Calculator emitting cosine waves over a stock chart city, symbolizing precise option pricing.

Decoding Option Pricing: How Many Calculations Are Really Needed?

Avery Sinclair in Business & Economy December 2025 • 4 min read.

"Demystifying the COS Method for European Option Pricing"

In the high-stakes world of finance, accurately pricing options is crucial. European options, a common type of financial contract, give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price on a specific date. Getting the price right is essential for both buyers and sellers, influencing investment strategies and risk management.

One popular technique for pricing these options is the Fourier-Cosine Expansion (COS) method. Think of it as a super-efficient calculator that uses cosine functions to approximate the option price. However, like any tool, the COS method requires careful setup. Two key parameters must be chosen: the truncation range (how much of the data to consider) and the number of terms (how precise the approximation should be).

While the method itself is celebrated for its speed, deciding how many calculations to perform has been somewhat of a guessing game—until now. Recent research has cracked the code, providing clear guidelines on selecting the number of terms needed for accurate option pricing. Let’s dive into how this works and what it means for investors and financial professionals.

What's the Magic Number? Finding the Right Balance with the COS Method

Calculator emitting cosine waves over a stock chart city, symbolizing precise option pricing.

The COS method's efficiency hinges on representing the option's price using a series of cosine functions. The more terms you include in this series, the more accurate the approximation, but also the more computationally intensive the process becomes. So, how do you strike the right balance? The key lies in understanding the characteristics of the underlying asset's price behavior.

The latest research introduces a practical formula for determining the optimal number of terms, denoted as 'N'. This formula considers several factors:

The desired level of accuracy: How close do you want your approximation to be to the true price? A smaller error tolerance demands a larger N.
The smoothness of the underlying asset's price distribution: If the price movements are well-behaved and predictable, you can get away with a smaller N.
The presence of "heavy tails": Some assets exhibit extreme price swings more frequently than others. These "heavy tails" require a larger N to capture their impact on option prices.

By plugging these factors into the formula, you can arrive at a specific value for N that ensures accurate option pricing without unnecessary computational burden. This is a significant improvement over the traditional trial-and-error approach.

The Future of Option Pricing: Faster, More Accurate, More Accessible

The new insights into the COS method represent a significant step forward in option pricing. By providing a clear, mathematically sound way to determine the optimal number of terms, the research makes this powerful technique more accessible to both practitioners and academics. This translates to faster, more accurate option prices, leading to better investment decisions and more efficient risk management.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2303.16012,

Title: On The Number Of Terms In The Cos Method For European Option Pricing

Subject: q-fin.cp

Authors: Gero Junike

Published: 28-03-2023

Everything You Need To Know

What is the COS method and why is it used in option pricing?

The COS method, or Fourier-Cosine Expansion method, is a technique used to price European options. It's essentially a fast and efficient way to calculate option prices by approximating them using a series of cosine functions. This method is preferred because it provides a balance between accuracy and computational speed, making it suitable for complex financial calculations. It helps financial professionals and investors accurately determine the price of options, which is crucial for making sound investment decisions and managing risk effectively.

What are the key parameters in the COS method, and how do they affect the accuracy of option pricing?

In the COS method, two critical parameters are the truncation range and the number of terms (N). The truncation range defines the portion of the data considered, influencing the scope of the calculations. The number of terms, represented as 'N', directly impacts the accuracy of the option price approximation. A higher 'N' generally leads to greater accuracy but also increases the computational effort. The choice of 'N' is crucial and, as outlined by the formula introduced by recent research, should consider the desired level of accuracy, the smoothness of the underlying asset's price distribution, and the presence of heavy tails.

How is the optimal number of terms, 'N', determined in the COS method?

The optimal number of terms, 'N', in the COS method is determined using a formula that considers the desired level of accuracy, the smoothness of the underlying asset's price distribution, and the presence of heavy tails. The level of accuracy determines how close the approximation must be to the true price, influencing 'N'. Assets with well-behaved price movements may require a smaller 'N', while those with heavy tails, exhibiting extreme price swings, necessitate a larger 'N' to capture their impact, leading to more accurate option pricing.

What are the implications of using the COS method for investors and financial professionals?

The COS method enables faster and more accurate option pricing, leading to better investment decisions and more efficient risk management for both investors and financial professionals. This means that investment strategies can be developed and executed with greater confidence. Risk management becomes more effective as options can be priced more precisely, allowing for better hedging strategies and protection against market volatility. The increased accessibility of this method also empowers a broader audience to understand and utilize sophisticated financial tools.

How does the formula for 'N' represent an advancement in option pricing using the COS method?

The formula for determining 'N' in the COS method is a significant advancement because it moves away from the traditional trial-and-error approach. It provides a clear, mathematically-sound way to determine the optimal number of terms, which ensures accurate option pricing without unnecessary computational burden. By considering factors like the desired accuracy, asset price distribution smoothness, and the presence of heavy tails, the formula allows practitioners to tailor the method to specific assets and market conditions, leading to improved precision and efficiency in option pricing. This also makes the COS method more accessible to a broader audience.