A CGMY-Hawkes Clock Model: Esscher Martingale Measures and Model-Preserving Minimal Entropy under Self-Exciting Jump Activity

In the world of finance, predicting how asset prices move is a constant struggle against the unpredictable. Traditional models often assume that price changes happen smoothly and independently, like raindrops falling at a steady rate. However, real markets behave differently. They are prone to sudden, violent bursts of activity where large price swings arrive in clusters, one after another, rather than in isolation. This phenomenon, known as jump clustering, creates a heavy-tailed distribution of returns, meaning extreme events happen far more often than standard theories predict. To capture this reality, researchers have long relied on two distinct mathematical tools. One tool, the CGMY process, is excellent at describing the size and frequency of these jumps, accounting for the heavy tails and the "fat" probability of extreme moves. The other tool, the Hawkes process, is designed to model self-excitation, where an event triggers a higher likelihood of another event happening soon after, creating the observed bursts of activity. For years, these two concepts existed in separate silos, and combining them into a single, workable model for pricing financial assets like options had remained a significant challenge.

A new study by Imen Ben Abdelwahed bridges this gap by introducing a unified framework called the CGMY–Hawkes clock model. The core idea is to treat time itself as a variable that speeds up and slows down based on market activity. Imagine a standard clock ticking away the passage of time; in this model, that clock is replaced by a "stochastic clock" that runs faster when the market is agitated and slower when it is calm. This clock is driven by the Hawkes process, which dictates the intensity of market activity. The actual price movements are then generated by the CGMY process, but they are evaluated not against the standard clock, but against this variable, self-exciting clock. The result is a model where the jumps in price naturally cluster together because the clock speeds up during periods of high intensity, while the statistical law governing the size of each individual jump remains consistent. This approach allows the model to capture the messy, clustered reality of financial markets while remaining mathematically tractable enough to be useful for pricing.

The researchers then tackled a critical question: how do we assign a fair price to financial contracts, such as European options, within this complex environment? In incomplete markets, where risks cannot be perfectly hedged, there is no single unique way to choose a pricing measure. The team focused on a specific method called the Esscher transform, which adjusts the probabilities of different outcomes to ensure the model is consistent with the absence of arbitrage opportunities. A major breakthrough in their work is the discovery that this adjustment can be made without destroying the model's structure. They proved that by shifting certain parameters that control the tail behavior of the jumps, they could change the pricing measure while leaving the underlying self-exciting nature of the Hawkes clock completely untouched. This means the "clustering" behavior of the market is preserved as a fundamental property of the market itself, rather than being an artifact of the pricing method.

To make the model work for pricing, the author introduced a risk premium that is proportional to the current intensity of the market. This specific choice led to a surprising simplification: the condition required to make the model mathematically consistent reduced to a single equation that does not depend on the specific parameters of the Hawkes process. In other words, the adjustment needed to price the asset correctly is determined solely by the characteristics of the jumps themselves, not by how the market excitement is generated. This finding allows for a unique solution within a specific class of measures, which the author identifies as the model-preserving minimal entropy martingale measure. This measure is special because it introduces the least amount of "surprise" or deviation from the original market model while still satisfying the necessary pricing rules. The density of this measure includes a specific stochastic term that accounts for the fluctuations of the integrated intensity, capturing the randomness of the clock itself.

The study also explored what happens when the researchers allow for more flexibility by using a two-parameter adjustment instead of a single one. They found that the most efficient way to minimize entropy in this broader setting involves adjusting the positive and negative tails of the jump distribution differently, rather than treating them symmetrically. This two-parameter solution yields a slightly lower entropy than the single-parameter version, indicating a more precise fit, though it is not located on the diagonal where the parameters are equal. When the author applied these findings to calculate the prices of standard European call options, they compared their exact results against a common approximation known as the mean-field approach. The mean-field method simplifies the problem by assuming the market intensity is constant and equal to its average value. The researchers demonstrated that this simplification is insufficient. Their exact calculations showed that option prices depend significantly on the specific parameters that govern the excitation of the market, such as how strongly one event triggers the next.

In numerical tests using benchmark parameters, the difference between the exact model and the mean-field approximation was measurable and meaningful. For a standard option with a one-year maturity, the exact price decreased as the branching ratio—the measure of how much one event excites the next—increased. For instance, with a specific set of parameters, increasing the excitation level from zero to a very high level caused the option price to drop by approximately 2.6 percent compared to the mean-field prediction. This occurs because higher excitation spreads the distribution of the time clock more widely around its average, and the pricing formula reacts to this spread in a non-linear way. The mean-field approximation, by ignoring these fluctuations, fails to capture this effect. The author concludes that while their model preserves the essential structure of the market and provides a unique, efficient pricing measure, the details of how market excitement evolves matter deeply for valuation. They suggest that future work could extend this framework to more complex scenarios, such as models where the price jumps themselves trigger further intensity, or to the calibration of these models against real-world market data to see how well they perform in practice.