220
 (0.0589)(0.0183)(0.25)(0.00963)
 [0.00106][0.0000777][0.000149][0.000245]
inline image 3.590.533.150.53
 (0.0313)(0.0328)(0.0327)(0.0201)
 [0.000153][0.0000982][0.0000959][0.000102]
inline image 0.0009520.000789−0.0140−0.0109
 (0.0000361)(0.0000359)(0.000357)(0.000254)
 [0.000000379][0.000000200][0.00000120][0.00000128]
inline image 0.02010.0202−0.00891−0.00901
 (0.000151)(0.000160)(0.0000477)(0.0000466)
 [0.000000965][0.000000565][0.000000529][0.000000439]
inline image 0.02050.01990.02630.0222
 (0.000295)(0.000290)(0.000438)(0.000359)
 [0.00000205][0.000000837][0.00000129][0.000000814]
inline image −1.05−1.07−0.79−0.83
 (0.00864)(0.00839)(0.00739)(0.00784)
 [0.0000963][0.0000509][0.0000537][0.0000444]
inline image n.a.3.75n.a.3.53
  (0.0326) (0.0359)
  [0.000122] [0.0000943]
inline image −2.16−2.20−2.12−2.24
 (0.0238)(0.0238)(0.0238)(0.0236)
 [0.000121][0.000130][0.0000913][0.0000974]
inline image 18.9019.4118.4519.06
 (0.26)(0.27)(0.26)(0.26)
 [0.00234][0.00138][0.00145][0.00131]
inline image 0.05920.05930.06090.0605
 (8.64)(8.72)(8.41)(8.48)
 [0.0833][0.0550][0.0399][0.0484]
inline image −85.47−85.08−88.98−86.96
 (0.55)(0.55)(0.54)(0.54)
 [0.00158][0.00130][0.00105][0.00133]
inline image 0.04510.04510.04750.0468
 (2.76)(2.77)(2.77)(2.77)
 [0.00936][0.00604][0.00871][0.00738]
loglik.40385.340840.040971.741589.0
 [0.30][0.31][0.12][0.17]

The comparison between the asymptotic and the numerical standard errors weights the relative importance of the ‘statistical’ sampling uncertainty versus the ‘numerical’ one induced by simulations. The latter is smaller than the former by at least two orders of magnitude; apparently our IS strategy succeeds in reducing simulation variance to an acceptable level. We also conducted the tests of the null that the IS estimate (4.5) has finite variance proposed by Monahan (1993) and Koopman et al. (2009), which consider an inequality restriction on a parameter of a Generalized Pareto density, and can be based on a non-parametric procedure (Monahan, 1993) or on the trilogy of ML tests (Koopman et al., 2009). In both cases, these tests must be conducted separately for each integral, which means that in our sample the procedure must be repeated T=2516. A joint statistic can then be formed by aggregating the univariate tests results as in Rao (1952, p. 44). To save space, we do not report the results, but they support the null hypothesis of finite variance of the sample weights.

The estimates of the parameters are in line with results reported elsewhere, obtained using different estimation procedures and data. Some differences can be spotted for the parameters that appear both in inline image and in inline image, such as those in the volatility diffusion coefficient, in the correlation coefficient inline image or in the jump size variance; in general, our estimates are closer to those obtained using only option prices (see e.g. Bakshi et al., 1997), than to those obtained when the analysis is limited to just one option per day. The reason for this is simply the larger size of the option sample w.r.t. the stock price sample. The same feature implies that risk neutral parameters are estimated more accurately than their historical counterparts.

With the only exception of inline image, all estimates are very accurate and highly significant, in particular for the risk-neutral parameters. This is due to the fact that we completely exploit the large cross-sectional dimension of the sample, instead of focussing only on a single option per day. The estimates imply a stationary volatility process under both inline image and inline image and for every specification, which stands in contrast with the non-stationarity of the CEV-J Vt process under inline image suggested by the parameter estimates reported in Table 2 using the approach of Section 3.

The estimated jump intensities and jump size distributions are quite different under inline image and inline image. The comparison between the baseline LOG-J and CEV-J models, and their augmented LOG-Ja and CEV-Ja versions, highlights the importance of allowing different jump size variances under the two measures. The loglikelihood of the augmented specification is higher by 455 points in the CEV-Ja case, and by 618 points in the LOG-Ja case; the constraint inline image is clearly rejected by a LR test. The augmented specifications suggest that the jumps are very frequent, essentially zero on average with low dispersion under inline image, and much rarer, but around 1% on average and with a standard deviation of about 3–3.5% under inline image. Overall, these results suggest that the the jump component is less important than what is usually acknowledged in previous works, with the partial exception of Durham (2010). This discrepancy could be due to a different sample or model, as non-affine models could capture features of the distribution that a less flexible affine specification would attribute to jumps.

The comparison between the LOG-Ja and the CEV-Ja models is not as simple as that between the baseline and the augmented specifications, since the former are not nested. We prefer the log volatility model, not only because of the higher loglikelihood value, but also on the basis of its goodness of fit and of the diagnostic checks based on filtered option prices residuals and log stock price generalized residuals described in Section 4.2. For this reason, we limit our discussion of analysis of the filtered (generalized) residuals to the LOG-Ja model.

5.2. Predicted values and diagnostic tests

Figure 1 illustrates the updated latent volatility Vt in the LOG-Ja model for the sample of options on the S&P 500 index, 4 January 1996 to 30 December 2005. The plot also depicts the 95% interquantile ranges of the updated volatilities, which witness the accuracy of the filtering procedure; actually, the narrowness of the range makes it necessary to plot it as a separate line, instead of a band surrounding the sample average. The shape of the filtered volatility trajectory is in line with similar plots relative to the same period reported elsewhere.

image

Figure 1. Average and 95% interquantile range for filtered volatility in the LOG-Ja model.

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As a way to cross-validate the filtered volatility trajectory, Figure 2 compares the observed VIX contract price over the same period (in daily percentage volatility) with the same quantity implied by the LOG-Ja model and computed numerically on the basis of the average filtered volatility in the LOG-Ja model for the sample of options on the S&P 500 index, 4 January 1996 to 30 December 2005 as in Jiang and Tian (2005) and Durham (2010). The two trajectories are strikingly similar in shape and very close to each other, although some differences can be spotted at some high-volatility periods: the Asian currency crisis in July 1997 and the mini-crash of 27 October 1997; the LTCM bailout around mid-1998 and the Russian Default in August of the same year; and the period from October 2002 to April 2003, marked by the escalation of the Iraq crisis and the break out of the second Gulf War. Despite these discrepancies, the correlation between the two series is 0.973.

image

Figure 2. Observed daily VIX and predicted daily VIX computed using the LOG-Ja model.

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Inspection of the time series of the predicted generalized residuals of the log index price highlights a few notable outliers, the most striking of which is the large negative value corresponding to the 27 October 1997 mini-crash. This anomaly is shared by all the specifications we examined, and points to the need of a more flexible specification. Given the outliers, it is not surprising that a Jarque–Bera test of normality rejects the null overwhelmingly (the test statistic is equal to 73.1, with P value equal to inline image). Figure 3 illustrates the Gaussian Q–Q plot of the S&P predicted generalised residuals from 4 January 1996 to 30 December 2005, in order to get a visual interpretation of this conclusion. The plot clearly points to the left tail as the major source of misspecification.

image

Figure 3. Gaussian Q–Q plot of the S & P 500 log stock index price predicted residuals.

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The correlograms (with 95% confidence intervals) of the 500 log stock index price predicted generalised residuals and of their squares are illustrated in Figure 4 from 4 January 1996 to 30 December 2005. The residuals look fairly uncorrelated over time, but not their squares, which is coherent with unexplained conditional heteroscedasticity. A more sophisticated specification of the volatility process—e.g. a two factor volatility model—could help solving this issue.

image

Figure 4. Autocorrelation functions of the S & P 500 log stock index price predicted residuals (top) and their squares (bottom).

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Finally, we examine the various generalized residuals given by the difference between observed option NPs and model implied NPs estimated conditionally on four information sets. The most useful for diagnostic checking are the predicted residuals, computed conditionally on the previous period information set; for pricing purposes, conditioning on enlarged information sets may be more interesting. Given the wealth of our sample, the descriptive statistics of the options NPs residuals take several tables. Tables 6 and 7 report the sample averages and standard deviations computed for the residuals conditioned on the four information sets, and disaggregated by time to maturity and discounted moneyness. In general, average residuals are small and close to results reported in other work—see e.g. Table 3 of Bates (2000), which is closest in spirit with the fourth panel. Average residuals decrease with the amount of conditioning information (moving from the top to the bottom panel); moreover, the predicted residuals suggest that the model systematically overprices (underprices) out-of-the-money (in-the-money) options, but this bias seems to vanish in the remaining panels.

Table 6.  Sample averages of option pricing residuals.
Time to maturity, inline imageDiscounted moneyness, X
(−5%,−3%)(−3%,−1%)(−1%,1%)(1%,3%)(3%,5%)All
  1. Note: The averages (inline image) of option pricing residuals were computed according to different filtering rules by discounted moneyness and maturity for the LOG-Ja model.

Predicted residuals
(15,24)−2.5−7.2−5.9 6.4 15.4 −0.9
(25,33)−5.5−9.2−7.9 3.8 8.7−3.6
(34,42)−3.8−8.2−4.9 3.0 7.9−2.7
(43,51)−2.5−4.6 0.2 7.7 4.5 0.1
(52,60)−7.2−8.2−7.6 1.6 4.6−5.2
All−4.2−7.7−5.6 4.6 9.4−2.4
Updated residuals given Pt
(15,24) 3.0 0.5 0.4 2.9 4.7 1.9
(25,33)−1.6−2.2−1.2 0.3−0.2−1.1
(34,42)−0.1−0.5 0.3−0.5−1.8−0.3
(43,51) 0.0 0.4 1.8−0.2−3.7 0.2
(52,60)−3.3−2.6−1.2−4.2−6.7−3.0
All−0.1−0.8−0.1 0.2−0.3−0.3
Updated residuals given Pt and inline image
(15,24) 2.5 0.0−0.2 2.2 4.0 1.4
(25,33)−1.8−2.3−1.2−0.5−1.2−1.5
(34,42)−0.9−1.2−0.6−2.1−3.7−1.5
(43,51)−0.9−0.5 0.8−2.0−5.3−1.0
(52,60)−4.0−3.0−1.8−5.8−8.4−3.8
All−0.7−1.3−0.7−1.0−1.6−1.0
Fully updated residuals
(15,24) 2.9 0.3−0.1 1.9 3.5 1.4
(25,33)−0.2−0.6 0.5 0.9 0.1 0.1
(34,42) 0.1−0.1 0.6−1.1−2.9−0.3
(43,51)−0.2 0.0 0.7−2.2−5.4−0.7
(52,60)−2.1−1.1 0.0−4.6−7.3−2.1
All 0.4−0.3 0.3−0.4−1.1−0.1
Table 7.  Sample standard deviations of option pricing residuals.
Time to maturity, inline imageDiscounted moneyness, X
(−5%,−3%)(−3%,−1%)(−1%,1%)(1%,3%)(3%,5%)All
  1. Note: The standard deviations (inline image) of option pricing residuals were computed according to different filtering rules by discounted moneyness and maturity for the LOG-Ja model.

Predicted residuals
(15,24)31.243.760.979.789.060.6
(25,33)34.545.462.176.384.860.4
(34,42)39.148.862.678.784.161.8
(43,51)41.650.761.376.383.660.9
(52,60)46.951.063.478.681.562.1
All37.847.362.178.085.461.1
Updated residuals given Pt
(15,24)10.411.914.516.819.014.3
(25,33)11.012.715.017.119.614.8
(34,42)11.513.114.917.618.614.8
(43,51)13.416.117.720.021.417.4
(52,60)16.418.321.224.323.620.4
All12.514.216.518.820.416.1
Updated residuals given Pt and inline image
(15,24) 8.1 7.6 9.212.015.010.2
(25,33) 8.2 6.6 6.8 9.315.9 9.2
(34,42) 7.2 5.9 5.8 8.411.5 7.6
(43,51)10.511.413.114.415.512.8
(52,60)18.215.518.018.020.217.8
All10.9 9.611.712.516.111.8
Fully updated residuals
(15,24) 7.1 6.2 6.8 9.513.0 8.3
(25,33) 6.0 4.5 4.6 7.912.2 6.9
(34,42) 5.0 3.1 3.3 6.510.5 5.6
(43,51) 5.9 4.9 5.5 8.411.4 7.0
(52,60)11.710.912.314.016.012.7
All 7.6 6.3 7.0 9.613.2 8.3

The effect of an augmented information set is clearer for standard deviations. On average, standard deviations of option residuals decrease by 73% by conditioning on the current stock index price in addition to last period’s information. This decrease is further enhanced to 81% by adding an option, and reaches 86% for fully updated values. All panels show that pricing errors for options with longer time to maturity are more dispersed; the same is true w.r.t. discounted moneyness, but this is probably due to higher prices of deep in-the-money options.

Table 8 reports sample correlations between contemporaneous option residuals and their squares by time to maturity and discounted moneyness. To save space, we limit the table to the predicted residuals and to three classes of time to maturity and discounted moneyness. Consider the top left entry in the top panel: 0.94 is the sample correlation between pricing residuals observed at each date for short-lived deep out-of-the-money options. Since this computation is not limited to one option per date, the result is not equal to 1, as it is for standard correlation matrices; notice however that these matrices are symmetric. Any other entry should be interpreted as the sample correlation between contemporaneous pricing residuals associated to contracts with characteristics belonging to different classes.

Table 8.  Sample contemporaneous correlations of predicted option pricing residuals and squared residuals.
inline image X Time to maturity, inline image
inline image inline image inline image
 
Discounted moneyness, X
Xl Xm Xh Xl Xm Xh Xl Xm Xh
  1. Notes: The correlations were computed by discounted moneyness and maturity for the LOG-Ja model. inline image, k=h, m, l means that time to maturity belongs to (15,30), (31,45), (46,60), respectively. Xk, k=h, m, l means that discounted moneyness belongs to (−5%,−1.66%), (−1.66%,1.66%), (1.66%,5%), respectively.

Residuals
  Xl 0.940.900.860.880.860.830.910.870.82
inline image Xm  0.970.960.920.960.950.920.950.93
  Xh   0.990.920.970.980.870.950.97
  Xl    0.960.940.900.910.900.90
inline image Xm     0.980.970.930.960.97
  Xh      1.000.910.960.98
  Xl       0.980.950.88
inline image Xm        0.990.97
  Xh         1.00
Squared residuals
  Xl 0.910.830.610.760.740.650.860.760.50
inline image Xm  0.930.780.760.870.830.790.860.63
  Xh   0.960.800.890.940.550.750.91
  Xl    0.930.880.770.850.790.81
inline image Xm     0.960.890.870.900.92
  Xh      0.970.820.900.94
  Xl       0.940.890.54
inline image Xm        0.950.78
  Xh         0.98

The predicted residuals and their squares are very correlated; the lowest entry in the top panel is 0.82, corresponding to contracts with completely opposite characteristics. The correlations are lower for squared predicted residuals, but still very high. A plausible explanation of these results is that the model lacks jumps in volatility. In this case, predictive filtering cannot anticipate the occurrence of a jump in the latent state, inducing large price residuals of the same sign, which in turn are reflected in large and positive correlations across all contracts.

The first two panels of Table 9 consider the sample skewness and excess kurtosis of the pricing residuals. The estimates based on predicted residuals can be used to test the normality hypothesis of the measurement errors. Normality does not seem to be completely at odd with the data, with the exception of out-of-the-money options, for which the symptoms of negative skewness and leptokurtosis are clear.

Table 9.  Sample statistics of predicted option pricing residuals.
Time to maturity, inline imageDiscounted moneyness, X
(−5%,−3%)(−3%,−1%)(−1%,1%)(1%,3%)(3%,5%)All
  1. Note: The statistics were computed by discounted moneyness and maturity for the LOG-Ja model.

Skewness
(15,24)−2.1−1.9−0.6−0.30.0−1.1
(25,33)−0.3−0.9−0.6−0.30.0−0.5
(34,42) 0.4−0.9−0.8−0.40.3−0.4
(43,51)−0.8−0.9−0.4−0.30.2−0.5
(52,60)−0.9−0.6−0.5−0.10.2−0.5
All−0.9−1.1−0.6−0.30.1−0.6
Excess kurtosis
(15,24)19.113.4 1.80.70.38.0
(25,33) 8.42.71.00.40.12.8
(34,42)13.35.03.21.20.85.1
(43,51)12.85.71.20.70.14.6
(52,60)12.94.11.90.10.14.5
All13.56.71.80.60.35.1
First lag autocorrelations
(15,24)0.120.030.00−0.03−0.03 0.02
(25,33)0.130.02−0.01 −0.030.030.02
(34,42)0.07−0.01 0.01 0.010.040.02
(43,51)0.060.00−0.10 −0.010.00−0.02 
(52,60)0.070.000.01−0.04−0.01 0.01
All0.100.01−0.01 −0.020.000.02
First lag autocorrelations of the squares
(15,24)0.130.070.090.100.010.09
(25,33)0.270.130.130.070.060.14
(34,42)0.110.060.050.030.060.06
(43,51)0.080.140.120.10−0.11 0.10
(52,60)0.200.060.090.120.070.11
All0.170.090.100.080.020.10

Finally, the last two panels of Table 9 report the first order sample autocorrelations between the pricing errors and their values lagged one period, as well as autocorrelations between consecutive squared pricing residuals. The results for predictive residuals highlight some dynamic misspecification for deep out-of-the-money options; the evidence is fairly coherent to the i.i.d. hypothesis in the other cases. Autocorrelations are much higher for residuals filtered conditionally on augmented information sets (not reported), but our results are again coherent with those reported elsewhere—see e.g. the two columns under the heading ‘Autocorrelations’ in Table 2 of Bates (2000).

To summarize: the analysis of the generalized residuals suggests that the stock price model is probably misspecified, and should allow at least for a more flexible form of heteroscedasticity. The filtered predicted option pricing residuals are not incompatible with the i.i.d. assumption, but point to the need to extend the model in order to explain the very high contemporaneous correlations reported in Table 8. Finally, dramatic improvements in predicting option prices can be obtained by pricing options relative to other options at the same date. Using the results in Tables 6 and 7, it can be seen that conditioning on all the other options at the same date reduces the RMSE by a factor ranging from 9% to 58%, depending on the class of moneyness and time to maturity, w.r.t. prices computed conditioning on just one option.

6. CONCLUSIONS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

In this paper, we consider joint estimation of objective and risk-neutral parameters for SV option pricing models using both stock and option prices. A common strategy simplifies the task by limiting the analysis to just one option per date. We first discuss its drawbacks on the basis of model interpretation, estimation results and pricing exercises. We then turn the attention to a more flexible approach, that successfully exploits the wealth of information contained in large heterogeneous panels of options, and we apply it to actual S&P 500 index and index call options data.

Our approach has two crucial features. First, we break the stochastic singularity between contemporaneous option prices by assuming that every observation is affected by measurement error. We deem this assumption much more appealing that the alternative one, in which at each date one specific option is observed without measurement error. The price to pay for this increased flexibility is that the evaluation of the likelihood function poses some non-trivial numerical challenges, but we successfully overcome them using a MC-IS strategy, combined with a Particle Filter algorithm. Second, we approximate the theoretical model-implied option prices using a highly flexible parametric model, which allows us to compute very quickly and accurately a huge number of implied volatilities.

The results we obtain suggests that the model is misspecified, but that some significant improvements could probably be obtained by extending it in the direction of including jumps or regime switching in the volatility dynamics. Other extensions can be envisioned, but we leave them to future research.

ACKNOWLEDGMENTS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

We thank seminar participants at the 2011 ICEEE for helpful comments and suggestions. We are also grateful to the Editor, Jianqing Fan, and two anonymous referees for their comments. Any remaining errors are our own.

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices
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Appendices

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICING UNDER JUMP-DIFFUSION SV
  5. 3. VOLATILITY FILTERING BY OPTION PRICES INVERSION
  6. 4. ESTIMATION BY NON-LINEAR FILTERING
  7. 5. IMPLEMENTING THE SML ESTIMATOR
  8. 6. CONCLUSIONS
  9. ACKNOWLEDGMENTS
  10. REFERENCES
  11. Appendices

APPENDIX A: THE DATA

The sample contains data on the spot price of the S&P500 index and daily call prices on the index. It consists of 2,517 daily observations from January 4, 1996 to December 30, 2005. For each date, we extracted from the population of all exchanged call options on the index (179,176 over the whole 10 years interval) those with discounted moneyness inline image, time to maturity inline image days, and transaction volume of at least 5 contracts, leaving us with a total of 40,211 observed options. The constraints were imposed to exclude from the sample illiquid or seldom traded contracts. The number of call options observed at each date varies from 3 to 44, with an average of almost 16. It also appears that the cross-sectional dimension steadily grew over the last four observation years.

Table A1 reports the sample frequencies of options observations by discounted moneyness and maturity. This table highlights that the sample contains a clear prevalence of short maturity options, and an even clearer majority of at- and out-of-the-money options.

Table A1.  Sample frequencies of options observations.
Time to maturity, inline imageDiscounted moneyness, X
(−5%,−3%)(−3%,−1%)(−1%,1%)(1%,3%)(3%,5%)All
  1. Note: The frequencies correspond to different sample selection rules by discounted moneyness and maturity.

(15,24)2384264726931965121910908
(25,33)2205241425401878121310250
(34,42)14721702187212497667061
(43,51)1200142215889665345710
(52,60)14061564176010185346282
All86679749104537076426640211

APPENDIX B: THE MC-IS APPROXIMATION OF THE TRANSITION DENSITY

To illustrate the IS approach we use, we follow Durham (2010, sect. 3.2).

Let inline image be a partition of the interval [t−1, t], where for simplicity we assume that for all inline image, where inline image is the length of the interval between observations at dates t−1 and t. Let us for simplicity denote inline image and inline image, inline image, and inline image and inline image. By the Chapman–Kolmogorov property and the Markov nature of the bivariate diffusion

  • image

This integral can be approximated using

  • image

where fa is the transition density implied by an approximate discretization scheme of (2.5) – (2.4); in this paper, we use the first order Euler scheme, which implies that fa is bivariate Gaussian. Apart from its simplicity, this choice also allows to analytically compute the integral w.r.t. inline image, leaving us with:

  • image((B.1))

where inline image is the product of M Gaussian densities:

  • image

where inline image is the Gaussian density, and inline image is the conditional distribution of PM given inline image and V0 implied by the Euler discretization. Under our assumptions about the jump process, this density is a mixture of Gaussian pdfs, with the number of jumps being the mixing variable:

  • image

where:

  • image

with:

  • image

The (M−1)-dimensional integral in (B.1) can be evaluated using a IS approach. Let inline image be a pdf on inline image, and rewrite (B.1) as follows:

  • image((B.2))

Let inline image, inline image, be independent draws from q. The IS estimate of (B.2) is then given by

  • image((B.3))

Durham and Gallant (2002) showed that to reduce the bias in (B.3) it is important to transform the original diffusion into another one characterized by a constant diffusion coefficient. In our framework, it is sufficient to consider the Lamperti transform of Vt, which is defined as:

  • image

where the lower bound is irrelevant. By Itô’s Lemma:

  • image

The variance of the simulation noise can be reduced by carefully choosing q. In this respect, a very efficient sampling strategy, labelled Modified Brownian Bridge (MBB) by Durham and Gallant (2002), suggests to recursively draw inline image for inline image from a Gaussian density based on the Euler discretization of the process and conditional to VM and Vm−1. With a minor approximation, this density is Gaussian with mean equal to Vm−1+(VMVm−1)/(Mm+1), and variance given by inline image. The product of these M−1 Gaussian densities defines the auxiliary density which is used as the denominator in (B.3), and as the distribution from which the simulated inline image, inline image are drawn. Sampling from it is extremely fast and can be combined with other variance reduction techniques. On the basis of the analysis in Durham (2010), we implement the MBB approach using M=8 subintervals, and L=256 simulated volatility trajectories. A comparison across parameter estimates of the unavoidable sampling noise with the simulation noise confirms the adequacy of these settings.

APPENDIX C: NUMERICAL EVALUATION OF OPTION PRICES IN NON-AFFINE JUMP DIFFUSION MODELS

Apart a few special cases, theoretical option prices are not known in closed-form in a general jump-diffusion SV model, and need to be evaluated numerically. In the absence of jumps, a particularly efficient pricing strategy was advanced by Willard (1997), based on conditional Monte Carlo technique combined with quasi random sequences. This strategy is based on the observation that, in a pure SV model, the price of a call can be computed by first conditioning on the trajectory of the volatility Brownian motion during the life of the option, and then numerically evaluating the expectation w.r.t. the distribution of the trajectory. The resulting pricing formula CSV (St, Vt) is then given by

  • image

where inline image is the Black and Scholes formula for price S and volatility inline image (and where for simplicity we neglect the remaining arguments), and:

  • image

Merton (1976) used a similar argument to price options with (i) Poisson jumps in the St process with intensity inline image and size independent from the Brownian processes and (ii) relative jump size equal to inline image, where inline image. In our case, it is possible to proceed by conditioning on n of jumps during the life inline image of the option, compute the price using the Willard (1997) approach with modified arguments, and then compute the expectation w.r.t. a Poisson distribution with parameter inline image

  • image((C.1))

where

  • image

A truncated version of (C.1) provides a viable strategy to compute the option price.