Jacquier-Jarrow (2000) Bayesian Analysis of Contingent Claim Model Error

Summary

Jacquier and Jarrow formally incorporate parameter uncertainty and model error into the estimation of option pricing models. They construct a Bayesian Markov chain Monte Carlo (MCMC) estimator that delivers exact finite-sample inference on option values, hedge ratios, and pricing error distributions — avoiding the delta-method approximations that fail in the small samples typical of daily re-estimation. Applied to Black-Scholes and polynomial non-parametric extensions on TOYS'R US call options (Dec 1989–Mar 1990), the extensions improve in-sample fit but yield no out-of-sample improvement over the basic B-S, contrasting sharply with implied-tree methods that overfit in-sample.

Key Claims

• Model error formalization: Two error structures — multiplicative: $\log C_i = \log b(x_i, \theta) + \eta_i$ (guarantees positivity); additive: $C_i = b(x_i, \theta) + \eta_i$ (dollar terms). Error $\eta_i \sim N(0, \sigma_\eta^2)$, possibly heteroskedastic across moneyness groups.
• MCMC estimator: Metropolis step for $\sigma$ (underlying volatility, non-conjugate posterior); Gibbs cycle for $(\beta, \sigma_\eta)$ given $\sigma$. No analytical posterior even for B-S. Delivers draws of any nonlinear function of parameters (model price, hedge ratio $\Delta$) without approximation.
• Fit vs. predictive density: Fit density integrates out parameter uncertainty only; predictive density additionally integrates out future model error $\eta^*$ — the correct benchmark for out-of-sample tests. Fit interquartile range (IQR) covers actual quotes only ~2% of the time (should be 50%), massively overstating precision. Predictive IQR with heteroskedastic errors covers 50–75% in-sample.
• Non-parametric extensions: B-S nested in polynomial expansions of moneyness $z = \log(S/Xe^{-rT})$ and maturity $\tau$ (models 0–4, up to 12 parameters; Jarrow-Rudd 1982). In-sample: model 2 reduces root mean squared error (RMSE) from 16% to 13% (log); larger expansions fail even in-sample due to multicollinearity. Out-of-sample: no extension beats basic B-S.
• Heteroskedastic errors: Strong moneyness-dependence of $\sigma_\eta$. Log model: error variance highest out-of-the-money, OTM (relative errors large for cheap options); level model: highest in-the-money, ITM (dollar errors large for expensive options). Allowing three $\sigma_\eta$ levels dramatically improves predictive density coverage; crucial for level models.
• Non-normality: Small-sample posterior of $\sigma_\eta$ is strongly right-skewed. Asymptotic normality would give flawed inference even with 452 quotes.
• DFW comparison: Dumas-Fleming-Whaley (1998) implied trees: near-perfect in-sample, catastrophic out-of-sample (OOS). JJ estimator's out-of-sample RMSE is close to in-sample RMSE (log model 2: 7% → 10%), because it penalizes model error instead of eliminating it by overfitting.
• Estimation method as crucial as model: Common nonlinear least squares (NLS) practice uses the fit density implicitly; the predictive density is what matters for trading decisions.

Concepts Introduced or Extended

• Contingent Claim Model Error
• Gibbs Sampler

Entities Mentioned

• Eric Jacquier
• Robert Jarrow

Quotes

"The estimation method is as crucial as the model itself."

"Specification tests on the basis of the fit and predictive densities of the call price show that the failure to include model error in prediction intervals leads to an underestimation of the variability of the quotes."

"Out-of-sample, the non parametric expansions most always fail to improve on the B-S."

My Take

The central methodological contribution — fit density vs. predictive density — is underappreciated in the options pricing literature and transfers directly to any domain where a deterministic model is used for prediction (term structure, credit). The finding that estimation method matters as much as model choice is important but perhaps unsurprising in retrospect: overfitting is overfitting regardless of how theoretically motivated the model is. The limitation is that the paper applies only to static (constant-parameter) models; as the authors note, out-of-sample failure of non-parametric expansions is partly a consequence of time-varying smile dynamics that a static polynomial can't track. The natural extension — dynamic stochastic-volatility (SV)-augmented Bayesian estimation — was not implemented here.