Improving Heston Option Pricing with Fourier-Cosine Expansions
Summary
The document discusses numerical pricing accuracy in the Heston stochastic volatility model, especially for very far out-of-the-money options near expiry. It contrasts a reported Gaussian quadrature error with option prices smaller than that error, raising concerns about using such prices in calibration. One proposed alternative is the Fourier-cosine expansion method, which the response describes as simple to implement and rapidly convergent. Another answer notes that accuracy depends on implementation choices, including the integration method and number of quadrature points.
The method’s truncation interval requires care, particularly for far out-of-the-money options, and the document mentions adaptive Filon quadrature as another approach. It also distinguishes numerical pricing error from model fit: Heston’s limited parameterization may fail to match market prices even when prices are computed accurately. The claims about speed and precision are respondent observations, not a controlled benchmark in the text, so the best method and calibration treatment depend on the target accuracy and validation setup.
Key ideas
- Fourier-cosine expansions are proposed as an alternative for computing Heston option prices.
- The choice and configuration of quadrature affect numerical accuracy, especially for far out-of-the-money options.
- Selecting a suitable truncation interval is an important part of applying the cosine method.
- Pricing error against a reference model is distinct from Heston’s ability to fit market prices.
- Adaptive Filon quadrature is mentioned as another numerical approach.
Tags
Full text
# How can I improve the numerical integration accuracy in Heston model? # How can I improve the numerical integration accuracy in Heston model? I am trying to perform the numerical integration in the Heston using Gaussian quadrature but I obtain an error of 4e-3 while some of the deep out-of-the-money near expiry Call prices are smaller than 1e-5. Is there any way I can improve the accuracy without being extremely slow or should I exclude those prices from my calibration ? ## Answer by James Spencer-Lavan (score 2, accepted) https://quant.stackexchange.com/a/35555 Use fourier-cosine expansions, first paper by Fang-Oosterlee in 2008. Very simple to code and exponential convergence affords working accuracy within sub-seconds vs order of magnitude slower under Carr-Madan. For reference precision you can add another order of magnitude. My VBA implementation is faster and more stable than Carr-Madan in C++ for OTM options ## Answer by jherek (score 1) https://quant.stackexchange.com/a/44707 The long list of comments suggests two different issues: - Are you measuring your error against market prices? or in other words, are you trying to calibrate Heston parameters to market prices. If yes, it is well known that Heston is not going to match well, which is not too surprising since it has only 5 free parameters. - If you measure against some reference Heston model prices (given in the literature), what kind of Gauss quadrature are you referring to? There are many different. Is it adaptive? How many points are used? Regarding (2), @James suggestion is good. The Cos method is very simple to implement and produces satisfactory results most of the time. What is not always so trivial is to find a good estimate of the truncation interval (the suggestions from the Cos paper are reasonable good starting point). This impacts mostly very out-of-the-money options. A recent fast quadrature has been proposed in An adaptive Filon quadrature for stochastic volatility models, along with comparison against various other quadratures methods.
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