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Option Pricing under Stochastic Volatility and Jumps: A PIDE Framework with Empirical Evidence
\noindentKeywords: stochastic volatility; jump-diffusion; partial integro-differential equations; option pricing; CGMY; L\'{e}vy measure.
\noindentJEL Classification: C58; C63; G12; G13.
A central question in modern option pricing is the relative importance of diffusive volatility dynamics versus discontinuous jump risk in explaining the observed structure of implied-volatility surfaces. Option pricing models confront a persistent tension between analytical tractability and empirical realism. The Black--Scholes framework black1973,merton1973 prices contingent claims under geometric Brownian motion but systematically underprices out-of-the-money contracts and cannot account for volatility clustering or discontinuous price movements. These deficiencies are structural, reflecting well-documented properties of equity returns, including heavy tails, persistent variance dynamics, and jump behavior associated with macroeconomic stress, that are incompatible with the lognormal assumption.
Two principal extensions address these limitations. Stochastic-volatility models allow the instantaneous variance to evolve as a mean-reverting diffusion, generating richer term-structure dynamics and a leverage-driven smile. Jump-diffusion models introduce discontinuous increments, providing a direct mechanism for tail risk and short-maturity skewness. At the same time, the economic roles of these mechanisms differ. Stochastic volatility primarily captures variance persistence, leverage effects, and term-structure dynamics, whereas jump specifications are intended to reproduce discontinuities, heavy tails, and crash-sensitive skew behavior. Distinguishing their relative empirical contribution is therefore important not only for pricing accuracy, but also for understanding how markets price tail risk across different volatility environments.
Rather than treating option pricing primarily as a model-comparison exercise, this paper studies the economic decomposition of implied-volatility structure across stochastic-volatility and jump components. The objective is not merely to compare calibration errors across specifications, but to identify which features of the volatility surface are explained by variance persistence, leverage effects, and discontinuous jump dynamics across maturities and moneyness regions.
When jumps are incorporated into the risk-neutral dynamics, option values satisfy a partial integro-differential equation (PIDE) rather than a standard PDE. The nonlocal integral operator couples option values across the entire state space, reflecting the fact that jump risk is not locally diversifiable. This nonlocal structure introduces both mathematical and computational challenges, chief among them the efficient evaluation of the jump integral in calibration settings where the pricing equation must be solved across large contract cross-sections.
This paper makes three contributions. First, we present a PIDE formulation for stochastic-volatility jump-diffusion (SVJD) pricing derived explicitly from the infinitesimal generator, with emphasis on the nonlocal jump operator's role in the pricing hierarchy. Second, we develop a numerically stable implementation combining Crank--Nicolson finite-difference discretization, FFT-based convolution for the jump integral, and operator-splitting techniques. Third, we provide a systematic empirical decomposition of the incremental pricing value of stochastic volatility and jumps across maturities and moneyness regions, using a cross-section of S&P 500 index options.
Early evidence on non-Gaussian return distributions is due to mandelbrot1963 and fama1965; cont2001 provides a comprehensive synthesis of stylized facts including volatility clustering and the leverage effect. The stochastic-volatility literature traces to hull1987 and stein1991, with the affine tractable formulation of heston1993 serving as the dominant empirical benchmark. Jump specifications originate with merton1976 and kou2002, while the CGMY family carr2002 encompasses both finite- and infinite-activity behavior. Numerical methods for PIDE-based pricing are developed in andersen2000, cont2005, and dhalluin2005.
The empirical analysis calibrates all four models, Black--Scholes, Heston SV, Heston+CGMY, and SVJD (Merton), to SPX contracts with maturities of 27, 89, and 153 days. The results show that stochastic volatility accounts for the preponderance of pricing improvement, reducing IV RMSE by 39% relative to Black--Scholes. The Heston specification captures most of the economically relevant structure of the implied-volatility surface through variance persistence and leverage-driven asymmetry. Jump augmentation produces marginal gains concentrated in the deep out-of-the-money region for CGMY and at longer maturities for Merton. The calibrated CGMY activity index places the model firmly in the compound-Poisson regime, consistent with high-frequency evidence in aitsahalia2009 and cont2011. These findings suggest that the sample period was characterized primarily by elevated but largely diffusive volatility, although jump components would likely assume greater importance during periods of severe market stress, liquidity disruption, or heightened crash-risk pricing.
Section (ref) reviews the literature. Section (ref) presents the theoretical framework. Section (ref) describes the numerical methods. Section (ref) covers the empirical design. Section (ref) reports results. Section (ref) concludes.
The literature underlying this paper spans three domains: empirical evidence on return distributions and implied-volatility surfaces, structural models incorporating stochastic volatility and jumps, and numerical methods for PIDE-based pricing. We summarize each in turn.
Systematic departures from lognormality are well-established in the empirical asset-pricing literature. mandelbrot1963 and fama1965 document excess kurtosis in equity returns across asset classes and time periods; cont2001 synthesizes these findings into a set of stylized facts that includes volatility clustering, the leverage effect, and heavy tails. Although the Black--Scholes model remains analytically tractable, it is structurally inconsistent with each of these properties.
The option-market implications are correspondingly severe. Black--Scholes prices out-of-the-money options at systematically low levels and produces a flat implied-volatility surface. rubinstein1994 shows that post-1987 implied-volatility skews reflect a substantial premium for downside tail risk. bakshi1997 provide the most comprehensive model comparison to date, demonstrating that joint stochastic volatility and jump components are required to fit the observed cross-section; bates1996 establishes analogous results for currency options, and pan2002 documents that jump risk carries a significant and separately identified risk premium.
Stochastic-volatility models were introduced by hull1987 and stein1991; the affine specification of heston1993 remains the dominant empirical benchmark due to its semi-closed-form pricing solution via characteristic functions. duffie2000 characterize the general class of affine jump-diffusions and establish conditions under which characteristic-function pricing extends to jump-augmented specifications.
Jump-diffusion models originate with merton1976, who augmented Brownian motion with a compound Poisson process while preserving closed-form tractability. kou2002 refined this framework with a double-exponential jump distribution, improving asymmetric tail fit. L\'{e}vy process models provide a richer class: the Variance Gamma specification of madan1998 and the CGMY model of carr2002 allow joint control over tail decay and jump activity. carr2003 extend this class via stochastic time changes, while barndorff2001 introduce Ornstein--Uhlenbeck stochastic-volatility models driven by subordinators. Comprehensive treatments are in boyarchenko2002 and cont2004.
The introduction of jumps renders the pricing equation nonlocal. Standard finite-difference methods must be augmented to handle the integral term, which couples option values across the entire spatial domain. andersen2000 showed that PIDE-based jump-diffusion models can be solved efficiently by combining finite-difference schemes with Fourier-domain convolution, exploiting the shift structure of the jump operator in log-price coordinates. cont2005 established rigorous convergence results for L\'{e}vy-driven PIDEs. Operator-splitting methods offer an alternative that decouples the diffusion and jump components; dhalluin2005 demonstrate their efficiency and robustness in calibration settings. Transform-based methods, the FFT approach of carr1999 and the COS method of fang2008, are widely used for characteristic-function models but are less suited to early-exercise and path-dependent problems, where PIDE methods retain a structural advantage.
Existing contributions tend to emphasize one dimension of the problem: theoretical analysis and convergence cont2005, numerical scheme development dhalluin2005, or empirical model comparison bakshi1997. This paper integrates these threads by combining a PIDE-based formulation with a rigorous numerical implementation and a structured empirical decomposition of stochastic-volatility and jump contributions across maturities and moneyness regions.
Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\geq 0}, \mathbb{Q})$ denote a filtered probability space under the risk-neutral measure $\mathbb{Q}$. The asset price $S_t$ and variance process $V_t$ evolve as
where $\tilde{N}(dt,dy) = N(dt,dy) - \nu(dy)\,dt$ is the compensated Poisson random measure associated with L\'{e}vy measure $\nu(dy)$, and
is the risk-neutral compensator ensuring the discounted gains process is a local martingale. The Feller condition $2\alpha\beta \geq \eta^2$ ensures $V_t > 0$ a.s., and $\rho \in (-1,0)$ captures the leverage effect.
The structural decomposition is as follows. Stochastic volatility governs the term structure of implied volatility and leverage-driven asymmetry, while the jump component determines tail behavior at short maturities, where variance mean-reversion has had insufficient time to attenuate left-tail risk.
Let $C(s,v,t)$ denote the value of a European contingent claim with maturity $T$ and payoff $\Phi(S_T)$. Under risk-neutral valuation,
The pricing equation follows from the infinitesimal generator $\mathcal{L} = \mathcal{L}_{\mathrm{diff}} + \mathcal{L}_{\mathrm{jump}}$ of the Markov process $(S_t, V_t)$. The local diffusion component is
and the nonlocal jump operator is
The no-arbitrage pricing equation is then
subject to the terminal condition $C(s,v,T)=\Phi(s)$.
The operator $\mathcal{L}_{\mathrm{jump}}$ is nonlocal: the value at $(s,v)$ depends on values at all states $se^y$ reachable via jumps. The compensating term $-s(e^y-1)C_s$ ensures integrability and consistency with the compensated jump measure. Setting $\nu \equiv 0$ reduces (ref) to the Heston PDE; further setting $\eta \to 0$ and $v$ constant recovers the Black--Scholes equation.
We restrict attention to finite-activity jump specifications, characterized by
under which the jump component reduces to a compound Poisson process with intensity $\lambda$ and jump-size density $f_Y$, so that $\nu(dy) = \lambda\,f_Y(y)\,dy$. The integrability condition
ensures the compensator $\kappa$ is well defined. Canonical examples include the Gaussian jump model of merton1976 and the double-exponential specification of kou2002; both admit straightforward quadrature evaluation of the jump integral and economically interpretable parameters.
By contrast, infinite-activity L\'{e}vy models, the Variance Gamma madan1998 and CGMY carr2002 specifications, satisfy
and exhibit singular behavior near $y=0$ requiring regularization or Fourier-based treatment. The PIDE framework accommodates both classes through the L\'{e}vy measure $\nu(dy)$; the empirical analysis focuses on finite-activity models to preserve calibration tractability and parameter interpretability.
The PIDE (ref) is completed by the terminal condition $C(s,v,T) = \Phi(s)$ and boundary conditions \[ C(s,v,t) \to 0 \;\text{ as }\; s \to 0, \qquad C(s,v,t) \sim s e^{-q(T-t)} - K e^{-r(T-t)} \;\text{ as }\; s \to \infty. \] The nonlocal jump operator requires truncation of the computational domain at a threshold $|y| \leq y_{\max}$ chosen to render excluded tail mass negligible; for finite-activity models the jump-size density decays sufficiently fast that this condition is easily satisfied.
For American-style contracts, the valuation problem becomes the variational inequality
Although the present analysis concerns European options, the PIDE structure extends to (ref) without modification to the operator formulation.
The principal numerical challenges in PIDE-based pricing are the global coupling induced by the nonlocal jump operator and the repeated solves required for calibration. The approach below treats the diffusion component implicitly, ensuring stability with respect to the stiff local terms, and evaluates the jump integral via FFT-based convolution, exploiting the shift structure of the operator in log-price coordinates.
Under the log-price substitution $x = \log s$, $\tau = T - t$, the dynamics of $X_t = \log S_t$ are
The infinitesimal generator $\mathcal{A}$ acting on $C(x,v,\tau)$ is
and the pricing equation becomes $\partial_\tau C = \mathcal{A}C - rC$. The log-transformation converts multiplicative jumps into additive shifts, giving the integral term a convolution structure amenable to FFT evaluation.
Spatial derivatives are discretized using second-order central differences in the interior; the mixed $C_{xv}$ term uses a standard four-point stencil. Time integration uses the Crank--Nicolson scheme with Rannacher smoothing over the first two steps, which suppresses oscillations near payoff discontinuities and ensures smooth implied-volatility extraction.
In log-price coordinates the jump operator is
The convolution structure of (ref) reduces the evaluation cost from $\mathcal{O}(N^2)$ to $\mathcal{O}(N\log N)$ via FFT, a reduction that is decisive in calibration settings. After domain truncation to $[y_{\min}, y_{\max}]$, the integral is approximated by
where $\{y_m, w_m\}$ are quadrature nodes and weights. Truncation error is controlled by the exponential decay of the jump-size density and is negligible relative to discretization error for the finite-activity models considered.
The semi-discrete system is
where $A_{\mathrm{diff}}$ is sparse and $A_{\mathrm{jump}}$ is dense. Operator splitting treats these components separately. The second-order Strang scheme is
Alternatively, the IMEX scheme
treats the stiff diffusion operator implicitly and the dense jump operator explicitly, preserving stability without solving a dense linear system. Both schemes are suitable for the calibration grids used in the empirical analysis.
The numerical scheme is validated prior to calibration via three exercises.
\paragraph{Grid-refinement study.} Table (ref) reports convergence under grid refinement for ATM call and OTM put contracts at $T=27$ days ($S_0=6{,}506.48$, $r=4.5\%$, $q=1.3\%$). Pricing errors decrease monotonically as the grid is refined. At the calibration grid ($N_S=100$, $N_t=80$), relative errors are below 6%, which is within acceptable tolerance given market bid-ask noise. The finest grid ($N_S=200$, $N_t=160$) serves as the reference.
\paragraph{Nested Black--Scholes validation.} Table (ref) compares the numerical PIDE solution against the analytical Black--Scholes price when $\eta \to 0$, collapsing the Heston diffusion to a one-dimensional PDE. The absolute error of 4.29 reflects grid coarseness and lies within the range consistent with calibration accuracy requirements.
\paragraph{Monte Carlo convergence.} Figure (ref) reports MC pricing error against path count for the at-the-money option at $T=89$ days. The convergence rate closely tracks the theoretical $O(N^{-1/2})$ benchmark; at 10,000 paths, pricing error falls below \$0.50. This confirms correct implementation of the simulation engine used within the GMM estimation step.
The PIDE solver provides high-accuracy reference prices for validation. For calibration, Black--Scholes and Heston prices are computed via transform-based FFT methods; SVJD and Heston+CGMY prices use the Carr--Madan FFT framework applied to the combined characteristic function. Monte Carlo with 10,000 paths is used as a secondary check within the GMM objective. Agreement across methods confirms that empirical differences across models reflect genuine specification effects rather than numerical artifacts.
The empirical analysis uses end-of-day S&P 500 (SPX) index option data. The dataset contains 1,280 contracts across three maturities: $T=27$, 89, and 153 days. The underlying index level is $S_0 = 6{,}506.48$, the risk-free rate is $r=4.5\%$, and the continuous dividend yield is $q=1.3\%$. Standard filters are applied to remove contracts with excessively wide bid-ask spreads, near-zero prices, or apparent no-arbitrage violations.
After filtering, the sample contains 582 short-maturity, 453 medium-maturity, and 245 long-maturity contracts. Moneyness $M = K/S$ ranges from approximately 0.80 to 1.20. At-the-money implied volatilities average 25.99%, 23.51%, and 22.96% for the three maturities, respectively, indicating a modest downward-sloping term structure. This maturity structure is central for identification: short-maturity options are sensitive primarily to jump risk, while medium and long maturities identify variance persistence and mean reversion.
Market implied volatilities are extracted by numerically inverting the Black--Scholes formula using Newton--Raphson iteration, with bisection as a fallback for deep moneyness contracts. Calibration targets implied volatility rather than raw prices to obtain a scale-free error measure that weights the smile geometry, the primary dimension of model differentiation, appropriately.
Parameters are estimated by GMM, minimizing the weighted sum of squared implied-volatility pricing errors
with $W=I$ in the baseline. Robust standard errors are computed via the sandwich estimator based on the numerical Jacobian of $m(\theta)$. Global optimization uses differential evolution for exploration, followed by Nelder--Mead refinement to mitigate convergence to local optima.
Model pricing is implemented as follows: Black--Scholes in closed form; Heston via FFT applied to the affine characteristic function; SVJD (Merton) and Heston+CGMY via the Carr--Madan FFT framework applied to the combined characteristic function. Specifically:
The L\'{e}vy measure $\nu(dy)$ is infinite-dimensional; without parametric restriction it is identified from a finite cross-section only through finitely many linear functionals $\int g(y)\,\nu(dy)$. Point-identification requires a parametric family. We adopt the CGMY specification of carr2002,
which nests finite-activity ($Y<0$), Variance Gamma ($Y=0$), and infinite-variation ($Y\in[1,2)$) specifications as special cases. Under mild regularity conditions, Proposition 11.4 of cont2004 implies that the map $(C,G,M,Y)\mapsto\{\text{IV surface}\}$ is injective, ensuring point-identification from the cross-section.
\paragraph{Signal content.} Each parameter affects a distinct feature of the implied-volatility surface. $C$ scales $\nu(dy)$ uniformly and is identified by the level of implied volatility relative to the diffusive benchmark. $G$ governs left-tail decay and is identified by the slope of the OTM put wing at short maturities, where jump risk is least attenuated by time averaging. $M$ plays the symmetric role for the call wing and is constrained to $M>1$ by the requirement that the compensator $\kappa_{\mathrm{CGMY}}$ (ref) be finite, an economic condition whose violation implies infinite $\mathbb{E}_{\mathbb{Q}}[S_T]$ and arbitrage. $Y$ determines the fine structure of jump activity and is identified by the joint variation in smile curvature across maturities.
\paragraph{Role of the maturity cross-section.} A single maturity is insufficient to separately identify all CGMY parameters alongside the five Heston parameters. The multi-maturity design ($T = 27$, 89, 153 days) is therefore essential: short-maturity contracts are maximally informative about jump parameters given minimal variance mean-reversion, while medium and long maturities identify variance persistence and leverage dynamics. This identification strategy follows bakshi1997 and bates1996.
\paragraph{Prior constraints.} Table (ref) maps each parameter to its identification signal, the direction of its effect on implied volatility, admissible bounds, and literature anchors. The constraint $M>1$ is imposed throughout estimation; for Merton specifications, $\kappa_{\mathrm{M}} = \lambda(e^{\mu_j+\frac{1}{2}\sigma_j^2}-1)$ is always finite, requiring only $\lambda \geq 0$.
\paragraph{Compensator and martingale constraint.} For the CGMY specification, the risk-neutral compensator is
finite if and only if $M>1$. The constraint is an economic requirement: $M \leq 1$ assigns positive probability to arbitrarily large upward jumps under $\mathbb{Q}$, implying $\mathbb{E}_{\mathbb{Q}}[S_T] = \infty$ and violating no-arbitrage. The constraint $M>1$ is imposed throughout estimation.
\paragraph{Testability: the activity index.} aitsahalia2009 and cont2011 provide nonparametric tests for jump activity from high-frequency return data; their evidence for SPX broadly supports a finite or near-finite activity regime. The calibrated estimate $\hat{Y} = -2.893$ is consistent with this evidence, placing the model in the compound-Poisson regime. Under this calibrated measure, the CGMY parameters $(C,G,M)$ admit the same economic interpretation as in the classical Merton model, with greater distributional flexibility.
The empirical analysis compares four nested specifications: Black--Scholes, Heston SV, Heston+CGMY, and SVJD (Merton). This hierarchy enables a direct decomposition of the incremental pricing contribution of stochastic volatility and alternative jump specifications. Performance is evaluated by RMSE and MAE in both implied-volatility (pp) and price (\$) units, reported for the full sample and decomposed by maturity and moneyness.
Table (ref) reports GMM estimates for all four models. Estimates are broadly consistent with the empirical option-pricing literature.
For the Heston specification, the estimated mean-reversion speed $\hat{\alpha}=1.356$ yr$^{-1}$ implies moderate adjustment toward the long-run variance. The long-run variance $\hat{\beta}=0.109$ ($\sqrt{\hat{\beta}} \approx 33.1\%$ annualized) exceeds the initial variance $\hat{V}_0=0.071$ ($\approx 26.6\%$), indicating that short-run implied volatility lies below its unconditional benchmark on the sample date. The leverage estimate $\hat{\rho}=-0.891$ is consistent with the well-documented negative correlation between equity returns and conditional variance. The Feller ratio $2\hat{\alpha}\hat{\beta}/\hat{\eta}^2 = 0.224 < 1$ indicates that the variance process reaches zero with positive probability under the estimated parameters, a feature common in short-horizon calibrations that does not impair option-price fit.
For the Heston+CGMY specification, the calibrated activity index $\hat{Y}=-2.893$ places the model firmly in the finite-activity regime ($Y<0$), consistent with the high-frequency evidence discussed in Section (ref). The parameters $\hat{C}=0.0024$, $\hat{G}=1.450$, $\hat{M}=2.733$ describe a L\'{e}vy measure that is asymmetric and concentrated on downside movements, with the constraint $\hat{M}>1$ satisfied. The near-zero compensator $\kappa_{\mathrm{CGMY}}=-0.0005$ indicates that the CGMY component contributes marginally to drift adjustment; pricing performance is driven primarily by the Heston diffusion backbone.
The Merton SVJD specification yields an estimated intensity $\hat{\lambda}=0.01$, reflecting near-zero jump activity under the calibration. The standard errors on $\mu_j$ and $\sigma_j$ are large, indicating that these parameters are weakly identified in the presence of the stochastic-volatility component. Consequently, SVJD (Merton) pricing is dominated by the Heston diffusion.
Table (ref) reports RMSE and MAE by maturity. Figures (ref) and (ref) present the fitted smiles and residual patterns.
The results support a clear model hierarchy. Black--Scholes produces the largest errors at all maturities. Heston SV reduces aggregate IV RMSE from 10.13\,pp to 6.18\,pp, a 39% improvement attributable to variance persistence and leverage-driven asymmetry. Jump-augmented specifications improve marginally on Heston at the aggregate level: IV RMSE falls to 6.14\,pp (Heston+CGMY) and 6.15\,pp (SVJD), reductions of 0.6% and 0.5%, respectively, relative to Heston SV. The near-equivalence of the two jump models reflects the fact that both produce small effective jump contributions (the calibrated $\hat{Y}=-2.893$ and $\hat{\lambda}=0.01$ place both specifications near the compound-Poisson limit with minimal activity) and that pricing performance is driven predominantly by the Heston backbone in each case.
Table (ref) reports in-sample implied-volatility RMSE by moneyness bucket, pooled across maturities.
In the OTM and Deep OTM regions, Heston+CGMY holds a small advantage, consistent with the left-tail asymmetry of the calibrated CGMY measure ($\hat{G}<\hat{M}$). Heston SV performs best at ATM and in the ITM region, where the leverage effect is the dominant driver of smile asymmetry. Heston+CGMY achieves the lowest error in the Deep ITM region (0.68\,pp versus 1.17\,pp for Heston SV), indicating that the CGMY right-tail structure provides a useful refinement for deep in-the-money contracts. The modest absolute differences across Heston-backbone models in the OTM region are consistent with the near-zero effective jump contribution under the calibrated parameters: the stochastic-volatility channel already captures most of the observed OTM skew.
Table (ref) compares in-sample calibration ($T=27$, 89 days) with out-of-sample performance on $T=153$ day contracts excluded from estimation. The SVJD (Merton) model retains its small advantage at the long maturity, confirming that the performance differential does not reflect overfitting.
Figure (ref) plots the term structure of ATM implied volatility. All Heston-backbone models reproduce the observed downward slope with comparable accuracy.
Figure (ref) compares simulated terminal distributions for Heston SV and Heston+CGMY at $T=27$ days using 10,000 paths. Table (ref) reports distributional moments. Heston SV and Heston+CGMY produce near-identical dispersion, skewness, and excess kurtosis under the calibrated parameters, confirming that the small CGMY contribution does not materially alter tail behavior. The mean log-return difference of 0.012\,pp reflects the CGMY drift correction.
Figure (ref) compares model and market prices for all 1,280 contracts.
All Heston-backbone specifications achieve $R^2 > 0.998$ and price RMSE below \$15. The marginal improvement of jump specifications over Heston SV is visible but small in absolute terms, consistent with the aggregate RMSE evidence.
Figure (ref) plots the calibrated CGMY L\'{e}vy density on both tails. The near-zero $C$ and large negative $Y$ confirm finite-activity behavior: the measure assigns density primarily to moderate jumps, not to the infinitely fine small-jump structure characteristic of processes with $Y\geq 0$. The asymmetry between left ($\hat{G}=1.450$) and right ($\hat{M}=2.733$) tails reflects the downside orientation of SPX jump risk: negative movements decay more slowly under the calibrated $\mathbb{Q}$.
The empirical pattern admits a coherent economic interpretation. The Black--Scholes model performs poorly because its constant-volatility assumption cannot reproduce two dominant empirical features of option markets: the persistence of volatility through time and the pronounced cross-sectional asymmetry of implied volatilities across strike prices. In particular, Black--Scholes generates a flat implied-volatility surface, whereas the observed data exhibit both skewness and term-structure effects that vary systematically across moneyness and maturity. As a result, the model systematically underprices downside risk and fails to capture the curvature embedded in market-implied distributions.
The Heston stochastic-volatility specification corrects these deficiencies by allowing variance itself to evolve as a stochastic process and by incorporating a strong negative correlation between returns and volatility innovations through the leverage effect. Economically, this mechanism reflects the empirical tendency for volatility to rise when equity prices decline. The introduction of stochastic variance generates richer dynamics for both the term structure and the asymmetry of implied volatilities, allowing the model to reproduce the downward-sloping volatility skew observed in equity index options. In the present sample, these features account for the majority of the improvement in pricing accuracy relative to Black--Scholes, particularly for near-the-money and moderately out-of-the-money contracts where diffusive volatility dynamics dominate pricing behavior.
The jump-augmented specifications provide more localized incremental improvements. The Heston+CGMY framework performs somewhat better in the Deep ITM region because the CGMY jump structure permits greater flexibility in modeling heavy tails and asymmetric return distributions. Likewise, the Merton jump-diffusion specification contributes modestly at longer maturities, where the cumulative effect of infrequent but non-negligible jump events becomes more relevant to option valuation. Nevertheless, neither jump extension materially outperforms the baseline Heston stochastic-volatility model in the current sample. The incremental gains remain relatively small compared with the substantial improvement already achieved by introducing stochastic variance and leverage dynamics.
These findings suggest that the sample period was characterized by elevated but predominantly diffusive volatility rather than by markets strongly dominated by discrete discontinuities or crash-risk repricing. In such an environment, the stochastic-volatility backbone embedded in the Heston framework captures most of the economically relevant structure of the implied-volatility surface. As a result, the pricing evidence indicates that continuous variance fluctuations, rather than large jump arrivals, were the principal drivers of option-implied risk premia during the estimation period.
At the same time, the results do not imply that jump risk is unimportant in general. Under market conditions in which discontinuous price movements are more heavily priced, such as periods of geopolitical stress, liquidity dislocation, systemic contagion, or severe market fragmentation, the contribution of jump components is expected to increase materially. In those environments, models incorporating richer jump structures may become substantially more valuable for reproducing tail asymmetries, steep short-maturity skews, and extreme downside insurance premia. The comparatively modest contribution of jumps in the present analysis should therefore be interpreted as conditional on the prevailing market regime rather than as a universal conclusion regarding option pricing dynamics.
Finally, the robustness exercises reported in Appendix (ref) reinforce the stability of these conclusions. Sensitivity analyses and out-of-sample evaluations indicate that the ranking of model performance remains largely unchanged across parameter perturbations, alternative calibration windows, and validation samples. This stability suggests that the superiority of the stochastic-volatility framework is not merely an artifact of overfitting or sample-specific parameterization, but rather reflects persistent structural features of the implied-volatility surface that are captured most effectively by the Heston specification.
This paper develops and implements a PIDE-based pricing framework for options under joint stochastic volatility and jump dynamics, and evaluates the empirical decomposition of these components using a cross-section of SPX contracts across three maturities.
The principal empirical finding is that stochastic volatility accounts for the dominant share of pricing improvement. Relative to Black--Scholes, Heston SV reduces aggregate IV RMSE by 39% by capturing variance persistence, term-structure effects, and leverage-driven asymmetry. Jump augmentation via either CGMY or Merton specifications produces marginal additional improvements: aggregate IV RMSE declines by 0.04\,pp (Heston+CGMY) and 0.03\,pp (Merton) relative to Heston SV. These gains are concentrated in the Deep ITM region (CGMY) and at longer maturities (Merton). The calibrated CGMY activity index is consistent with compound-Poisson jump behavior, as supported by high-frequency evidence in aitsahalia2009 and cont2011.
The overall empirical pattern suggests that the sample period was characterized primarily by elevated but largely diffusive volatility, in which stochastic variance dynamics captured most of the economically relevant structure of the implied-volatility surface. At the same time, the results do not imply that jump risk is unimportant more generally. Under market conditions characterized by severe liquidity stress, geopolitical uncertainty, or heightened crash-risk pricing, jump components would likely play a substantially larger role in explaining implied-volatility dynamics and tail-risk premia.
From a numerical standpoint, the combination of Crank--Nicolson discretization, FFT-based jump integral evaluation, and operator-splitting time-stepping provides a stable and accurate PIDE implementation. Grid-refinement studies and nested validation exercises confirm that empirical results reflect model specification rather than numerical artifacts. The robustness exercises additionally indicate that the relative model hierarchy remains stable across parameter perturbations and out-of-sample validation, suggesting that the superiority of the stochastic-volatility specification is structural rather than sample-specific.
Three extensions merit future investigation. First, American and barrier contracts, where the nonlocal PIDE operator interacts directly with free-boundary conditions, would provide a richer testbed for the numerical scheme. Second, joint time-series and cross-sectional estimation would enable separate identification of the physical and risk-neutral measures, permitting inference on jump risk premia. Such an approach would also allow a more precise decomposition of variance and jump compensation across market regimes and volatility environments. Third, the multi-asset extension, where stochastic volatility, jumps, and dependence structure interact in the pricing of index derivatives, is a natural direction given the framework's generality. In particular, extending the framework to correlated multi-asset settings could provide insight into systemic tail dependence, contagion dynamics, and the pricing of cross-asset volatility transmission.