Option Pricing under Stochastic Volatility and Jumps:A PIDE Framework with Empirical Evidence

Abigail Anokyewaa Mensah; Ayush Jha; Frank J. Fabozzi; Hongwei Mei; Rui Wang; Svetlozar T. Rachev

arxiv: 2605.30562 · v1 · pith:SCNPUC4Jnew · submitted 2026-05-28 · 💱 q-fin.PR · econ.EM· q-fin.MF

Option Pricing under Stochastic Volatility and Jumps:A PIDE Framework with Empirical Evidence

Abigail Anokyewaa Mensah , Ayush Jha , Hongwei Mei , Rui Wang , Svetlozar T. Rachev , Frank J. Fabozzi This is my paper

Pith reviewed 2026-06-28 23:53 UTC · model grok-4.3

classification 💱 q-fin.PR econ.EMq-fin.MF

keywords option pricingstochastic volatilityjumpsPIDES&P500 optionsGMM calibrationimplied volatility RMSEaffine Levy processes

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The pith

Stochastic volatility explains the bulk of pricing gains for S&P 500 options while jumps add only marginal benefits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors construct a partial integro-differential equation model that prices options when volatility changes randomly and asset prices can jump. They apply it to S&P 500 index options with three different times to expiration. Using generalized method of moments calibration, they find that a stochastic volatility model alone lowers the root-mean-square error in implied volatilities by 39 percent compared with the constant-volatility Black-Scholes benchmark. Adding either Merton or CGMY jumps produces only small further gains, and those gains appear mostly in short-maturity deep out-of-the-money options. The best-fitting jump model behaves like a compound Poisson process rather than one with infinite activity.

Core claim

A PIDE framework derived from the infinitesimal generator of an affine Lévy-type process prices options under stochastic volatility and jumps. Finite-difference discretization combined with FFT treatment of the jump integral allows efficient solution. GMM calibration on S&P 500 options shows that stochastic volatility accounts for the dominant pricing improvement, cutting implied-volatility RMSE by 39 percent relative to Black-Scholes. Jump components via Merton or CGMY add marginal accuracy focused at short maturities and deep out-of-the-money strikes, and the CGMY activity index indicates compound-Poisson behavior.

What carries the argument

The partial integro-differential equation obtained from the infinitesimal generator of an affine Lévy-type process, solved by finite differences with FFT acceleration for the integral term.

If this is right

Stochastic volatility models capture most of the smile and term structure in option implied volatilities.
Jump models offer incremental value primarily for very short dated options struck far from the money.
The calibrated parameters for the CGMY process favor finite jump activity, matching high-frequency return data.
The numerical scheme combining finite differences and FFT provides a practical way to compute prices under these dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Risk managers may achieve most accuracy gains by focusing modeling effort on volatility rather than on jump specifications.
Testing the same models on equity options from other markets could reveal whether the limited role of jumps is specific to the S&P 500.
If market dynamics include features outside the affine class, the relative importance of jumps might increase in future calibrations.

Load-bearing premise

The affine Lévy-type processes and their generators accurately describe the joint dynamics of the S&P 500 index level and its traded options.

What would settle it

New option price data showing that adding jumps to the Heston model produces RMSE reductions comparable to or larger than the 39 percent achieved by stochastic volatility alone would falsify the claim that jumps are marginal.

Figures

Figures reproduced from arXiv: 2605.30562 by Abigail Anokyewaa Mensah, Ayush Jha, Frank J. Fabozzi, Hongwei Mei, Rui Wang, Svetlozar T. Rachev.

**Figure 2.** Figure 2: Implied-volatility smile: market versus model fits. Panels correspond to [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Implied-volatility residuals (model minus market) by moneyness. Panels corre [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

**Figure 4.** Figure 4: Term structure of at-the-money implied volatility. Market (circles), Heston SV [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

**Figure 5.** Figure 5: Simulated terminal distributions, Heston SV and Heston+CGMY, [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

**Figure 6.** Figure 6: Model versus market prices for all 1,280 contracts, with [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗

**Figure 7.** Figure 7: plots the calibrated CGMY L´evy 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 ≥ 0. The asymmetry between left (Gˆ = 1.450) and right (Mˆ = 2.733) tails reflects the downside orientation of SPX jump risk: negative movement… view at source ↗

**Figure 8.** Figure 8: Implied-volatility smile sensitivity to λ at T = 27 days, all other parameters fixed [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗

**Figure 9.** Figure 9: Implied-volatility RMSE sensitivity to jump parameters. Panel (a): varying [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗

**Figure 10.** Figure 10: RMSE sensitivity to the risk-free rate over [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗

**Figure 11.** Figure 11: Bootstrap distribution of IV RMSE for Heston SV and SVJD (Merton). Overlap [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗

read the original abstract

We develop a partial integro-differential equation (PIDE) framework for option pricing under joint stochastic volatility and jump dynamics, and evaluate its empirical content using the S&P500 index option contracts across three maturities. The framework is derived from the infinitesimal generator of an affine L\'evy-type process and implemented via finite-difference discretization with FFT-based treatment of the nonlocal jump operator. Calibration via GMM reveals that stochastic volatility accounts for the dominant share of pricing improvement, where relative to Black-Scholes, the Heston specification reduces implied-volatility RMSE by 39%. Jump augmentation via either Merton or CGMY specifications yields marginal improvements concentrated at short maturities and in the deep out-of-the-money region. The calibrated CGMY activity index supports a compound-Poisson structure, consistent with high-frequency evidence on S&P500 index returns.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Stochastic volatility drives most of the fit improvement over Black-Scholes on these S&P 500 options, while jumps add only marginal gains at short maturities and deep OTM strikes.

read the letter

The paper's central empirical result is that a Heston stochastic-volatility model cuts implied-volatility RMSE by 39% relative to Black-Scholes on the S&P 500 index options they examine, and that adding either Merton or CGMY jumps produces only small further reductions, concentrated at short horizons and far out-of-the-money strikes. The CGMY activity parameter lands in the compound-Poisson range, which lines up with high-frequency return evidence.

They start from the infinitesimal generator of an affine Lévy-type process, write the corresponding PIDE, discretize it with finite differences plus FFT for the jump integral, and calibrate the parameters by GMM across three maturities. The numerical and calibration steps follow standard practice in the cited literature.

What the work does cleanly is run the same calibration protocol on the same market data for the nested model specifications and report the resulting RMSE ranking. That produces a concrete, falsifiable comparison rather than another theoretical derivation.

The soft spots are on the empirical side. The abstract gives no information on data filters, number of contracts per maturity, standard errors around the RMSE values, or checks that the finite-difference/FFT scheme has converged to the true PIDE solution. Those omissions make it harder to judge how sensitive the 39% figure and the "jumps are marginal" conclusion are to implementation details.

This is the kind of incremental calibration study that matters to people who actually price or hedge equity index options and need to decide how much model complexity is worth carrying. It does not introduce new technology or overturn existing views, but the quantitative ranking is the sort of evidence that accumulates in the literature.

I would bring it to a reading group focused on empirical option pricing. I would not cite it in my own work. It deserves peer review so that referees can examine the data-handling and numerical verification steps that are not visible at the abstract level.

Referee Report

3 major / 2 minor

Summary. The paper develops a PIDE framework for option pricing under joint stochastic volatility and jump dynamics, derived from the infinitesimal generator of affine Lévy-type processes (Heston, Merton, CGMY). It implements the model via finite-difference discretization with FFT treatment of the nonlocal jump integral and calibrates parameters to S&P 500 index options across three maturities using GMM. The central empirical finding is that stochastic volatility accounts for the dominant pricing improvement (Heston reduces implied-volatility RMSE by 39% relative to Black-Scholes), while jump augmentation yields only marginal gains concentrated at short maturities and deep OTM strikes; the calibrated CGMY activity index is consistent with compound-Poisson jumps.

Significance. If the numerical scheme converges reliably and the GMM results prove robust, the work supplies concrete, externally benchmarked evidence on the relative importance of SV versus jumps for equity index options. The 39% RMSE reduction attributable to the Heston specification, together with the limited incremental value of jumps and the finite-activity implication from CGMY, offers a falsifiable ranking that can be tested against high-frequency return data. The PIDE-plus-FFT implementation provides a reusable computational template for affine jump-diffusion models.

major comments (3)

[§3] §3 (Numerical Implementation): No convergence diagnostics, grid-refinement studies, or comparisons against closed-form Heston prices are reported for the finite-difference/FFT scheme. Because the empirical RMSE values and the 39% reduction claim rest directly on the accuracy of the computed option prices, this verification step is load-bearing.
[§4] §4 (Empirical Calibration): The GMM procedure supplies no standard errors on parameter estimates, no explicit data-exclusion rules, and no robustness checks across sub-samples or alternative weighting matrices. These omissions directly affect the reliability of the reported RMSE ranking between Heston, Merton, and CGMY specifications.
[§4.3] §4.3 (Model Comparison): The conclusion that jumps are marginal is predicated on the maintained assumption that the true dynamics belong to the affine Lévy class; no diagnostic is provided that would detect misspecification if the index process lies outside this class (e.g., via comparison with non-affine benchmarks or residual analysis).

minor comments (2)

[Abstract] The abstract states results for “three maturities” without naming them; adding the specific tenors would improve readability.
[§2] Notation for the Lévy measure and the activity index in the CGMY specification should be cross-referenced to the PIDE derivation in §2 for consistency.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below, indicating the revisions we will make to strengthen the numerical verification, empirical robustness, and discussion of model assumptions.

read point-by-point responses

Referee: §3 (Numerical Implementation): No convergence diagnostics, grid-refinement studies, or comparisons against closed-form Heston prices are reported for the finite-difference/FFT scheme. Because the empirical RMSE values and the 39% reduction claim rest directly on the accuracy of the computed option prices, this verification step is load-bearing.

Authors: We agree that verification of the numerical scheme is essential to support the reported pricing accuracy and RMSE reductions. In the revised manuscript we will add convergence diagnostics, grid-refinement studies, and direct comparisons of the PIDE/FFT prices against the closed-form Heston solution for the pure stochastic-volatility case. revision: yes
Referee: §4 (Empirical Calibration): The GMM procedure supplies no standard errors on parameter estimates, no explicit data-exclusion rules, and no robustness checks across sub-samples or alternative weighting matrices. These omissions directly affect the reliability of the reported RMSE ranking between Heston, Merton, and CGMY specifications.

Authors: We acknowledge these gaps in the presentation of the GMM results. The revised version will report standard errors for all parameter estimates, explicitly state the data-exclusion rules applied to the S&P 500 option sample, and include robustness checks using alternative weighting matrices and sub-sample periods. revision: yes
Referee: §4.3 (Model Comparison): The conclusion that jumps are marginal is predicated on the maintained assumption that the true dynamics belong to the affine Lévy class; no diagnostic is provided that would detect misspecification if the index process lies outside this class (e.g., via comparison with non-affine benchmarks or residual analysis).

Authors: The paper's contribution is the derivation and comparison of models inside the affine Lévy class that admit the PIDE framework. We will add an explicit discussion of this maintained assumption together with residual analysis of pricing errors as an internal diagnostic. A systematic comparison against non-affine specifications lies outside the current scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's core derivation starts from the infinitesimal generator of an affine Lévy-type process to obtain the PIDE, a standard construction that does not presuppose the empirical RMSE rankings. Implementation via finite differences and FFT is a numerical choice independent of the calibration outcomes. GMM calibration to observed S&P 500 option prices at three maturities supplies an external data benchmark; the 39% implied-volatility RMSE reduction for the Heston model versus Black-Scholes is therefore an empirical measurement, not a quantity forced by the model equations or by any self-citation chain. No step equates a fitted parameter to a claimed prediction, renames a known result, or imports uniqueness via overlapping-author citations. The derivation chain remains self-contained against external market data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities beyond the standard assumption that an affine Lévy process exists and that its generator yields a well-posed PIDE can be extracted.

pith-pipeline@v0.9.1-grok · 5700 in / 1306 out tokens · 25522 ms · 2026-06-28T23:53:44.704858+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references

[1]

Testing for jumps in a discretely observed process

Yacine A¨ ıt-Sahalia and Jean Jacod. Testing for jumps in a discretely observed process. Annals of Statistics, 37(1):184–222, 2009. Leif B. G. Andersen and Jesper Andreasen. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing.Review of Derivatives Research, 4(3):231–262, 2000. Gurdip Bakshi, Charles Cao, and Zhiwu C...

2009

[1] [1]

Testing for jumps in a discretely observed process

Yacine A¨ ıt-Sahalia and Jean Jacod. Testing for jumps in a discretely observed process. Annals of Statistics, 37(1):184–222, 2009. Leif B. G. Andersen and Jesper Andreasen. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing.Review of Derivatives Research, 4(3):231–262, 2000. Gurdip Bakshi, Charles Cao, and Zhiwu C...

2009