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Equivalent Behavioural Martingale Measure

This paper integrates Samuelson-Merton's utility-based pricing into a continuous-time equivalent martingale measure framework by deriving a compensated stochastic-exponential representation of the behavioral density, which empirically decomposes option-implied densities into risk attitudes and demonstrates superior out-of-sample performance over SVI in a panel of liquid options.

Original authors: G. Charles-Cadogan

Published 2026-10-08✓ Author reviewed ⓘ

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: G. Charles-Cadogan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of finance, prices are not just numbers; they are signals. When an investor buys an option, a contract that gives them the right to buy or sell a stock at a future date, the price they pay reveals what the market collectively believes about the future. Specifically, it reveals how much the market fears a crash or hopes for a boom. For decades, economists have used a standard mathematical framework to decode these prices, treating them as if the market were a single, rational machine. However, real markets are made of people, and people do not always act like machines. They feel fear, they get greedy, and they weigh risks differently depending on whether they are looking at a potential loss or a potential gain. The question has long been: can we take the messy, emotional reality of human behavior and fit it into the strict, logical rules that govern financial pricing without breaking the system?

This is the territory explored in a new working paper by G. Charles-Cadogan, which seeks to bridge the gap between human psychology and mathematical finance. The paper focuses on a specific pattern seen in option markets: the "smile" or "smirk." If you were to plot the prices of options against how far they are from the current stock price, you would often see a curved line rather than a flat one. This curve suggests that the market prices extreme events—both crashes and massive rallies—differently than standard theory predicts. The author asks whether this curve is simply a mathematical artifact or if it is a direct map of human risk attitudes, specifically how people feel about risk when they are losing money versus when they are winning. The goal is to show that these emotional reactions can be translated into a rigorous mathematical language that respects the fundamental rule of finance: that there is no such thing as a free lunch, or in technical terms, no arbitrage.

The paper introduces a new way of looking at these market prices, calling it the "Equivalent Behavioral Martingale Measure." To understand this, imagine that the market assigns a different weight to every possible future outcome. A standard model might treat all outcomes equally, but a behavioral model acknowledges that investors care more about a bad outcome than a good one of the same size. The author demonstrates that you can turn this weighted, emotional view of the future into a valid mathematical tool, but only if you apply a specific correction. Think of it like adjusting a scale: if you want to weigh a feather and a rock on the same scale, you cannot just put them on; you must first account for the difference in how they sit on the pan. Similarly, the paper shows that you cannot simply use raw human feelings about risk to price options. You must first remove a specific mathematical "drift" caused by the combination of how risk-averse people are and how much the market is moving. Once this drift is compensated for, the resulting formula becomes a valid, stable way to price assets that still captures the essence of human behavior.

The core finding is that this corrected formula works. The author proves that if you take the marginal utility of wealth—which is a measure of how much an extra dollar means to an investor at a specific moment—and adjust it for the risks of the market, it transforms into a valid probability measure. This means it can be used to calculate prices without creating contradictions or impossible profits. The paper identifies the exact mathematical term that must be removed to make this work, linking it directly to the Arrow-Pratt measure of risk aversion, a classic concept in economics that quantifies how much someone dislikes risk. The author shows that this risk aversion, when multiplied by the volatility of the market, creates the necessary adjustment. Without this adjustment, the model would fail the basic tests of financial logic.

Beyond the theory, the paper tests this idea against real-world data. The author gathered a large set of option prices from major US stocks and exchange-traded funds, including giants like Apple, Microsoft, and Tesla, as well as broad market indices. They filtered out noisy or unreliable data and then tried to fit their new behavioral model to the observed prices. They compared their results against other standard models used by professionals, such as the SVI model, which is designed specifically to fit the shape of option prices. The results were mixed but revealing. When the new behavioral model was given the same strict rules to prevent arbitrage as the standard models, it performed just as well, and in some pooled tests, even slightly better, at predicting future prices. However, the paper is careful to note that this is not a replacement for the complex tools currently used by traders to build entire trading surfaces. Instead, it serves as a diagnostic tool.

The most significant insight from the data comes from breaking down the "smile" into its components. The author found that the curve is not just a single shape but a combination of two distinct forces. One part is symmetric, representing a general fear of risk that affects all outcomes equally, creating a standard "smile." The other part is asymmetric, tilting the curve to one side, which the author interprets as a "smirk" of speculation. This tilt suggests that in certain market conditions, investors are not just afraid of losing; they are actively seeking out the possibility of a massive, lottery-like gain. The data showed that this speculative tilt was a strong and consistent feature in the market, often more pronounced than the general fear of risk. In fact, the strength of this speculative tilt appeared to have a small but measurable ability to predict how much the market would move in the near future.

The paper also addresses what happens when different types of investors are mixed together. In a real market, some people are very cautious while others are very bold. The author shows that you can combine the pricing views of these different groups into a single, valid model, provided you weight them correctly. If you simply add their views together without adjusting for their relative size or importance, the math breaks down. But if you normalize the weights, treating them as shares of the total market, the combined model remains valid. This provides a way to understand how a market with diverse personalities can still arrive at a single, coherent price for an option.

Ultimately, the paper does not claim to have solved the mystery of why markets move or to have found a new way to make money. Instead, it offers a clearer lens through which to view the prices that already exist. It confirms that the strange shapes of option prices are not just random noise or mathematical errors, but are deeply rooted in how humans perceive risk and reward. By showing exactly how to translate these human feelings into a mathematically sound framework, the author provides a way to separate the signal of genuine behavioral preference from the noise of market friction. The work suggests that while the market is complex, the underlying drivers of its pricing are consistent with the fundamental laws of probability, provided we take the time to correct for the human tendency to overreact to risk. This allows economists and traders to look at a price and understand not just what the market thinks will happen, but how the market feels about the possibility of it happening.

Technical Summary: Equivalent Behavioral Martingale Measure

Problem Statement
The paper addresses a foundational gap in the pricing of options and the interpretation of volatility smiles and smirks. While the Samuelson–Merton "util-prob" construction (where a pricing kernel is formed by marginal utility weighting physical probabilities) is a standard theoretical concept, it is not automatically an equivalent martingale measure (EMM). The raw marginal-utility density often fails to satisfy the strict mathematical conditions required for an EMM, specifically the requirement that the density process be a true martingale (i.e., having an expectation of one and no drift).

Consequently, interpreting observed option-implied volatility patterns (smiles and smirks) as direct reflections of investor risk aversion is mathematically precarious without a mechanism to ensure the resulting density is admissible. The paper seeks to formalize the conditions under which the "util-prob" density can be rigorously represented as a compensated stochastic exponential, thereby validating a "behavioral" interpretation of option prices within a no-arbitrage framework.

Methodology
The paper constructs a continuous-time framework using a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P})$ .

Theoretical Construction:
- Compensated Stochastic Exponential: The core mathematical contribution is the derivation of a compensator $\Gamma_t$ that transforms the raw marginal utility $U'(Y_t)$ into a true martingale density $L_t$ . Using Itô's formula, the paper shows that the drift generated by risk aversion ( $A_t$ ) and prudence ( $a_t = U'''/U'$ ) must be removed.
- The Kernel: The resulting behavioral density is expressed as a Doléans–Dade stochastic exponential:
  $L_t = \mathcal{E}\left(-\int_0^t A_s \sigma_s dW_s\right)$
  where $A_t$ is the Arrow–Pratt absolute risk-aversion process and $\sigma_t$ is the state volatility. The Brownian risk price is identified as $\lambda_t = A_t \sigma_t$ .
- Admissibility Conditions: The paper establishes that for $L_t$ to define an Equivalent Behavioral Martingale Measure (EBMM), it must satisfy positivity, integrability, normalization, and a sufficient martingale condition (e.g., Novikov's condition).
Structural Representation of Smiles and Smirks:
- The paper proposes a "τ-welded" hyperbolic basis to decompose the risk-attitude process $A_t$ across moneyness ( $m = \log(K/S)$ ):
  $A_\theta(m, \tau) = \beta_0(\tau) + A_{\text{averse}}(\tau)[\cosh(m) - 1] + A_{\text{seek}}(\tau)\sinh(m)$
- Here, the even component $[\cosh(m)-1]$ captures symmetric curvature (smiles), while the odd component $\sinh(m)$ captures asymmetric skew (smirks). $A_{\text{seek}}$ is allowed to be negative, representing local risk-seeking or speculative behavior.
Empirical Calibration:
- Data: A panel of liquid US index ETFs (SPY, QQQ, IWM) and single-name options (AAPL, MSFT, NVDA, AMZN, TSLA) from ThetaData Terminal.
- Procedure:
  1. Recover the risk-neutral density ( $q_\tau$ ) from option prices using an arbitrage-penalized SVI surface.
  2. Estimate the physical density ( $p_\tau$ ) from historical returns.
  3. Compute the empirical pricing-kernel ratio $\hat{L}_\tau = q_\tau / p_\tau$ .
  4. Fit the EBMM parameters ( $A_{\text{averse}}, A_{\text{seek}}$ ) to minimize the distance between the model-implied density and the recovered ratio, subject to constraints ( $A_{\text{averse}} > 0, A_{\text{seek}} < 0$ ).
- Diagnostics: The study employs a finite-sample Novikov consistency screen to check if the fitted parameters produce explosive exponential terms, which would violate martingale admissibility.

Key Results

Mathematical Equivalence: The paper proves that the Samuelson–Merton "util-prob" density becomes an equivalent martingale measure only after compensating for the drift terms arising from risk aversion and prudence. The Brownian risk price is explicitly linked to the product of Arrow–Pratt risk attitude and state volatility.
Cox–Ross Identity: Under the EBMM framework, the paper re-derives the Cox–Ross risk-premium identity, showing that the ratio of option-to-underlying risk premia equals the ratio of their volatilities, with the common pricing kernel canceling out.
Empirical Fit:
- In a pooled holdout exercise, the arbitrage-penalized EBMM achieves a lower squared-error loss than both the raw EBMM and the arbitrage-penalized SVI benchmark.
- However, at the slice level (individual symbol/maturity), results are mixed, with SVI often outperforming EBMM in specific slices.
- Risk-Seeking Component: The empirical estimates show that the signed risk-seeking component ( $A_{\text{seek}}$ ) is consistently negative and statistically significant, whereas the symmetric risk-aversion component ( $A_{\text{averse}}$ ) is often near the numerical lower bound (zero), suggesting the data in this sample primarily identifies speculative tilts rather than symmetric curvature.
Predictive Content: A preliminary predictive regression indicates that the signed risk-seeking component has some ability to forecast next-horizon realized variance, though the risk-aversion component does not.

Significance and Claims
The paper explicitly frames its contribution as modest and methodological rather than a replacement for existing volatility surface estimation techniques.

Admissibility Filter: The primary value of the EBMM is not to produce a "better" volatility surface than SVI or SABR, but to serve as a density-ratio admissibility test. It provides a structural filter to determine whether an observed option-implied smile or smirk can be legitimately interpreted as a compensated marginal-utility density satisfying martingale conditions.
Interpretive Bridge: It bridges the gap between the older "util-prob" literature and modern state-price density recovery. It clarifies that "behavioral" in this context refers to state-dependent marginal utility weighting, not a violation of no-arbitrage or rank-dependent probability distortion.
Decomposition: The framework allows for the decomposition of volatility surfaces into signed Arrow–Pratt components, distinguishing between symmetric risk aversion (smiles) and asymmetric risk-seeking/speculative behavior (smirks).
Limitations: The author cautions that the empirical results are based on a small panel and that the symmetric risk-aversion factor is often unidentified in the data. They emphasize that the EBMM is an interpretive tool for candidate density ratios, not a production-grade estimator for volatility surfaces. The paper concludes that future work should test these admissibility restrictions on broader datasets (commodities, crypto) and compare them against standard parametrizations.

Technical Summary: Equivalent Behavioral Martingale Measure

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