# A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

```@article{Lord2007AFA,
title={A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes},
author={R. Lord and Fang Fang and F. Bervoets and Cornelis W. Oosterlee},
journal={Risk Management eJournal},
year={2007}
}```
• Published 27 February 2007
• Mathematics
• Risk Management eJournal
A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for…
164 Citations
Efficient Options Pricing Using the Fast Fourier Transform
• 2010
We review the commonly used numerical algorithms for option pricing under Levy process via Fast Fourier transform (FFT) calculations. By treating option price analogous to a probability density
Look-Back Option Pricing Using the Fourier Transform B-Spline Method
• Mathematics
• 2013
We derive a new, efficient closed-form formula approximating the price of discrete look-back options, whose underlying asset price is driven by an exponential semi-martingale process including (jump)
Fourier methods for pricing early-exercise options under levy dynamics
The pricing of plain vanilla options, including early exercise options, such as Bermudan and American options, forms the basis for the calibration of financial models. As such, it is important to be
Lookback option pricing using the Fourier transform B-spline method
• Mathematics
• 2014
We derive a new, efficient closed-form formula approximating the price of discrete lookback options, whose underlying asset price is driven by an exponential semimartingale process, which includes (
Modeling the Short Rate as a Levy Process and Option Pricing with the FFT
This paper describes a practical algorithm for modeling interest rate derivatives with the short rate following a Levy process using the fast Fourier transform algorithm (FFT). It can be used with
Hedging and Pricing European-Type, Early-Exercise and Discrete Barrier Options Using an Algorithm for the Convolution of Legendre Series
• 2018
This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend 2014a), to pricing/hedging European-type, early-exercise and
Efficient Option Pricing by Frame Duality with the Fast Fourier Transform
• J. Kirkby
• Mathematics
SIAM J. Financial Math.
• 2015
A method for efficiently inverting analytic characteristic functions using frame projection, as in the case of Heston's model and exponential Levy models, is developed and convergence is demonstrated for geometric Asian options as well as the pricing of baskets of European options.
Option pricing with Legendre polynomials
• Mathematics
J. Comput. Appl. Math.
• 2017
Pricing Discretely Monitored Barrier Options and Credit Default Swaps under Lévy Processes
We introduce a new, fast and accurate method to calculate prices and sensitivities of European vanilla and digital options under the Variance Gamma model. For near at-the-money options of short
Pricing early-exercise and discrete barrier options by fourier-cosine series expansions
• Computer Science
Numerische Mathematik
• 2009
This paper is the follow-up of (Fang and Oosterlee in SIAM J Sci Comput 31(2):826–848, 2008) in which the impressive performance of the Fourier-cosine series method for European options was presented.

## References

SHOWING 1-10 OF 59 REFERENCES
A Double-Exponential Fast Gauss Transform Algorithm for Pricing Discrete Path-Dependent Options
• Mathematics
Oper. Res.
• 2005
Algorithms for the pricing of discretely sampled barrier, lookback, and hindsight options and discretely exercisable American options are developed using the double-exponential integration formula and the fast Gauss transform.
A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes
Option values are well-known to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'S-space', where S is the terminal
Numerical Valuation of European and American Options under Kou's Jump-Diffusion Model
Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model, which assumes the price of the underlying asset to behave like a geometrical Brownian
Robust numerical valuation of European and American options under the CGMY process
• Mathematics
• 2007
We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Levy process. For processes of finite variation,
Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing
• Mathematics
• 2000
This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly
A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS
• Mathematics
• 2006
This paper gives a tree‐based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price
Valuing Bermudan options when asset returns are Lévy processes
• Mathematics, Economics
• 2004
Abstract Evidence from the financial markets suggests that empirical returns distributions, both historical and implied, do not arise from diffusion processes. A growing literature models the returns
Accurate Evaluation of European and American Options Under the CGMY Process
• Computer Science
SIAM J. Sci. Comput.
• 2007
A finite-difference method for integro-differential equations arising from Le´vy driven asset processes in finance is discussed and the discretization is shown to be second-order accurate for a relevant parameter range determining the degree of the singularity in the Levy measure.
Lévy Processes in Finance: Theory, Numerics, and Empirical Facts
Preface Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible
On American Options Under the Variance Gamma Process
• Mathematics
• 2007
American options are considered in a market where the underlying asset follows a Variance Gamma process. A sufficient condition is given for the failure of the smooth fit principle for finite horizon