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The standard Black-Scholes-Merton model is valuable in both theory and practice. However, in certain situations, more advanced models are preferable. In this post, I explore stochastic volatility models.
 

Stock and Volatility Simulation: A Comparative Study of Stochastic Models

Stochastic volatility models, unlike constant volatility models, which assume a fixed level of volatility, allow volatility to change. By incorporating factors like mean reversion and volatility of volatility, stochastic volatility models offer a robust framework for pricing derivatives, managing risks, and improving investment strategies.

Reference [1] investigates several stochastic models for simulating stock and volatility paths that can be used in stress testing and scenario analysis. It also proposes a method for evaluating these stochastic models. The models studied include

-Geometric Brownian Motion (GBM),

-Generalized Autoregressive Conditional Heteroskedasticity (GARCH),

-Heston stochastic volatility,

-Stochastic Volatility with Jumps (SVJD), and a novel

-Multi-Scale Volatility with Jumps (MSVJ).
 

Findings

-The paper compares several stochastic models for simulating leveraged ETF (LETF) price paths, using TQQQ as the case study.

-The MSVJ model captures both fast and slow volatility components and demonstrates superior performance in modeling volatility dynamics and price range estimation.

-The evaluation framework tests price and volatility characteristics against actual TQQQ data under different market conditions, including the COVID-19 crash and the 2022 drawdown.

-GBM and Heston models are most effective in simulating market crashes, as they reproduce historical drawdowns and capture tail risk well.

-The MSVJ model is the most suitable for option pricing because it provides the best fit for both price and volatility, as measured by its highest WMCR.

-The SVJD model performs best in generating realistic price and volatility paths, as it incorporates both stochastic volatility and jump processes.

-SVJD’s realism makes it useful for portfolio managers in backtesting trading strategies and assessing portfolio risk across different market conditions.

In short, each model has distinct strengths, so the optimal choice depends on whether the goal is risk management, option pricing, or portfolio simulation.

Reference

[1] Kartikay Goyle, Comparative analysis of stochastic models for simulating leveraged ETF price paths, Journal of Mathematics and Modeling in Finance (JMMF) Vol. 5, No. 1, Winter & Spring 2025
 

Modeling Short-term Implied Volatilities in Heston Model

Despite their advantages, stochastic volatility models have difficulty in accurately characterizing both the flatness of long-term implied volatility (IV) curves and the steep curvature of short-term ones. Reference [2] addresses this issue by introducing a term-structure-based correction to the volatility of volatility (vol-vol) term in the classical Heston stochastic volatility model.
 

Findings

-Existing financial models struggle to capture implied volatility (IV) shapes across all option maturities simultaneously. This paper introduces a term-structure-based correction to the volatility of volatility (vol-vol) term in the classical Heston stochastic volatility model.

-The correction is modeled as an exponential increase function of the option expiry.

-An approximate formula for IV is derived using the perturbation method and applied to Shanghai Stock Exchange 50 ETF options.

-Numerical and empirical results show that the correction significantly improves the Heston model’s ability to capture short-term IVs.

-The corrected model enhances both IV forecasting and option quoting performance compared to the classical Heston model.

-While demonstrated on the Heston model, the method can be extended to other stochastic volatility models.

-Future research could include embedding strike into the correction function to better capture the entire implied volatility surface.

In brief, both short- and long-term IVs are accurately modeled in the new Heston variant.

This paper improves the existing Heston model. Thus, it helps portfolio managers and risk managers to better manage the risks of investment portfolios.

Reference

[2] Youfa Sun, Yishan Gong, Xinyuan Wang & Caiyan Liu, A novel term-structure-based Heston model for implied volatility surface, International Journal of Computer Mathematics, 1–24.
 

Closing Thoughts

Both studies advance volatility modeling in financial markets. The first highlights how different stochastic models, including a novel multi-scale volatility with jumps framework, can better simulate leveraged ETF dynamics under varying conditions, with specific strengths depending on the application. The second shows that enhancing the Heston model with a term-structure correction improves the fit of implied volatility surfaces across maturities, especially for short-dated options. Together, these findings underscore the importance of refining volatility models to capture market complexity and improve applications in risk management, option pricing, and forecasting.

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