Intertwined currency symbols on a circuit board, symbolizing complex currency modeling.

Decoding Currency Fluctuations: Can a New Model Outsmart the Markets?

Samir D’Costa in Business & Economy December 2025 • 4 min read.

"Explore how CBI-time-changed Lévy processes could revolutionize foreign exchange modeling and option pricing."

The foreign exchange (FX) market stands as one of the largest and most dynamic financial arenas globally. Successfully modeling its complexities poses significant challenges for quantitative analysts. These challenges range from respecting the inherent symmetric structure of FX rates to accurately capturing the market's unique risk characteristics, like stochastic volatility and jump risk. As the FX market continues to evolve, the need for sophisticated models becomes ever more critical.

Traditional models often fall short of fully capturing the intricacies of FX rate behavior. For instance, a core requirement of any multi-currency model is its ability to respect FX rate symmetries, such as inversion (where the reciprocal of one currency pair should match the inverse pair) and triangulation (ensuring consistency across different currency triangles). Ignoring these symmetries can lead to inaccurate pricing and risk management.

In a groundbreaking paper, researchers Claudio Fontana, Alessandro Gnoatto, and Guillaume Szulda introduce a novel stochastic volatility framework that employs CBI-time-changed Lévy processes. This approach not only respects FX rate symmetries but also captures self-exciting volatility dynamics. Let’s dive into how this framework aims to revolutionize our understanding—and modeling—of currency fluctuations.

What are CBI-Time-Changed Lévy Processes and Why Do They Matter for FX Modeling?

Intertwined currency symbols on a circuit board, symbolizing complex currency modeling.

At the heart of this innovative framework lies the use of CBI-time-changed Lévy (CBITCL) processes. These processes represent a sophisticated way to model random fluctuations over time, making them particularly well-suited to capture the volatile nature of FX markets. Unlike simpler models, CBITCL processes can account for sudden jumps and stochastic volatility—two key features observed in real-world currency price movements.

To truly appreciate the power of this approach, let's break down the key components:

Lévy Processes: These form the base, describing random movements with independent, identically-distributed increments over time. They can capture jumps, making them more realistic than standard Brownian motion.
CBI (Continuous-State Branching with Immigration) Processes: CBI processes introduce a "self-exciting" element. They allow the intensity of the jumps to depend on the current state of the process, mirroring how volatility clusters in financial markets. When volatility is high, it tends to stay high for a while.
Time-Changed: By changing the time clock of the Lévy process using a CBI process, the model can create stochastic volatility.

By combining these elements, CBITCL processes offer a flexible and powerful framework that overcomes the limitations of traditional models. The framework captures the key characteristics that affect FX markets, offering a more accurate depiction of currency dynamics.

The Future of FX Modeling: Embracing Complexity

The research by Fontana, Gnoatto, and Szulda offers a significant step forward in the quest to model currency fluctuations more accurately. By embracing the complexity of CBI-time-changed Lévy processes, they have created a framework that not only respects the fundamental symmetries of FX rates but also captures the crucial characteristics of market behavior, such as stochastic volatility and jump risk. As the FX market continues to evolve, models like this will be essential for informed decision-making and effective risk management.

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Everything You Need To Know

What are the main challenges in accurately modeling the foreign exchange (FX) market?

Modeling the FX market presents several challenges. These include respecting FX rate symmetries like inversion and triangulation, and capturing unique risk characteristics such as stochastic volatility and jump risk. Traditional models often fail to fully incorporate these complexities, leading to inaccurate pricing and risk management.

What are CBI-time-changed Lévy processes, and why are they important for modeling currency fluctuations?

CBI-time-changed Lévy (CBITCL) processes are a sophisticated method for modeling random fluctuations over time, well-suited for capturing the volatile nature of FX markets. They combine Lévy processes, which model random movements with jumps, with CBI (Continuous-State Branching with Immigration) processes, which introduce a self-exciting element that mirrors how volatility clusters in financial markets. By changing the time clock of the Lévy process using a CBI process, the model can create stochastic volatility. This combination allows the model to capture sudden jumps and stochastic volatility—two key features observed in real-world currency price movements.

How do CBI-time-changed Lévy processes capture the self-exciting volatility dynamics in FX markets?

CBI processes introduce a "self-exciting" element that mirrors how volatility clusters in financial markets. When volatility is high, it tends to stay high for a while. CBI processes allow the intensity of jumps to depend on the current state of the process, mirroring the volatility clusters in financial markets. This "self-exciting" feature is crucial for capturing the dynamics where high volatility periods tend to persist.

How does the stochastic volatility framework introduced by Fontana, Gnoatto, and Szulda improve upon traditional FX modeling approaches?

The stochastic volatility framework introduced by Fontana, Gnoatto, and Szulda utilizes CBI-time-changed Lévy processes, which addresses the limitations of traditional models by respecting FX rate symmetries and capturing self-exciting volatility dynamics. This framework provides a more accurate depiction of currency dynamics by incorporating key characteristics such as stochastic volatility and jump risk, which are often overlooked or simplified in conventional models. Their approach embraces complexity, which is essential for informed decision-making and effective risk management in the evolving FX market.

What are the practical implications of using a model that incorporates CBI-time-changed Lévy processes for FX option pricing and risk management?

Using a model that incorporates CBI-time-changed Lévy processes has significant practical implications for FX option pricing and risk management. By more accurately capturing stochastic volatility and jump risk, such models can lead to more precise option pricing, reducing the risk of mispricing and potential losses. Additionally, better capturing market dynamics enhances risk management strategies, allowing for more informed decisions and better protection against unexpected currency fluctuations. This advanced modeling approach is essential for navigating the complexities of the FX market and making sound financial decisions.