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Entropic Dynamics of Jump-Diffusion Option Pricing

This paper derives the Merton jump-diffusion model and its associated option pricing framework from an entropic inference approach that utilizes information constraints on log-price dynamics and no-arbitrage conditions, thereby unifying continuous and jump processes while recovering the Black-Scholes limit when jumps are absent.

Original authors: Mohammad Abedi

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammad Abedi

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 or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the path of a hiker walking through a forest.

Most traditional financial models start by guessing the rules of the forest. They say, "Let's assume the hiker walks in a smooth, wiggly line like a drunkard, and occasionally gets pushed by a sudden gust of wind." They build their entire house of cards on these assumptions.

This paper, however, takes a completely different approach. Instead of guessing the rules, it asks: "What is the most logical way to update our beliefs about the hiker's path if we only know a few specific facts?"

The authors use a method called Entropic Dynamics. Think of this as a "logic engine" that takes whatever information you have and spits out the most probable scenario, without adding any extra guesses. Here is how they apply this to the stock market:

1. The Right Ruler: Measuring Returns, Not Prices

First, the authors ask: "What are we actually measuring?"
If a stock goes from \ $10 to \$ 11, that's a 10% gain. If it goes from \ $100 to \$ 101, that's only a 1% gain. The market cares about the percentage (the return), not the dollar amount.

The Analogy: Imagine measuring a hiker's progress. If you measure in "steps," a step is a step. But if you measure in "miles," a step means something different depending on how big the hiker is.
The Result: The paper proves that to make the math fair and consistent, we must measure the stock in logarithms (a special kind of math scale). On this scale, multiplying prices becomes simply adding numbers. This is the only way to respect the market's symmetry.

2. The Two Lanes of Traffic

The paper argues that stock prices move in two distinct "lanes" or channels.

Lane A (The Smooth Path): This is the normal, everyday trading. Prices drift slowly and wiggle a bit. This is the "continuous" part.
Lane B (The Jump): Sometimes, big news hits (like an earnings report or a war). The price doesn't wiggle; it teleports to a new level. This is the "jump" part.

The Big Discovery: The authors show that because these two lanes carry different types of information (one is about smoothness, the other is about sudden shocks), they don't interfere with each other. They can be calculated separately and then combined.

The Result: When they run their "logic engine" with these two lanes, the smooth lane naturally becomes the famous Geometric Brownian Motion (the standard model used for decades), and the jump lane becomes the Merton Jump-Diffusion model.
Why it matters: They didn't assume these models were true. They derived them. It's like discovering gravity by observing how apples fall, rather than just saying "let's assume gravity exists."

3. The "Smile" of the Market

In the real world, stock options (bets on future prices) have a weird pattern called a "volatility smile." It means that people are willing to pay more for bets that protect against huge crashes than the standard models predict.

The Paper's Explanation: The standard models assume the hiker only walks smoothly. But the real hiker jumps! When you add the "Jump Lane" to the model, the math naturally creates this "smile." The market isn't broken; the old models were just missing the jump lane.

4. Pricing the Options: The "No-Stealing" Rule

Now, how do you price an option if the market has jumps?
In a smooth world, there is only one correct price. But in a world with jumps, the market is "incomplete"—there are infinitely many ways to price the option without breaking the rules.

The Rule: The only rule everyone agrees on is No Arbitrage (no "free lunch"). You can't set up a trade that costs nothing, has no risk, and guarantees a profit.
The Solution: The authors use their logic engine again. They ask: "What is the most logical way to adjust our probabilities so that the 'No Free Lunch' rule is satisfied?"
The Result: The engine selects a specific mathematical tool called the Esscher Transform. This isn't a rule they made up; it's the only logical answer that comes out of the machine when you feed it the "No Free Lunch" constraint.

The Bottom Line

The paper's main message is about information.

Old Way: "Let's assume the stock moves like X. If X is wrong, our model fails."
New Way: "Here is the information we have (prices move smoothly, but sometimes jump; no free lunches allowed). What is the most logical model that fits this?"

The authors show that if you feed the right information into this logic engine, you get the standard models (Black-Scholes) as a special case when jumps disappear, and the more complex models (Merton) when jumps are present. The "magic" isn't in the math; it's in the fact that the math is just a reflection of the information you put in.

In short: They didn't invent a new way for stocks to move. They built a machine that proves the way stocks actually move is the only logical outcome of the information the market provides.

Technical Summary: Entropic Dynamics of Jump-Diffusion Option Pricing

Problem Statement
Standard financial models, such as Geometric Brownian Motion (GBM) and the Black–Scholes framework, rely on postulating specific stochastic processes for stock prices. While these models successfully describe continuous price evolution, they fail to capture empirical realities: asset returns exhibit leptokurtosis, negative skewness, and heavy tails, while option markets display persistent volatility smiles and smirks. These features are attributed to discontinuous price jumps triggered by discrete information events (e.g., earnings, policy decisions). Furthermore, the presence of jumps renders markets incomplete, meaning the no-arbitrage condition alone is insufficient to uniquely determine a risk-neutral pricing measure. Existing literature typically assumes a jump-diffusion process (e.g., Merton's model) and a specific pricing measure (e.g., the Esscher transform) as axioms. This paper addresses the need to derive these dynamics and pricing measures from first principles using incomplete information, rather than assuming them.

Methodology: Entropic Inference
The paper employs Entropic Dynamics (ED), an inductive framework based on the method of Maximum Entropy (MaxEnt) and relative entropy. The core philosophy is that dynamics are not mechanical laws but inferential updates of probability distributions as new information arrives. The methodology proceeds in two distinct phases:

Derivation of Dynamics (The Stock Model):
- Variable Selection: The framework begins by identifying the correct dynamical variable. By imposing return symmetry (invariance of transition probabilities under uniform price rescaling), the paper derives that the logarithm of the price ( $\ln S$ ) is the unique variable that renders multiplicative rescalings additive.
- The Continuous Channel: The dynamics of $\ln S$ are constructed by maximizing relative entropy subject to two constraints: continuity (the next price is close to the current one, encoded via a mean-square displacement constraint) and directionality (a systematic drift, encoded via a mean log-return constraint). This yields a driftless Gaussian prior and a shifted Gaussian posterior, recovering GBM without assuming it.
- The Jump Channel: To account for discontinuities, the microstate is enlarged to include a jump count variable ( $n$ ) and jump size ( $Y$ ). The framework introduces three additional constraints: the expected arrival rate ( $\lambda$ ), and the first two moments of the jump size ( $\kappa, \nu$ ).
- Factorization: Crucially, because the constraints on the jump channel are disjoint from those on the continuous channel, the MaxEnt procedure forces the posterior to factorize. The continuous component remains GBM, while the jump component is derived as a Gaussian distribution for the jump size. The combination yields the Merton jump-diffusion process as a theorem, not an assumption.
- Forward Equation: The evolution of the log-price density is shown to obey the Kolmogorov–Feller equation, which generalizes the Fokker–Planck equation by including an integral term for jumps.
Derivation of Pricing (Option Valuation):
- Market Incompleteness: With jumps, the market is incomplete, and the equivalent martingale measure (EMM) is not unique. No-arbitrage alone cannot select a single measure.
- Selection Principle: The paper imposes no-arbitrage through the mean log-return constraint (replacing the subjective drift with the risk-free rate adjusted for convexity). It then maximizes the relative entropy of the pricing measure with respect to the physical jump-diffusion measure.
- The Esscher Transform: This inference procedure uniquely selects the Esscher transform as the pricing measure. The paper argues this is not an external imposition but a consequence of updating beliefs based on the mean log-return constraint.
- Structure Preservation: The log-return constraint ensures the pricing measure remains within the jump-diffusion family, merely shifting the drift, intensity, and jump size distribution.
- Pricing Equation: The resulting option premium satisfies Merton's partial integro-differential equation (PIDE). The risk-neutral distribution is a Poisson-weighted mixture of lognormals, which naturally generates the implied-volatility smile.

Key Contributions and Results

Derivation of Merton's Model: The paper derives the Merton jump-diffusion process and its associated Kolmogorov–Feller equation strictly from entropic inference principles, replacing the standard postulation of these models.
Derivation of the Esscher Transform: It demonstrates that the Esscher transform, widely used in jump-diffusion pricing, is the unique measure selected by maximizing relative entropy under a mean log-return no-arbitrage constraint.
Unification of Models: The framework shows that Black–Scholes (GBM) and Black–Scholes–Merton (Jump-Diffusion) are not competing theories but limiting cases of a single inferential machinery, distinguished only by the information (constraints) supplied to the system.
Volatility Smile: The model naturally produces the implied-volatility smile and smirk observed in equity markets as a direct consequence of the jump constraints, without requiring ad-hoc volatility functions.
Resolution of Non-Uniqueness: The paper reframes the non-uniqueness of the pricing measure in incomplete markets as the non-uniqueness of the informational constraint used to impose no-arbitrage. Choosing the mean log-return yields the Esscher measure; choosing the gross return would yield a different (canonical) measure.

Significance and Claims
The paper claims its central contribution is methodological. It posits that the distinction between different financial models lies not in the inference engine itself, but in the specific information (constraints) fed into it. By treating dynamics as "mechanics without a mechanism," the authors argue that the entropic framework provides a unified, principled foundation for financial modeling.

The work asserts that:

Dynamics are inferred, not assumed: Processes like GBM and Jump-Diffusion are the necessary outcomes of maximizing entropy under specific economic symmetries and information constraints.
Pricing is an inference problem: The selection of a risk-neutral measure is an inductive step driven by the choice of the observable used to enforce no-arbitrage.
Modularity: The framework allows for the seamless integration of new market features (like jumps) by simply enlarging the microstate and adding corresponding constraints, without discarding the underlying inferential logic.

The paper concludes that the "real difficulty" in modeling lies not in the updating mechanism (which is fixed by the MaxEnt principle) but in correctly identifying the relevant variables and the information genuinely pertinent to the problem at hand.