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

This research proposes a computational framework for option pricing that utilizes Monte Carlo simulations with GARCH, Heston, and Merton jump-diffusion models to overcome the limitations of the Black-Scholes framework, demonstrating that the Heston model generally yields the most accurate market price estimates while the Merton and GARCH models excel for volatile assets and volatility forecasting, respectively.

The Gist

Imagine you are trying to predict the price of a ticket to a concert that won't happen for six months. This ticket is an Option. In the financial world, figuring out what that ticket is worth today is a massive challenge.

For decades, the financial world relied on a famous recipe called the Black-Scholes model. Think of this model as a perfectly smooth, glassy highway. It assumes that:

1. The road is always straight (prices move smoothly).
2. The weather is always the same (volatility is constant).
3. There are no sudden potholes or detours (no price jumps).

The problem? Real life isn't a glassy highway. It's a bumpy, chaotic dirt road with sudden storms, potholes, and traffic jams. When the 2008 financial crisis hit, it was like a massive earthquake on that road, and the old Black-Scholes recipe failed miserably because it couldn't handle the bumps.

This paper, written by two researchers from USC, proposes a new, high-tech navigation system to replace that old recipe. They built a computational framework using three advanced tools to simulate the messy reality of the stock market.

Here is how they did it, explained simply:

1. The Engine: Monte Carlo Simulation

Instead of trying to predict the one exact path a stock will take, the researchers used a method called Monte Carlo Simulation.

• The Analogy: Imagine you are trying to guess where a drunk person will end up after walking for an hour. You can't predict their exact steps. So, you run a computer simulation 10,000 times, each time giving the person a slightly different random path. You then look at where most of them ended up to make your best guess.
• In the Paper: They simulated thousands of possible future paths for stocks like Tesla, Meta, and AMC. This allowed them to see a "cloud" of possibilities rather than just one straight line.

2. The Upgrades: Three New Models

The researchers realized that even with 10,000 simulations, if you use the old "smooth road" rules, you still get the wrong answer. So, they added three specific upgrades to their simulation engine:

A. The GARCH Model (The Weather Forecaster)

• The Problem: The old model assumed the "wind" (volatility) was always the same. In reality, if there is a storm today, there's likely to be a storm tomorrow. This is called "volatility clustering."
• The Solution: They used the GARCH model to act like a weather forecaster. Instead of assuming a constant breeze, it looks at the past few days of "wind" to predict how stormy the next few days will be. This helps them price options more accurately when the market is nervous.

B. The Heston Model (The Shaky Hand)

• The Problem: The old model assumed the "wind" (volatility) was a fixed number. But in the real world, the wind changes speed and direction constantly.
• The Solution: The Heston model treats volatility as a living, breathing thing that fluctuates randomly. It's like driving a car where the steering wheel shakes on its own. By letting the "wind" change randomly during the simulation, the model captures the "fear" and "greed" of the market much better than the old static model.
• The Result: When they tested this on Tesla and AMC stocks, the Heston model's predictions were much closer to the actual market prices than the old method.

C. The Merton Jump-Diffusion Model (The Sudden Earthquake)

• The Problem: The old model assumed prices move smoothly, step-by-step. But sometimes, a stock price doesn't walk; it teleports. A sudden news headline (like a war or a CEO scandal) can cause a stock to jump up or down instantly.
• The Solution: The Merton model adds a "Jump" button to the simulation. It allows the stock price to suddenly leap forward or backward, just like a frog jumping in a pond.
• The Result: This was crucial for volatile stocks like MARA (a crypto-mining company). The old model kept underestimating the price because it didn't expect the jumps. The Merton model accounted for the chaos and got much closer to the real price.

3. The Secret Sauce: Machine Learning

To make these complex models work perfectly, the researchers didn't just guess the settings. They used Machine Learning (specifically an optimizer called L-BFGS-B) to act like a tuning fork.

• The Analogy: Imagine you have a complex radio with 50 knobs. You want to tune it to get the clearest signal. Instead of turning the knobs randomly, you use a robot that listens to the static and automatically adjusts every single knob until the music is perfect.
• In the Paper: The robot adjusted the math parameters of the Heston and Merton models until they matched the real-world market data as closely as possible.

The Verdict: Did it Work?

Yes. The researchers tested their new framework against real stocks (Tesla, Meta, AMC, Shopify, MARA) using live data from November 2024.

• The Old Way (Black-Scholes): Often got the price wrong, especially when the market was scary or when prices jumped suddenly. It was like trying to drive a Formula 1 car on a dirt road with a map that only showed highways.
• The New Way (Their Framework): The Heston model was the overall winner, capturing the "shaky" nature of the market best. The Merton model was great for stocks that jump around wildly. The GARCH model helped predict how the market would behave in the coming days.

Conclusion

This paper is essentially saying: "Stop pretending the stock market is a calm, predictable ocean. It's a stormy sea with sudden waves."

By combining Monte Carlo simulations (running thousands of scenarios), advanced math models (to handle shaking and jumping), and AI (to fine-tune the settings), they created a much more accurate tool for pricing options. This helps investors and banks understand risk better and make smarter decisions, rather than relying on an outdated map that doesn't match the terrain.

Technical Summary

1. Problem Statement

The research addresses the limitations of the traditional Black-Scholes (BS) model in accurately pricing financial options in dynamic real-world markets. While the BS model offers a closed-form solution, it relies on restrictive assumptions that often fail in practice:

• Constant Volatility: It assumes volatility remains constant over the option's life, ignoring "volatility clustering" and the volatility smile/skew observed in markets.
• Continuous Price Paths: It assumes asset prices follow Geometric Brownian Motion (GBM), ignoring sudden, discontinuous price jumps caused by geopolitical events or market shocks (e.g., the 2008 financial crisis).
• European-Style Limitations: It is primarily designed for European options and struggles with path-dependent or American-style options.
• Static Parameters: It assumes constant risk-free rates and no dividends during the option's life.

The goal is to develop a robust, flexible computational framework that captures stochastic volatility, price jumps, and time-varying volatility to provide more accurate option valuations and hedging strategies.

2. Methodology

The authors developed a computational framework using Python and Monte Carlo (MC) simulations as the backbone. They implemented and compared four distinct modeling approaches, utilizing Yahoo Finance API for live and historical data from a diverse portfolio of stocks (Tesla, Meta, AMC, MARA, Shopify).

A. Baseline: Monte Carlo with Geometric Brownian Motion (GBM)

• Mechanism: Simulates thousands of asset price paths using the discretized GBM equation: $S_{t+\Delta t} = S_t \exp[(r - \frac{1}{2}\sigma^2)\Delta t + \sigma Z_t]$.
• Purpose: Serves as the control group to demonstrate the limitations of constant volatility assumptions.

B. Merton Jump-Diffusion Model

• Mechanism: Extends GBM by adding a jump component modeled via a Poisson process. The price dynamics include a jump term $(J-1)dN(t)$, where jump sizes $J$ are log-normally distributed.
• Optimization: To handle the non-linearity of jump parameters ($\lambda_j, \mu_j, \sigma_j$), the authors employed a Machine Learning-based optimizer (L-BFGS-B). This optimizer minimizes the Mean Squared Error (MSE) between model prices and actual market prices to calibrate the jump parameters empirically.

C. GARCH-Monte Carlo Model

• Mechanism: Uses the Generalized Autoregressive Conditional Heteroskedasticity (GARCH 1,1) model to forecast future volatility based on historical returns, capturing "volatility clustering."
• Integration: The forecasted conditional variance ($\sigma_t^2$) from GARCH is annualized and used as the input volatility for the Monte Carlo simulation.
• Goal: To predict option prices for future dates (1–3 days ahead) by accounting for time-dependent volatility.

D. Heston Stochastic Volatility Model

• Mechanism: Models volatility as a stochastic process correlated with the asset price. It uses two coupled stochastic differential equations:
  1. Asset Price: $dS(t) = S(t)(\mu dt + \sqrt{v(t)} dW_1(t))$
  2. Variance: $dv(t) = \kappa(\theta - v(t))dt + \sigma_v\sqrt{v(t)} dW_2(t)$
  • Where $W_1$ and $W_2$ are correlated Brownian motions.
• Optimization: Similar to the Merton model, the Heston parameters ($\kappa, \theta, \sigma_v$) are optimized using L-BFGS-B to minimize the difference between the model's predicted variance and historical realized variance.

3. Key Contributions

1. Unified Computational Framework: Successfully integrated four distinct pricing methodologies (Standard MC, Merton, GARCH, Heston) into a single Python-based framework.
2. Machine Learning Integration: Introduced L-BFGS-B optimization to empirically calibrate complex parameters for both the Merton (jump) and Heston (stochastic volatility) models, moving beyond theoretical parameter guessing.
3. Empirical Validation: Conducted extensive experiments in November 2024 using live market data across five stocks with varying volatility profiles (e.g., high-volatility crypto-miner MARA vs. stable tech giants).
4. Comparative Analysis: Provided a granular comparison of model performance across different strike prices (In-the-Money, At-the-Money, Out-of-the-Money) and market conditions.

4. Results

The study compared estimated option prices against actual market prices for Call options.

• Black-Scholes (GBM) vs. Reality: The standard MC model showed significant deviations, particularly for high-volatility assets (MARA) and deep Out-of-the-Money options. It tended to underprice options in volatile markets because it ignored jumps and volatility clustering.
• Merton Jump-Diffusion:
  • Performance: Improved accuracy for assets prone to sudden price shocks (e.g., MARA, AMC).
  • Observation: The model successfully captured the "fat tails" of the distribution. However, it slightly overestimated prices for high strike prices, likely due to limited historical data preventing full capture of stochastic volatility nuances.
• GARCH Model:
  • Performance: Successfully predicted future option prices (Nov 19–21) by modeling volatility clustering.
  • Observation: While trends matched market movements, discrepancies occurred at extreme strike prices due to liquidity issues and the model's inability to predict sudden sentiment shifts not present in historical data.
• Heston Model:
  • Performance: Demonstrated the highest overall accuracy, particularly for Tesla (TSLA) and AMC.
  • Observation: The Heston model's ability to model stochastic volatility and the volatility skew allowed it to track market prices more closely than the GBM baseline. The mean error between Heston estimates and market prices was consistently lower than the GBM model.
  • Deviation: Minor deviations were observed at specific strikes (e.g., $195 for TSLA), attributed to temporary market anomalies like low liquidity or large trade skewing.

5. Significance and Conclusion

• Bridging Theory and Practice: The research effectively bridges the gap between theoretical assumptions (constant volatility) and real-world market complexities (jumps, stochastic volatility).
• Practical Application: The framework provides financial managers and traders with superior tools for:
  • Accurate Valuation: Better pricing for complex derivatives and exotic options.
  • Risk Management: More robust hedging strategies that account for sudden market shocks and volatility clustering.
• Future Directions: The authors suggest extending this framework by utilizing Neural Networks (specifically LSTMs) for deeper parameter optimization and exploring Quantum Walks for simulating price paths, potentially offering computational advantages in high-dimensional financial modeling.

In conclusion, the paper demonstrates that while the Black-Scholes model remains a useful baseline, integrating stochastic volatility (Heston), jump diffusion (Merton), and time-varying volatility (GARCH) via Monte Carlo simulation significantly enhances pricing accuracy and market adaptability.