COS-TT-CHF: A Tensor-Train Characteristic-Function COS Method for Multi-Asset Option Pricing

In the world of finance, predicting the future value of a single stock is already a complex task, but modern markets often require investors to price options on baskets containing dozens of assets simultaneously. These financial contracts, known as options, give the holder the right to buy or sell a group of stocks at a specific price in the future. The difficulty arises when trying to calculate the fair price for these contracts as the number of assets grows. Traditional mathematical tools struggle immensely with this growth; every time a new asset is added to the basket, the amount of calculation required does not just increase slightly, it explodes exponentially. This phenomenon, known as the curse of dimensionality, has long forced analysts to rely on slow, approximate methods or to limit their analysis to very small groups of assets, leaving a significant gap in our ability to price complex, multi-asset financial products accurately and quickly.

Researchers at the University of Copenhagen have developed a new approach to bridge this gap, creating a method that allows for the rapid pricing of options on baskets containing up to thirty different assets. Their work, published in a paper titled "COS–TT–CHF," focuses on a specific type of financial model that describes how asset prices move using a mathematical tool called a characteristic function. Instead of trying to calculate the price by looking at every possible combination of asset movements at once—a task that becomes impossible as the number of assets rises—the team used a technique called tensor-train approximation. This method works by compressing the massive, high-dimensional data required for the calculation into a much smaller, manageable format. It is similar to how a high-resolution photograph can be compressed into a small file size without losing the essential details needed to recognize the image, allowing the computer to process the information efficiently without getting bogged down by the sheer volume of data.

The researchers tested their new method against several established benchmarks, including older, direct calculation techniques and sophisticated simulation methods that rely on random sampling. They found that for baskets with just two or three assets, their new method was not necessarily faster than existing techniques. However, as soon as the number of assets increased to four or more, the new method began to outperform the older direct calculation approaches significantly. In tests involving twenty assets, the method completed calculations in a matter of seconds, whereas the traditional direct methods would have taken minutes or even hours to reach a similar level of accuracy. The team also demonstrated that their approach works well for different types of market models, including those that account for sudden jumps in prices or changing volatility, which are common features in real-world financial markets.

Beyond simply finding the price of the option, the new method allows for the quick calculation of "Greeks," which are measures of how sensitive the option's price is to changes in the underlying assets or market conditions. This is crucial for risk management, as traders need to know how their portfolios will react to market shifts. The researchers showed that once the initial compressed data structure was built, calculating these sensitivities for different strike prices or market scenarios was nearly instantaneous. This reusability is a key advantage, as it means the heavy lifting is done only once, and the results can be applied to a wide range of questions without starting the calculation from scratch.

The study also addressed a specific type of option that pays out based on the lowest or highest performing asset in a group, known as min/max options. These are notoriously difficult to price because the payoff depends on the interaction between all assets in a way that does not fit standard calculation patterns. The researchers successfully adapted their method to handle these complex payoffs, proving that their compressed representation could capture the necessary joint behavior of the assets. In their tests, the method maintained high accuracy even when dealing with thirty assets, a scale that was previously out of reach for this class of pricing techniques. The results suggest that by moving the complexity of the calculation into a compressed format early in the process, it is possible to bypass the exponential growth of computational cost that has long hindered the pricing of multi-asset derivatives.

The researchers compared their findings with reference data generated by other high-precision methods, such as randomized sampling techniques that are considered the gold standard for accuracy but are very slow. The new method matched these reference prices with high precision while being orders of magnitude faster in higher dimensions. For instance, in a stress test involving a common financial model with twenty assets, the new method completed the calculation in roughly sixty-three seconds, while the reference simulation method took over five hundred seconds. The accuracy remained consistent across different types of assets and market conditions, including models with heavy-tailed distributions where extreme price movements are more likely.

This work does not claim to solve every problem in financial mathematics, nor does it suggest that all pricing challenges are now obsolete. The method relies on the assumption that the underlying data can be compressed effectively, which holds true for many common market models but may vary in more exotic scenarios. However, the results clearly demonstrate that for a wide range of practical applications, the barrier of dimensionality can be lowered. By transforming the problem into a compressed representation, the researchers have provided a tool that makes it feasible to price complex, multi-asset options with a speed and accuracy that was previously unattainable. This advancement offers a practical path forward for financial institutions that need to manage risk and price products in increasingly interconnected and complex markets.