Stochastic Volatility Models for Capturing ETF Dynamics and Option Term Structures

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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 issue, I explore stochastic volatility models.

In this issue:

• Latest Posts

• Stock and Volatility Simulation: A Comparative Stu …

  • Findings

• Modeling Short-term Implied Volatilities in Heston …

  • Findings

• Closing Thoughts

• Educational Video

  • Heston Model Calibration in the "Real" World with …

• Volatility Weekly Recap

• Around the Quantosphere

Latest Posts

• Cross-Sectional Momentum: Results from Commodities and Equities (11 min)

• Predictive Information of Options Volume in Equity Markets (11 min)

• The Impact of Market Regimes on Stop Loss Performance (12 min)

• The Limits of Out-of-Sample Testing (12 min)

• Sentiment as Signal: Forecasting with Alternative Data and Generative AI (12 min)

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.

Educational Video

Heston Model Calibration in the "Real" World with Python - S&P500 Index Options

This video shows how to price options with the Heston stochastic-volatility model under the risk-neutral measure and calibrate it to market data. The presenter derives the semi-analytical formula via the model’s characteristic function, notes common sign pitfalls when collapsing the two integrals to one, and implements pricing in Python using numerical integration. He then pulls SPX option quotes and fits a risk-free curve, defines an objective that minimizes weighted squared price errors, and estimates Heston parameters (v₀, κ, θ, σ, ρ, λ). The calibrated prices are compared to market surfaces for validation, and the workflow is positioned as a base for extending to American options via finite-difference methods.

Volatility Weekly Recap

The figure below shows the term structures for the VIX futures (in colour) and the spot VIX (in grey).

Stocks started the week relatively quietly. On Friday, a new Core PCE report suggested inflation remains a concern, triggering bearish sentiment into the long weekend. Overall for the week, the S&P 500 lost 0.10% and the Nasdaq lost 0.19%.

Oil prices were relatively quiet, slipping slightly on Friday. Gold posted its strongest week since June. Crypto saw a small rebound but fell again on Friday alongside equities, with Bitcoin briefly dipping below $108,000.

On the volatility front, the VIX rose slightly, closing around 15.36. Both spot and VIX futures curves remain in contango, and the roll yield continued to be elevated. We note reports in the financial media (see the Around Quantosphere section below) that hedge funds are capitalizing on the elevated roll yield by shorting VIX futures.

Around the Quantosphere

• Hedge funds are shorting the VIX at rates not seen since 2022 (bloomberg)

• Hedge fund flows hit decade highs (opalesque)

• A Niche Arbitrage Trade Is Gaining Traction Among Hedge Funds (bloomberg)

• Future of quant finance: JPMorgan’s $500M Numerai bet validates AI-driven alpha (ainvest)

• Ex–Bank of America quant finds agentic LLMs don’t like to trade (efinancialcareers)

• Becoming a quant trader: Jason’s career journey (optiver)

• Electronic trading firm offers $500K to retain elite engineers eyeing exits (efinancialcareers)

• Hedge funds’ insurance binge threatens catastrophe cover, warns Munich Re (ft)

• Hedge Fund Manager Ernie Chan: Use GenAI to Manage Risk, Not Predict Return (youtube)

Disclaimer

This newsletter is not investment advice. It is provided solely for entertainment and educational purposes. Always consult a financial professional before making any investment decisions.

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