Contingent Claim Model Error

Definition

Contingent claim model error is the stochastic gap between a theoretical option pricing formula and observed market prices, arising from model misspecification, market frictions, and occasional quote errors. Rather than eliminating this gap by overfitting (as in implied tree methods) or ignoring it (as in standard nonlinear least squares (NLS)), the model-error approach treats $\eta_i$ explicitly as a random variable with a specified distribution, enabling formal likelihood-based inference and well-calibrated out-of-sample predictions.

Key Ideas

Two error structures: Multiplicative — $\log C_i = \log b(x_i, \theta) + \eta_i$ — guarantees call price positivity for any error distribution. Additive (level) — $C_i = b(x_i, \theta) + \eta_i$ — measures dollar errors. The choice affects how heteroskedasticity manifests across moneyness.
Fit density vs. predictive density: The fit density $p(C_i|y, x_i)$ integrates out only parameter uncertainty (given model is correctly specified). The predictive density additionally integrates out a future error draw $\eta^* \sim p(\eta)$ — it is the correct benchmark for out-of-sample tests. The two can differ dramatically: fit interquartile range (IQR) covers quotes only ~2% of the time, predictive IQR 50–75%.
Heteroskedastic errors: Pricing error variance depends on moneyness. In log models $\sigma_\eta$ is largest out-of-the-money (OTM) (relative errors large for cheap options); in level models largest in-the-money (ITM) (dollar errors large for expensive options). Ignoring heteroskedasticity distorts predictive coverage across moneyness groups.
Non-parametric extensions: Following Jarrow-Rudd (1982), the basic model $b(x, \theta)$ can be nested in a polynomial expansion of moneyness $z = \log(S/Xe^{-rT})$ and maturity $\tau$ . Small expansions (model 2: add $z$ , $z^2$ , $\tau$ ) reduce in-sample root-mean-square error (RMSE); larger expansions fail even in-sample from multicollinearity. No extension beats the basic model out-of-sample.
Estimation method ≥ model choice: Implied tree methods (Dumas-Fleming-Whaley 1998) achieve near-perfect in-sample fit but catastrophic out-of-sample (OOS) performance precisely because they absorb all model error into the fit. An estimator that accounts for model error produces less impressive in-sample numbers but far more reliable OOS inference.

How It Works

Likelihood and Posterior

Given $N$ quotes $C_i$ with model $b(x_i, \theta)$ , the likelihood is:

$\ell(\theta|y) = \prod_{i=1}^N f_\eta(C_i - b(x_i,\theta))$

(or the log version for multiplicative errors). Even for Black-Scholes with homoskedastic normal errors, the posterior $p(\theta|y) \propto \ell(\theta|y)p(\theta)$ has no analytical form because $\sigma$ (underlying volatility) enters the model nonlinearly through the Black-Scholes formula.

Markov Chain Monte Carlo (MCMC) Algorithm

Decompose $\theta = (\sigma, \beta, \sigma_\eta)$ :

Metropolis for $\sigma$ : Choose blanketing density $q$ (truncated normal at mode of posterior). Accept/reject candidate draws proportional to $p^*(\sigma)/q(\sigma)$ .
Gibbs for $(\beta, \sigma_\eta)$ : Given $\sigma$ , the model is linear regression — draw directly from Normal-Gamma conjugate conditionals.
Heteroskedastic extension: Add a Gibbs cycle over $\sigma_{\eta,j}$ , $j = 1, \ldots, J$ (one per moneyness group).

Draws of any function $g(\theta)$ — model price, hedge ratio $\Delta$ — are obtained by direct computation $g(\theta^{(s)})$ for each MCMC draw. No delta method.

Predictive Density

For each draw $\theta^{(s)}$ , draw $\eta^{(s)} \sim \mathcal{N}(0, (\sigma_\eta^{(s)})^2)$ and compute $C^{(s)} = b(x, \theta^{(s)}) \cdot \exp(\eta^{(s)})$ (log model). The empirical distribution of $\{C^{(s)}\}$ is the predictive density. Coverage tests check: what fraction of actual quotes $C$ falls in the predictive IQR?

Why It Matters

Proper uncertainty quantification for practitioners: A trader needs to know not just the model price but whether a market quote is abnormal. Only the predictive density provides a well-calibrated interval; the fit density understates variability by an order of magnitude.
Connects to stochastic volatility: The MCMC apparatus is the same as Jacquier-Polson-Rossi (1994). The natural extension is to place a stochastic volatility (SV) model inside the box (randomizing $\sigma$ via an Ornstein-Uhlenbeck (OU) or Cox-Ingersoll-Ross (CIR) process) rather than adding polynomial corrections outside. This maintains no-arbitrage restrictions.
Small-sample validity: Daily re-estimation windows yield O(100) quotes; asymptotics are unreliable. Posterior densities of $\sigma_\eta$ are strongly non-normal even at 452 quotes — exact Bayesian inference is essential.
Overfitting diagnosis: The divergence between in-sample and out-of-sample RMSE directly measures overfitting. The Jacquier-Jarrow (JJ) estimator forces this divergence to be small by making the model-error term absorb systematic bias.

Open Questions

Dynamic extensions: Static non-parametric expansions fail OOS because smile dynamics change over time. An SV-augmented estimator that models $\sigma_t$ as a latent state (Heston, Hull-White) is the natural next step.
Errors inside vs. outside the model: Adding $\eta$ outside the Black-Scholes (B-S) formula may conflict with no-arbitrage. Randomizing $\sigma$ inside (e.g., $\sigma = \sigma_0 + \sigma_1 z + \sigma_2 z^2$ ) avoids this but is harder to estimate.
Non-synchroneity: Recording stock and option prices at different times generates errors correlated with moneyness (Renault 1995), violating the independent and identically distributed (iid) assumption on $\eta_i$ .