The Black-Scholes-Merton model (BSM) is a mathematical framework used to value options. Developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s, it provides a formula for calculating the fair price of a European call or put option. The model assumes that markets are efficient, asset returns follow a lognormal distribution, volatility and interest rates are constant, and no arbitrage opportunities exist.
Several modifications have been made to address the assumptions of the BSM model. Reference [1] is the latest effort, aiming to incorporate reflexivity into the BSM framework. Reflexivity theory, originally developed by Soros, is formally incorporated in this version of the model. Specifically, the author proposes three modifications,
- Predictability of Brownian motion: Trader dependence on predictive models induces path dependence, breaking the martingale property of the Wiener process (Wt).
- Drift coefficient deviation from risk free rate: Large-scale institutional model use shifts the drift away from the risk-free rate by a reflexivity-induced term.
- Volatility endogeneity: The volatility at time t depends on lagged volatility and hedging intensity.
These modifications were implemented by introducing a variable α, which represents the proportion of traders who rely on mathematical models for investment. The remaining (1–α) do not use such models and instead depend on trends or other factors. As a result, the stochastic differential equation for the stock price is adjusted. The author pointed out,
This paper tries to advance the theory of option pricing by embedding reflexivity into the Black–Scholes model and formalizing how feedback mechanisms reshape asset price dynamics. By modifying the drift, diffusion, and volatility terms to account for the influence of widespread mathematical model-based trading, this framework challenges three cornerstones of classical financial theory: the equivalence of risk-neutral drift to the risk-free rate, the assumption of exogenous volatility, and the presumption of market completeness. The resulting dynamics provide a more realistic representation of modern financial markets, where feedback from collective model use shapes the very variables on which models depend.
The framework highlights how reflexivity amplifies systemic risk and fragility, offering a structural explanation for extreme events in financial markets.
In short, this paper presents a promising extension of the BSM model. The next step is to calibrate it with extensive real-world data and to develop and test strategies based on it.
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References
[1] Atharva Singh Rathore, Reflexivity in Financial Markets: A Theoretical Framework Extending the Black–Scholes Model, 2025
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