Bayesian analysis of contingent claim model error☆
Introduction
Since the original Black and Scholes (1973) and Merton (1973) papers, the theory of option pricing has advanced considerably. The Black–Scholes (hereafter B–S) has been extended to allow for stochastic volatility and jumps. The newer equivalent martingale technique allows to solve complex models more simply. At the same time it appears that the finance literature has not spent as much energy on the effect of the econometric method used to estimate the models.1 Whaley (1982) used several options to compute the B–S implied volatility by minimizing a sum of squared pricing errors. The non linear least squares method is still the most often used in empirical work. The method of moments is sometimes used, e.g., Bossaerts and Hillion (1994a), Acharya and Madan (1995). Giannetti and Jacquier (1998) report potential problems with the asymptotic approximation. Rubinstein (1994)'s paper started a strain of non parametric empirical work with aim to retrieve the risk neutral pricing density implied by option prices. This literature is reluctant to assume a model error which is inconsistent with the no-arbitrage deterministic models within which it works. The results in Dumas et al. (1995) hereafter DFW, show that this can lead to catastrophic out-of-sample performance even if, or maybe because the in-sample fit is quasi perfect.
Absolute consistency with the no arbitrage framework is not consistent with the data. Deterministic pricing models ignore market frictions and institutional features too hard to model. The overfitted implied density is affected by model error. There is no tool to asses the effect of model error on the estimate. The results of DFW are consistent with an overfitting scenario: The B–S models is too restrictive to suffer from overfitting. So it performs much better out-of-sample than the more general non parametric models. Maybe DFW did not test the models as much as the estimation method.
We develop an estimator to allow its user to assess the effect of model error on the inference. It allows specification tests. The cost of this is that we must make explicit hypotheses on the pricing error distribution. We can then write the likelihood function of the parameters.2 Like the existing methods, non-linear least squares and methods of moments, the maximum likelihood estimator only has an asymptotic justification. We develop a Bayesian estimator which is also justified in small sample. This is useful for the small samples typical due to the common practice of updating model parameters daily.
The Bayesian estimator does not have an analytical solution even for the B–S and simple pricing error structures. We solve this problem by constructing a hierarchical Markov chain Monte Carlo estimator (hereafter MCMC). This simulation based estimator provides the second improvement over standard methods. It delivers exact inference for any non linear function of the parameters instead of relying on the usual combination of delta method and normality assumption. The simulation based estimator produces draws of the desired posterior or predictive densities, e.g., option price, hedge ratio, correlation structure of the errors, all non linear in the parameters.3 Before taking a position, an agent wants to assess if a quote is far from the price predicted by a model. For this type of tail based diagnostic, it is crucial to use an exact rather than an approximate density. This is because predictive and posterior densities are often non normal.
The Bayesian estimator allows for prior information on the parameters, which standard methods do not. This is useful for several reasons. First, the underlying time series can be used as prior information on the parameters. Second, imperfect models often result in time varying parameters and need to be implemented in a setup allowing for periodic reestimation. In the resulting small samples, priors may be used to improve precision. The result of a previous sample can serve as prior information for the next sample. Finally, priors can help resolve potential multicollinearities when nesting competing models. We also extend the MCMC estimator to heteroskedastic pricing errors and intermittent mispricing where the error variance is sometimes larger than usual. The latter extension is more than a way to model fat tail errors. It parameterizes a quantity of economic interest to the user, the probability of a quote being an outlier. This is in line with common views where models are thought to work most of the time, and some quotes may occasionally be out of equilibrium and include a market error.
A key here is that we put a distribution on the errors. Let us see what this allows us to do which is not done in the empirical finance literature. Usually, once a point estimate for the parameters is obtained by non linear least squares, model prices are computed by substitution into the price formula. These price estimates are then compared to market prices. Mispricings, e.g., smiles, are characterized. Competing models are compared on the basis of these price estimates and smiles. What is not produced is a specification test based upon confidence intervals around the price reflecting parameter uncertainty and model error. Quotes, whether in or out of sample, should be within this interval with the expected frequency. Without further distributional assumptions, the distribution of the model error in a method of moment or least squares estimation is not clear. Standard methods can, but this is not done in the literature, use the asymptotic distribution of the parameter estimator to get an approximate density for the model price reflecting parameter uncertainty alone. Coverage tests could follow. First, this (necessary) approximation can lead to flawed inference. Second, it does not allow the incorporation of the pricing error to form a predictive rather than a fit density. We produce both fit and predictive densities for option prices and document their behavior. The extent to which they produce different results has not been documented.
Of course, the modelling of pricing errors arises only if the data include the option prices. We state the obvious to contrast the estimation of model parameters from option prices and from time series of the underlying. For models with risk premia, the time series may not give information on all the parameters of the model. Also, even for simpler models, the time series and the option prices may yield different inference. Lo (1986) computes coverage intervals and performs B–S specification tests. He uses the asymptotic distribution of the time series variance estimator to generate a sampling variability in option prices. The estimation has not used the information in option prices to infer the uncertainty on the parameter or the model error. For a model with constant variance, this is a problem since the length of the time series affects the precision of the estimator of variance. In the limit, a long sample implies very high precision and a short one implies to low precision, with the corresponding effect on the B–S confidence intervals. This test is hard to interpret.4 Our method can incorporate the underlying historical information if desired, but the core of the information is the option data.
One does not expect any model to have no errors. Rather, an acceptable model would admit errors which: (1) are as small as possible, (2) are unrelated to observable model inputs or variables, and (3) have distributional properties preserved out-of-sample. (1) and (2) say that the model uses the information set as best as possible. (3) says that the inference can be trusted out-of-sample. Our estimator can produce small sample diagnostics of these criteria. We document its behavior for the B–S model. We well know that the B–S does not fare well at least by criteria (1) and (2) above. DFW's results seem to imply that it fares better than some more complex models by criterion (3). There are known reasons for the B–S sytematic mispricings. The process of the underlying asset may have a stochastic volatility, or jumps. The inability to transact continuously affects option prices differently depending on their moneyness. Time varying hedging demand can also cause systematic mispricings. Practitioners’ common practice of gamma and vega hedging with the Black–Scholes reveals their awareness of model error.
It also shows their reluctance to use the more complex models available. Despite known problems, the B–S is still by far the most used option pricing model. One reason is that practitioners see more complex models as costly or risky. They may not have an intuition on their behavior. The estimation of the parameters can be complex, lead to unfamiliar hedge ratios or forecasts. Jarrow and Rudd (1982) argue that in such cases, auxiliary models such as a polynomial expansions of known inputs may be useful extensions to a basic model. Their argument could be relevant beyond the B–S case. At any point, there is a better understood basic model – the status-quo, and more complex models entertained which are harder to learn and implement. To improve upon the current basic model while the more complex model is not yet understood, the expansions need to capture the patterns omitted by the basic model. Recently, Hutchinson et al. (1993) showed that non parametric forms can succesfully recover existing B–S patterns.5 We add an extension of the B–S in this spirit and incorporate it to our estimator. We essentially will test if the expansions, similar to the larger models of DFW, capture the systematic patterns omitted by the B–S.
In the empirical analysis, we document the use of the estimator. We show the non-normality of the posterior densities. We then show that tests of the B–S model (and its expansions) which only account for parameter uncertainty (fit density) do not give a reliable view of the pricing uncertainty to be expected out-of-sample. The use of the predictive density markedly improves the quality of the forecasts. The fit density is of course tighter than the predictive density. It leads to massive rejection of all the models used. Its use by a practitioner would lead to an overestimation of the precision of a model price. We show that the non parametric extended models have different in-sample implications than the simple Black–Scholes. They also improve the model specification. However we also show that these improvements do not survive out-of-sample.
Section 2 introduces the basic model b(.) and its extended counterpart. It also introduces two candidate error structures, multiplicative (logarithm) and additive (level), both possibly heteroskedastic. Section 3 discusses the estimators and details the implementation of the method in the case of both basic and extended Black–Scholes models.6 Section 4 shows the results of the application of the method to stock options data. Section 5 concludes and discusses areas of future research.
Section snippets
Basic model, additive and multiplicative forms
Consider N observations of a contingent claim's market price, Ci, for . We think of Ci as a limited liability derivative, like a call or put option. Formally, we can assume that there exists an unobservable equilibrium or arbitrage free price ci for each observation. Then the observed price Ci should be equal to the equilibrium price ci. There is a basic model for the equilibrium price ci. The model depends on vectors of observables x1,i, and parameters θ. We assume that the
Monte Carlo estimation and prediction
We now briefly outline the methodology for Monte Carlo estimation. The time-t information set consists of observations of the option prices Cs,i collected for times s up to t, the vectors of relevant observable inputs xτ,i, and the history of the underlying asset price Sτ. We index quotes by t and i because for a time s, a cross-section of (quasi) simultaneous quotes can be collected. Let be the vector of histories at t, the data. θ is the vector of all the relevant parameters including ση
Data
The options data come from the Berkeley database. Quotes for call options on the stock TOYS'R US, are obtained from December 1989 to March 1990. We use TOY for two reasons. First, it does not pay dividends so a European model is appropriate. Second, TOY is an actively traded stock on the NYSE. We define the market price as the average of the Bid and the Ask. Quotes are always given in simultaneous pairs by the market makers. We filter out quotes with zero bids, and quotes which relative spread
Conclusion
We incorporate model error into the estimation of contingent claim models. We design a Bayesian estimator which to produce the posterior distribution of deterministic functions of the parameters, or the residual of an in-sample observation and assess its abnormality, and to determine the predictive density of an out-of-sample estimation. We document the potential non-normality of some posterior distributions. This shows the added value of the Bayesian approach.
We apply this method to quotes on
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Jacquier gratefully acknowledges essential support from the CIRANO, Montreal, www.cirano.umontreal.ca. This paper has benefitted from discussions with Warren Bailey, Peter Bossaerts, David Brown, Peter Carr, Bernard Dumas, Eric Ghysels, Antoine Giannetti, Dilip Madan, Rob McCulloch, Nick Polson, Eric Renault, Peter Schotman, and the participants of the seminars at Boston College, CIRANO, Cornell, Wharton, Iowa, Brown, Penn State and the 1997 Atlanta Fed. Conference on Market Risk. We are especially grateful to René Garcia and two referees whose comments helped dramatically improve the paper. The previous title of the paper was: Model Error in Contingent Claim Models: Dynamic Evaluation.