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Hedging and Pricing European-type, Early-Exercise and Discrete Barrier Options using Algorithm for the Convolution of Legendre Series

This paper introduces a quadrature-free, high-accuracy method for pricing and hedging European, early-exercise, and discrete barrier options under Lévy processes by applying the CONLeg algorithm for Legendre series convolution within the Chebfun framework, demonstrating superior performance across various market conditions and option types compared to existing techniques.

Original authors: Tat Lung Chan, Nicholas Hale

Published 2026-06-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tat Lung Chan, Nicholas Hale

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 a financial architect trying to predict the future value of a house (an option) based on how the price of the land (the stock) might move. In the world of finance, these predictions are often complicated by "jumps" in the market or weird, unpredictable behaviors that standard math models struggle to handle.

This paper introduces a new tool called CONLeg (short for CONvolution of Legendre series) to solve these pricing puzzles. Here is how it works, explained through everyday analogies:

1. The Problem: The "Jagged Edge" Issue

Traditional methods for pricing options often rely on a technique called the Fast Fourier Transform (FFT). Think of the FFT like trying to draw a smooth curve using a ruler that only knows how to draw straight lines. When the market data is "smooth" (predictable), this works fine. But when the market data is "jagged" or has sudden spikes (like a stock price that jumps unexpectedly), the ruler creates a "jagged edge" effect known as the Gibbs phenomenon. It's like trying to trace a photo of a mountain range with a ruler; you get a lot of unwanted noise and wiggles at the peaks and valleys, making your prediction inaccurate.

Other methods try to fix this by taking a "snapshot" of the price at a single point. But in finance, you don't just want to know the price of one house; you want to know the price of every house on the street at once.

2. The Solution: The "Smooth Painter" (CONLeg)

The authors propose using Legendre series and Chebyshev series. Imagine these not as rulers, but as a set of flexible, smooth paintbrushes.

The Analogy: Instead of trying to force a jagged line to fit a straight ruler, the CONLeg method uses these flexible brushes to "paint" the entire curve of the option price.
The Magic: It treats the pricing formula as a convolution. In simple terms, a convolution is like sliding one shape over another to see how they overlap. The paper shows how to slide the "payoff shape" (what you get if you exercise the option) over the "probability shape" (how likely the stock is to move) using these smooth brushes.

3. Why It's Better: The "Whole Map" vs. "Single Pin"

Most old methods give you a single pin on a map: "The price of this option is $5.00." If you want to know the price for a slightly different strike price, you have to run the whole calculation again to get a new pin.

The CONLeg method is different. It doesn't just give you a pin; it draws the entire map.

The Analogy: Instead of calculating the price for one specific house, CONLeg calculates the price for the entire neighborhood in one go. It produces a continuous curve (a function) that tells you the price for any stock price within a range instantly.
The Benefit: This allows traders to see the "Greek" curves (sensitivity measures like Delta and Gamma) as smooth lines rather than a series of disconnected dots. This is crucial for hedging (protecting against risk) because it shows you exactly how the risk changes as the market moves, not just at one specific moment.

4. Handling the "Messy" Markets

The paper tests this method on various types of markets, including those driven by Lévy processes.

The Analogy: Think of a calm lake (standard markets) vs. a stormy sea with sudden, violent waves (markets with jumps).
The Claim: The CONLeg method works well even when the "water" is stormy. It can handle "non-smooth" probability densities (the stormy waves) without losing accuracy. It can price options that are very far from the current market price ("deep in/out of the money") and options that expire very soon or very far in the future.

5. The Toolkit: Chebfun

The authors built this using a software package called Chebfun.

The Analogy: If standard coding is like doing math by hand with a calculator, Chebfun is like having a robot that understands functions as whole objects. The authors used this robot to automate the "painting" process, making the complex math of convolutions much easier to implement.

Summary of Claims

The paper claims that:

Accuracy: CONLeg is highly accurate, even when the market data is rough or jagged, outperforming or matching existing methods like COS, SWIFT, and CONV.
Efficiency: It generates a whole curve of prices and risk measures in a fraction of a second, rather than calculating point-by-point.
Versatility: It works for European options (fixed expiry), American options (can be exercised anytime), Bermudan options (can be exercised on specific dates), and Barrier options (which vanish if a price is hit).
Robustness: It handles both smooth markets and "jumpy" markets (like those with sudden crashes or spikes) effectively.

In short, the paper presents a new way to "paint" the future value of financial options that is smoother, more comprehensive, and more robust against market chaos than previous techniques.

Technical Summary: Hedging and Pricing European-type, Early-Exercise and Discrete Barrier Options using an Algorithm for the Convolution of Legendre Series

Problem Statement
The paper addresses the challenge of pricing and hedging a wide variety of financial derivatives—including European, early-exercise (American/Bermudan), and discrete-monitored barrier options—under general Lévy processes. Existing numerical techniques face specific limitations:

FFT-based methods (e.g., CONV): While popular, they often require a large number of Fourier terms to achieve sufficient accuracy, making them computationally expensive, particularly for approximating probability density functions (PDFs) or pricing options with non-smooth features.
Quadrature-based methods (e.g., QUAD): These often rely on closed-form PDFs or require the inversion of characteristic functions via FFT. Furthermore, they typically produce individual point values rather than continuous pricing curves, and the calculation of option Greeks (sensitivities) is often limited to first-order finite differences, which can lack accuracy.
Series expansion methods (e.g., COS, SWIFT): While the COS method offers exponential convergence, it requires a priori selection of a computational domain; if the domain is inadequate, errors propagate recursively in time, leading to incorrect prices for early-exercise options. The SWIFT method addresses early-exercise features but lacks a theoretical proof of exponential convergence, as its scale parameter is chosen heuristically. Additionally, these methods often do not address the computation of Greeks.

Methodology
The authors propose the CONLeg method, which utilizes the convolution of compactly supported Legendre series to approximate option pricing and hedging formulas. The methodology integrates the following components:

Convolution of Legendre Series: The core algorithm, adapted from Hale and Townsend (2014a), approximates the convolution of two functions (the payoff function and the reflected PDF) using finite Legendre series. This approach avoids the Gibbs phenomenon associated with FFT approximations of non-periodic functions by splitting the convolution domain into left and right pieces ( $h_L$ and $h_R$ ) and computing coefficients via a three-term recurrence relation.
Chebfun Framework: The method employs Chebfun, a MATLAB system for numerical computing with functions, to represent functions as Chebyshev series. This allows for high-precision (approx. 15 digits) adaptive approximations.
Handling Non-Smooth PDFs: For processes where the PDF is not smooth or lacks a closed form (e.g., Variance Gamma, CGMY), the paper utilizes:
- Complex Fourier Series (CFS): To represent the PDF when a closed form is unavailable.
- Fourier–Padé Method: To locate singularities in the PDF and construct accurate piecewise approximations, thereby resolving Gibbs phenomena.
Formulation of Pricing and Greeks:
- Pricing: The option price is formulated as a convolution integral $V(x,K,t) \approx e^{-r(T-t)}K (f_M * g_N)(\tilde{x})$ . Unlike methods that compute point values, CONLeg generates a continuous pricing curve (a piecewise polynomial) over a range of strikes or spot prices.
- Hedging (Greeks): Because the price is represented as a Chebyshev/Legendre series, derivatives (Delta, Gamma, Vega) are computed analytically by differentiating the series, avoiding the inaccuracies of finite difference methods.
Application to Option Types:
- European/Bermudan/American: The method formulates recursive pricing equations. For American options, it employs Richardson extrapolation on Bermudan option prices.
- Barrier Options: The method handles discrete monitoring by treating barrier levels as breakpoints in the piecewise polynomial approximation.
Domain Selection: The computational interval $[c, d]$ is determined using the cumulants of the underlying Lévy process to ensure the interval captures the majority of the PDF mass.

Key Contributions

Curve Generation: The primary contribution is the ability to formulate option pricing and Greek curves (functions of $S$ or $K$ ) rather than individual point values. This is achieved by approximating the convolution integral directly.
Quadrature-Free Approach: The method provides a quadrature-free approach by leveraging the properties of orthogonal polynomials (Legendre and Chebyshev) and the convolution theorem, eliminating the need for numerical integration weights.
Robustness to Non-Smoothness: The method demonstrates high accuracy even when the risk-neutral PDF is non-smooth or contains singularities, provided the Fourier–Padé technique is used to locate and handle these singularities.
Flexibility: The approach is applicable to a broad range of stochastic processes (GBM, VG, NIG, CGMY) regardless of whether a closed-form PDF exists, as it can work with characteristic functions via CFS representation.
Implementation: The paper provides a proof-of-concept implementation using Chebfun, highlighting the convenience of using high-level function-based computing in financial modeling.

Numerical Results
The paper presents numerical experiments comparing CONLeg against established methods (COS, filter-COS, CONV, Lewis-FRFT, QUAD-CONV, SWIFT) across various models and option types:

European Options: Under the Black-Scholes (GBM) model, CONLeg achieves machine precision accuracy ( $R_\infty \approx 10^{-14}$ ) for prices, Delta, and Gamma in under 0.1 seconds for 1000 points. For long-maturity options (T=50, 100), it maintains high accuracy.
Short-Maturity/Non-Smooth PDFs: In the Variance Gamma (VG) model with a sharp-peaked PDF and singularity, CONLeg outperforms CONV, Lewis-FRFT, and QUAD-CONV, achieving accuracy comparable to COS and filter-COS methods.
Early-Exercise Options: For Bermudan and American options under NIG and CGMY models, CONLeg matches or slightly exceeds the accuracy of the COS method within similar computational times.
Barrier Options: For discrete-monitored barrier options (daily and monthly), CONLeg performs comparably to the SWIFT method in terms of accuracy and computational time.

Significance and Claims
The authors claim that the CONLeg method offers a favorable or comparable performance to existing techniques in numerical experiments. Its significance lies in:

Accuracy: It yields high accuracy for options deep in/out of the money and with varying maturities, even under non-smooth PDFs.
Efficiency in Curve Generation: It efficiently generates entire pricing and hedging curves, which is advantageous for model calibration and risk management, rather than calculating discrete points.
Theoretical Robustness: It avoids the domain selection pitfalls of the COS method for recursive pricing and provides a systematic way to handle singularities via Fourier–Padé.
Practicality: The authors acknowledge that while the current Chebfun-based implementation incurs some overhead due to adaptive procedures, the method is a "proof of concept" that demonstrates the viability of function-based convolution for financial modeling. They note that a non-adaptive version would be required for a fair efficiency comparison regarding convergence rates, but the current results are promising.

The paper concludes by stating that future work will extend this method to path-dependent options under time-changed Lévy processes and stochastic volatility models.