Bakshi-Cao-Chen (1997) Empirical Performance of Alternative Option Pricing Models

Summary

Bakshi, Cao, and Chen (1997) derive a general closed-form option pricing model — the SVSI-J model — that nests Black-Scholes (BS), stochastic-volatility (SV), stochastic-volatility with stochastic interest rates (SVSI), and stochastic-volatility with random jumps (SVJ) as special cases. They evaluate these competing models on 38,749 S&P 500 call options (June 1988–May 1991) along three dimensions: (1) internal consistency of implied structural parameters with time-series data, (2) out-of-sample pricing errors, and (3) hedging errors. The central finding is that stochastic volatility is the first-order improvement in every dimension; adding either stochastic interest rates or random jumps provides only second-order pricing improvements and does not improve hedging once SV is already present.

Key Claims

Nested model hierarchy: The SVSI-J model has stock dynamics $dS/S = [R(t) - \lambda\mu_J]\,dt + \sqrt{V(t)}\,d\omega_S + J(t)\,dq(t)$ , Cox-Ingersoll-Ross (CIR)-type volatility $dV = (\theta_v - \kappa_v V)\,dt + \sigma_v\sqrt{V}\,d\omega_v$ with $\mathrm{Cov}(d\omega_S, d\omega_v) = \rho\,dt$ , CIR interest rate $dR = (\theta_R - \kappa_R R)\,dt + \sigma_R\sqrt{R}\,d\omega_R$ , and Poisson-lognormal jumps (intensity $\lambda$ , mean $\mu_J$ , log-std $\sigma_J$ ). Admits a closed-form European call via characteristic function inversion: $C(t,\tau) = S\cdot\Pi_1 - K\cdot B(t,\tau)\cdot\Pi_2$ (eq. 8).
Internal consistency: All models are significantly misspecified. Option-implied $\rho \approx -0.64$ (SV) is 2–3 $\times$ larger in magnitude than its time-series estimate ( $\approx -0.28$ ). The implied volatility-of-volatility $\sigma_v$ is $\sim 4\times$ its maximum-likelihood (ML) estimate. Among SV models, SVJ is least misspecified; SVSI most misspecified despite more parameters.
Pricing rank: $\text{SVJ} \succ \text{SVSI} \succ \text{SV} \succ \text{BS}$ . SV reduces absolute pricing errors by 25–70% over BS across moneyness-maturity categories. Jumps improve short-term out-of-the-money (OTM) pricing; stochastic interest rates improve long-term pricing.
Hedging (single-instrument): $\text{SV} \succ \text{SVJ} \succ \text{SVSI} \succ \text{BS}$ . SVJ does not improve over SV because with jump intensity $\lambda \approx 0.59$ /year, the expected time between jumps ( $\sim$ 1.7 years) far exceeds the 1–5 day hedging interval — jump risk is simply not realized at these horizons.
Hedging (delta-neutral): $\text{SV} \approx \text{SVSI} \approx \text{SVJ} \gg \text{BS}$ . The BS delta-plus-vega-neutral strategy (BSDV) nearly closes the gap between BS and SV models (except deep in-the-money, ITM), suggesting the hedging improvement from SV models is largely due to using a second option instrument rather than model structure.
Implied volatility smile: SVJ exhibits the flattest implied-vol smile across moneyness, followed by SVSI, SV, and BS (sharpest). For medium- and long-term options the SV/SVSI/SVJ smiles are essentially flat; only BS retains a U-shaped smile at all maturities.
Regression diagnostics (Table VI): After including stochastic volatility, the adjusted $R^2$ of residual pricing-error regressions falls from 31% (BS) to 5–9% (SV, SVSI, SVJ), confirming that SV models absorb almost all contract-specific biases.
Stochastic interest rates: Incorporating SI does not significantly improve pricing or hedging beyond SV alone. The implied $\kappa_R$ and $\theta_R$ under SVSI are dramatically lower than time-series ML counterparts, suggesting a negative interest-rate risk premium that distorts implied parameters.
Implied parameters (All Options, Table III): For SVJ — $\kappa_v=2.03$ , $\theta_v/\kappa_v\approx 4\%$ , $\sigma_v=0.38$ , $\rho=-0.57$ , $\lambda=0.59$ /yr, $\mu_J=-5\%$ , $\sigma_J=7\%$ , implied vol $=19.38\%$ , sum of squared errors (SSE) $=6.46$ . For BS: implied vol $=18.23\%$ , SSE $=69.60$ .

Concepts Introduced or Extended

Entities Mentioned

Quotes

"Overall, our results support the claim that a model with stochastic volatility and random jumps is a better alternative to the BS formula, because the former not only performs far better but also is practically implementable." (p. 2043)

"Our empirical evidence indicates that regardless of performance yardstick, taking stochastic volatility into account is of the first-order importance." (p. 2042)

"Since the delta-neutral hedge for the BS does not use a second option contract whereas it does for the other three models, this may have biased the delta-neutral hedging results against the BS model." (p. 2039)

"These three performance yardsticks serve distinct purposes ... in-sample and out-of-sample pricing errors reflect a model's static performance, while hedging errors reflect the model's dynamic performance." (p. 2006)

My Take

The three-yardstick methodology is the paper's enduring contribution: pricing rank and hedging rank need not coincide, and BCC make this concrete. The SVJ prices best but hedges no better than SV; SVSI hedges worst among the SV models despite having more parameters. The key mechanism is transparent: at daily/weekly rebalancing frequencies, jump risk is not realized ( $\lambda \approx 0.59$ means one jump per 1.7 years), so adding the jump dimension to the hedging instrument set is irrelevant. Similarly, stochastic interest rates matter for option valuation (discounting future payoffs) but not for short-horizon hedging.

The internal consistency finding is sobering: all models require implausible implied $\rho$ (2–3 $\times$ the time-series estimate) and $\sigma_v$ (4 $\times$ its ML value), suggesting structural misspecification persists even in the best model. But as the paper shows, misspecification and empirical usefulness are orthogonal — the SVJ is simultaneously the most misspecified-in-parameter space and the best out-of-sample pricer.

The BSDV exercise (delta-plus-vega-neutral BS strategy) is an important methodological control: it separates the hedging improvement due to model specification from the improvement due to using a second option instrument. The finding that BSDV nearly matches SV/SVJ for non-ITM calls implies that the stochastic-volatility models' hedging advantage is largely mechanical — not a deep endorsement of their structural form.