Reflexivity is the process through which market participants’ actions influence the very market variables on which their decisions are based, creating feedback effects. In this post, we discuss how reflexivity manifests itself in options hedging and how it can be explicitly incorporated into option pricing models.
Feedback Effect in the Foreign Exchange Market
The Black–Scholes-Merton (BSM) model is the most frequently used option pricing model in the financial industry. However, it makes use of assumptions that are not always realistic. A crucial assumption of the BSM model is a frictionless and elastic market. In other words, it assumes that there are no barriers to buying or selling securities and that investors can instantly buy or sell an asset at the prevailing market price.
The markets are, however, not frictionless, and the buying and selling of securities can have an impact on the markets. This is often called a feedback loop. Reference [1] examined how delta hedging in the foreign exchange market can have an impact on market volatility.
Findings
-The model considers an aggregate option market maker (OMM) and an aggregate option market taker (OMT), whose exposures represent outstanding option positions across the market.
-Differences in the hedge ratios of market makers and market takers generate net delta-hedging activity in the spot market, creating market friction and affecting volatility.
-The theoretical model predicts that negative market-maker gamma exposure increases spot volatility, while positive gamma exposure decreases it.
-The magnitude of the volatility effect depends on the amount of net delta hedging executed in the spot market.
-Using trade repository data, the authors reconstruct aggregate market-maker gamma exposure and find that market makers have negative gamma exposure during the sample period.
-The empirical results strongly support the theoretical model, with market-maker gamma exposure being highly significant in explaining spot-market volatility.
-A negative gamma exposure of approximately −$1 trillion is estimated to increase volatility by 0.7% in EUR/USD and 0.9% in USD/JPY.
Overall, the results indicate that market-maker delta hedging can create a feedback effect in which short-gamma positioning amplifies volatility in the underlying foreign exchange market.
The findings are consistent with our experience, as it’s widely acknowledged that a feedback effect (or reflexivity) exists in the financial markets. This article provided an empirical proof. It would be interesting to see a similar study on leveraged Exchange-Traded Funds.
Reference
[1] B. Anderegga, F.Ulmann, D. Sornette, The impact of option hedging on the spot market volatility, Journal of International Money and Finance, 124, 2022, 102627
Incorporating Reflexivity into the Black–Scholes-Merton Framework
While the first paper studies reflexivity and its impact on financial market dynamics, Reference [2] explicitly incorporates reflexivity into an option pricing model, extending the Black-Scholes-Merton framework to account for feedback effects generated by model-based trading.
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.
Findings
-The framework challenges the assumption that financial-market dynamics are exogenous, arguing that widespread use of mathematical trading models can affect the same variables used as inputs to those models.
-The paper reformulates the stochastic differential equation for asset prices by modifying the drift, diffusion, and volatility terms to incorporate these feedback effects.
-The framework challenges several assumptions of classical financial theory, including risk-neutral drift equal to the risk-free rate, exogenous volatility, and market completeness.
-The author argues that collective use of mathematical models can create feedback mechanisms that reshape asset-price and option-pricing dynamics.
-The framework provides a formal mathematical structure linking George Soros’s concept of market reflexivity with option-pricing theory.
-The author suggests that reflexivity can amplify systemic risk and market fragility, potentially helping explain extreme market events.
-The paper highlights implications for volatility estimation, systemic risk, and the limitations of financial models that assume their inputs are unaffected by widespread model-based trading.
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.
Reference
[2] Atharva Singh Rathore, Reflexivity in Financial Markets: A Theoretical Framework Extending the Black–Scholes Model, 2025
Closing Thoughts
Together, these studies highlight reflexivity as an important feature of modern financial markets. The first demonstrates how market-maker delta hedging can feed back into the underlying market, with short-gamma exposure amplifying spot volatility. The second approaches the same phenomenon theoretically by explicitly incorporating reflexivity into option pricing, allowing model-based trading itself to influence asset-price dynamics and volatility.
Both studies challenge the conventional assumption that financial models operate on market variables that are independent of their own use.
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