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 post, 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.
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.
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