Beyond Black-Scholes: A Computational Framework for Option Pricing Using Heston, GARCH, and Jump Diffusion Models

Abstract: This research addresses accurate option pricing by employing models beyond the traditional Black-Scholes framework. While Black-Scholes provides a closed-form solution, it is limited by assumptions of constant volatility, no dividends, and continuous price movements. To overcome these limitations, we use Monte Carlo simulation alongside the GARCH model, Heston stochastic volatility model, and Merton jump-diffusion model. The Black-Scholes-Monte Carlo method simulates diverse stock price paths using geometric Brownian motion. The GARCH model forecasts time-varying volatility from historical data. The Heston model incorporates stochastic volatility to capture volatility clustering and skew. The Merton jump-diffusion model adds sudden price jumps via a Poisson process. Results show the Heston model consistently produces estimates closer to market prices, while the Merton model performs well for volatile assets with sudden price movements. The GARCH model provides improved volatility forecasts for future option price prediction. All experiments used live market data from November 2024.

• The paper presents a unified Python Monte Carlo framework that combines Black-Scholes, Heston stochastic volatility, Merton jumps, and GARCH forecasting using 10,000 simulations and optimizer-based calibration.
• Heston prices tracked Tesla market quotes more closely than GBM across strikes, while Merton improved representation of abrupt price moves but sometimes overestimated out-of-the-money options.
• The results suggest advanced models are most useful for volatility skew, jumps, and forecasting, but conclusions remain preliminary because the study lacks aggregate error metrics, confidence intervals, robust out-of-sample tests, and positivity controls for Heston variance.

Overview

This paper presents a computational framework for pricing European call options that extends beyond the Black-Scholes model by combining Monte Carlo simulation with three refinements: stochastic volatility (Heston), jump dynamics (Merton jump-diffusion), and time-varying conditional volatility forecasting (GARCH(1,1)). The authors—Sidhu and Saxena of USC's Viterbi School of Engineering—implement all models in Python against live market data fetched from Yahoo Finance in November 2024, evaluating them on five equities chosen for high options volume and heterogeneous business profiles: Tesla, Meta, AMC Entertainment, MARA Holdings, and Shopify. A machine-learning-based calibration layer, using L-BFGS-B optimization of squared-error objective functions, is introduced to fit both the Heston parameters $(\kappa, \theta, \sigma_v)$ and the Merton jump parameters $(\lambda_j, \mu_j, \sigma_j)$.

The paper's central empirical claim is that the Heston-Monte Carlo estimator tracks market option prices more closely than plain geometric Brownian motion (GBM) across strike prices, while the Merton extension improves fits for assets prone to abrupt price moves. These claims are supported by per-strike price comparisons rather than formal error statistics; no aggregate error metrics (e.g., RMSE or MAPE) are reported, which limits the strength of the comparative conclusions.

Methodological framework

All four estimators share a common simulation backbone: discretized GBM paths with 10,000 simulations and 100 time steps, risk-free rate fixed at 0.0427–0.047, and discounted average payoff $e^{-rT}\overline{\max(S_T - K, 0)}$ as the price estimate.

Black-Scholes-Monte Carlo baseline: paths follow

$$S_{t+\Delta t} = S_t \exp\left[\left(r - \tfrac{1}{2}\sigma^2\right)\Delta t + \sigma Z_t\right]$$

with implied volatility held constant along each path—the very assumption the later models relax.

Merton jump-diffusion: the SDE is augmented with a compound Poisson term $(J-1)\,dN(t)$, where jump sizes are lognormal ($J = e^Y$, $Y \sim N(\mu_j, \sigma_j^2)$) and arrivals are Poisson with intensity $\lambda_j$. The three jump parameters are calibrated by minimizing the sum of squared differences between model and market prices across strikes, using L-BFGS-B with box constraints to keep parameters in realistic ranges. The authors justify optimizer-based calibration on the grounds that the parameters affect option prices non-linearly and that purely theoretical parameterization is impractical in live markets.

GARCH(1,1)-Monte Carlo: rather than pricing current contracts, this variant forecasts option prices three trading days ahead. Conditional variance follows $$\sigma_t^2 = \alpha_0 + \alpha_1\epsilon_{t-1}^2 + \beta_1\sigma_{t-1}^2$$, fitted via the arch package on historical returns; the forecast variance is annualized ($\times\sqrt{252}$) and fed into the GBM simulator. This is the only component evaluated prospectively against realized future market prices.

Heston stochastic volatility: joint simulation of price and variance,

$(\lambda_j, \mu_j, \sigma_j)$0

with correlated Brownian shocks via $(\lambda_j, \mu_j, \sigma_j)$1. Calibration minimizes the squared difference between historically observed variances and variances predicted by the Heston dynamics, again via L-BFGS-B. Notably, the Euler discretization used here does not enforce positivity of the variance path, so negative-variance artifacts are possible at large steps—a standard concern the paper does not address.

Empirical results

The headline comparison is Heston versus GBM on Tesla. At the $(\lambda_j, \mu_j, \sigma_j)$2150.35 against a market price of $(\lambda_j, \mu_j, \sigma_j)$3151.69; at the $(\lambda_j, \mu_j, \sigma_j)$495.41 versus a market price of $(\lambda_j, \mu_j, \sigma_j)$596.34. Across the full strike ladder ($(\lambda_j, \mu_j, \sigma_j)$6225), GBM systematically overestimates low strikes and underestimates high strikes—consistent with unmodeled volatility skew—whereas Heston deviations remain small and roughly symmetric. The largest single Heston deviation occurs at the $(\lambda_j, \mu_j, \sigma_j)$71.63; the authors attribute this to transient liquidity conditions and limited historical data rather than model misspecification.</p> <p>For the Merton model, results are mixed in an instructive way. On Meta, the model underestimates at-the-money contracts ($(\lambda_j, \mu_j, \sigma_j)$8143.90 at the $(\lambda_j, \mu_j, \sigma_j)$93.74 at $e^{-rT}\overline{\max(S_T - K, 0)}$01.31 vs. $e^{-rT}\overline{\max(S_T - K, 0)}$13) and converges closely at $e^{-rT}\overline{\max(S_T - K, 0)}$20.35 vs. $0.35). The authors interpret the overpricing at higher strikes as a consequence of embedding jump risk that may not materialize in the actual underlying's behavior—an honest concession that jump intensity calibrated on one snapshot can misrepresent tail dynamics.

The GARCH forecasts show monotonically increasing option values across the three forecast dates, as expected from theta decay reversal under rising conditional variance. Validation against realized November 19–20 prices for AMC shows reasonable tracking at low strikes but persistent overprediction at $e^{-rT}\overline{\max(S_T - K, 0)}$35, where the authors note thin liquidity in deep out-of-the-money contracts distorts observed market prices downward relative to any model value.

A recurring qualitative finding deserves emphasis: the plain GBM baseline performs adequately when strikes sit below the spot price, contradicting the implication that sophisticated models dominate uniformly. The paper states this plainly—GBM "performed reasonably well when the strike price was lower than the current market price"—and positions Heston's advantage primarily in skew-sensitive regions and longer maturities.

Limitations and open questions

Several limitations are conceded explicitly. First, all calibrations rely on Yahoo Finance data with restricted historical depth (roughly one year); the authors note that Heston estimates slightly overprice options because longer histories are unavailable, and GARCH forecasts degrade for the same reason. Second, the evaluation protocol compares point estimates against mid-market prices without accounting for bid-ask spreads, transaction costs, or supply-demand imbalances—factors the authors repeatedly cite as explanations for residual deviation but do not quantify. Third, no statistical error bars accompany the Monte Carlo estimates; with 10,000 paths, standard errors are non-trivial relative to the cent-level deviations being interpreted. Fourth, the L-BFGS-B calibration minimizes in-sample squared error on the same strikes used for evaluation, raising an overfitting concern that is not tested via out-of-sample strike or date splits (the GARCH validation is the sole prospective check). Fifth, the Euler scheme for the Heston variance process admits negative values, and no absorption or reflection fix is described. Finally, the correlation parameter $e^{-rT}\overline{\max(S_T - K, 0)}$4 appears in the simulation equations but not in the calibrated parameter set, leaving its estimation procedure unspecified.

Concrete open questions include: whether the Heston advantage persists under formal out-of-sample backtesting across multiple calibration dates; how jump-intensity calibration behaves during regime shifts, given that a single snapshot of strikes drives $e^{-rT}\overline{\max(S_T - K, 0)}$5; and whether neural-network surrogates for the calibration map—as suggested in the cited literature on deep-learning volatility calibration—would reduce the computational cost of repeated L-BFGS-B runs.

Conclusion

The paper delivers a coherent, reproducible pipeline that layers stochastic volatility, jumps, and GARCH-forecasted variance onto a common Monte Carlo engine, with gradient-based calibration replacing hand-set parameters. Its strongest evidence is the Tesla strike-ladder comparison showing Heston tracking market prices more tightly than constant-volatility GBM, particularly where skew effects are pronounced. The contribution is primarily integrative and empirical rather than methodological: the individual models are standard, and the novelty lies in their unified implementation with optimizer-driven calibration against live quotes. The absence of aggregate error metrics, confidence intervals, and out-of-sample controls means the comparative claims should be read as indicative rather than definitive, and the framework's robustness across market regimes remains untested within the scope of this work.