Making Option Pricing Models More Practical

From Option Pricing to Portfolio Risk Management

Author

Nam Nguyen
July 26, 2026 • Estimated Reading Time: 5 minutes

The Black-Scholes-Merton model is one of the cornerstones of modern quantitative finance. Despite its elegance and widespread use, its simplifying assumptions limit its ability to capture many features of real financial markets. As a result, researchers continue to extend the model to make it more realistic and applicable in practice.

In this edition, we discuss two such extensions. The first incorporates a stochastic volatility model and intraday momentum into the option pricing framework. The second applies stochastic volatility models under the real-world measure to portfolio construction and volatility targeting, highlighting their practical use in risk management beyond derivative pricing.

In this issue:

Latest Posts

  • Algorithmic Trading, HFT, and Market Stability (10 min)

  • The Market Impact of Retail Options Trading (10 min)

  • Why System Validation Matters More Than Ever (10 min)

  • Does Regression Still Work in Modern Markets? (12 min)

  • Volatility Derivatives and VIX Market Dynamics (10 min)

Incorporating Momentum into Option Pricing Models

The Black–Scholes–Merton (BSM) model is a cornerstone of derivative pricing; however, it is not without limitations, and researchers continue to extend it. Reference [1] proposes an extension by incorporating intraday momentum into the BSM framework. This is achieved by introducing a drift term that represents intraday momentum, measured using a simple moving average of returns.

The model also adopts a modified Heston-type structure in which volatility follows a mean-reverting square-root process, allowing it to capture volatility clustering and remain consistent with empirical features such as volatility smiles. The momentum-driven drift adjustment influences the expected price path, while the stochastic volatility process models uncertainty around that path.

Findings

  • The study extends the BSM option pricing framework by incorporating intraday momentum into the drift term of a stochastic volatility-modified model.

  • It models time-varying volatility using a Heston-type stochastic volatility model and derives the momentum term from recent relative price changes.

  • The study analyzes the impact of intraday momentum on stock prices, volatility, and option valuations, with particular attention to high-momentum scenarios.

  • Numerical simulations show that positive momentum increases option valuations, while negative momentum decreases them.

  • The study finds that the proposed model converges to the classical Black-Scholes model under low-volatility or low-momentum conditions.

  • It concludes that incorporating momentum provides a theoretical framework for evaluating momentum-driven effects in derivative pricing and establishes quantitative metrics for empirical testing.

In short, the paper introduces a momentum term based on recent price changes to dynamically adjust the drift, capturing short-term intraday effects. Numerical results show that strong positive or negative momentum leads to substantial deviations from standard BSM prices, indicating that momentum is an important factor in option pricing.

This represents an interesting and potentially useful extension of the BSM model for traders and risk managers. However, as noted by the authors, the findings are based on simulated results rather than empirical data, and it would be valuable to see the model tested on real market data.

Reference

[1] Hossain, M.S., Yuan, X. & Sultan, S. Momentum-Driven Option Pricing: Integrating Intraday Trends into Financial Derivative Models. Comput Econ (2025).

Use of the Real-World Measure in Portfolio Management

In the realm of finance, the risk-neutral measure takes precedence in pricing financial derivatives. However, the real-world measure remains valuable and indispensable across various domains. It plays an important role in risk management and asset-liability applications, facilitating comprehensive evaluation and mitigation of risks.

Real-world measures are useful for simulation-based analyses of trading and investment strategies, offering insights into the practical implications of decisions in complex market environments. Reference [2] undertakes the calibration of stochastic volatility models as a means to estimate the real-world measure.

Employing the efficient method of moments (EMM), the authors perform calibration on the Heston and Bates SVJ models. Subsequently, the calibrated models are used to explore and analyze the risk and returns associated with volatility-target strategies.

Findings

  • The study shows how a real-world stochastic volatility model can be applied to test a simple volatility targeting strategy.

  • The results suggest that both stochastic volatility and jumps are required to characterize equity returns.

  • The results indicate that volatility targeting reduces the likelihood of extreme returns and lowers the volatility of volatility.

  • The study finds that portfolio risk and return both increase as the volatility target increases.

  • The 10% volatility target produces the lowest risk, measured by both the mean of volatility and the volatility of volatility, but also the lowest return.

  • An equity-only strategy produces the highest risk and the highest expected return.

  • The study states that volatility targeting provides an effective way to manage portfolio downside risk while limiting upside potential.

This article serves to exemplify the practical utility of the real-world measure by demonstrating its application in assessing investment strategies. Specifically, the study underscores the effectiveness of volatility targeting as a strategic approach that empowers investors to effectively manage and mitigate the downside risk inherent in portfolio management.

Reference

[2] Alexis Levendis and Eben Mare, On the calibration of stochastic volatility models to estimate the real-world measure used in option pricing, Orion, Volume 39(1), pp. 65 – 91

Closing Thoughts

Both papers highlight the practical importance of stochastic volatility models in real-world applications. While the first study extends the classical Black-Scholes framework by incorporating momentum into a Heston-type stochastic volatility model, the second demonstrates how stochastic volatility models calibrated under the real-world measure can be used to implement a practical volatility targeting strategy. Together, they illustrate how stochastic volatility models continue to evolve beyond theoretical option pricing and provide useful tools for derivative valuation and portfolio risk management.

Additional Reading

For further discussion on options pricing models, refer to the previous issues:

Educational Video

Computational Finance Lecture: Stochastic Volatility Models

The lecture explains why the Black-Scholes model is unable to reproduce the observed implied volatility smile and surface, making it inadequate for pricing many path-dependent and exotic derivatives. While jump-diffusion models improve the fit, the speaker notes that jumps are difficult to hedge in practice. Instead, the lecture focuses on stochastic volatility models, particularly the Heston model, in which volatility itself evolves randomly over time. By introducing a second stochastic process for variance, the model captures both the term structure and much of the shape of the implied volatility surface, making it one of the most widely used models in the derivatives industry.

The lecture also emphasizes the practical implementation of stochastic volatility models. It discusses model calibration to liquid market instruments such as vanilla options, the role of the risk-neutral measure for option pricing versus the real-world measure for historical analysis, and the importance of understanding how model parameters affect the volatility smile and skew for effective risk management.

Volatility Weekly Recap

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

Markets were volatile this week as optimism early in the week gave way to a broad selloff. Escalating tensions in the Middle East pushed oil prices higher and weighed on sentiment, while disappointing earnings from several large technology companies triggered a sharp decline late in the week. Energy, industrials, and financials outperformed, while technology stocks lagged.

Brent crude briefly traded above $100 per barrel before retreating on renewed ceasefire hopes. Gold was supported by safe-haven demand but capped by rising rate expectations, while Bitcoin initially rallied to a five-week high before giving back its gains as inflation concerns and broader market weakness returned.

On the volatility front, although spot volatility increased modestly, both the spot VIX and the VIX futures term structure remained in contango. The roll yield, however, reverted back toward zero. More interestingly, the longer-term trend in the roll yield (dashed black line in the figure below) has changed noticeably. After trending downward from the COVID period through mid-2025, it recovered somewhat before flattening, and has recently resumed a downward trend. Is this signaling a structural change?

Around the Quantosphere

  • How AI Is Affecting Careers in Electronic Trading and High-Frequency Trading (efinancialcareers)

  • Goldman Says Hedge Funds Sell US Tech Stocks at Record Pace (finance.yahoo)

  • Extreme Market Dispersion Prompts Hedge Funds to Position for Volatility Reversal (hedgeweek)

  • Vanishing CLO Profits Are Sparking Infighting (financialpost)

  • Putting AI to Work in Energy Trading (bcg)

  • A Day in the Life of a Trader at Citadel Securities (efinancialcareers)

  • Diversification Is the Only Free Lunch for Your Portfolio (afr)

  • China Hints at Regulations on Quants' AI Usage in State Paper (bloomberg)

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.

We are not responsible for any outcomes arising from the use of the content and codes provided in the outbound links. By continuing to read this newsletter, you acknowledge and agree to this disclaimer.