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Carr–Madan Option Pricing: FFT Versus Strike-Optimized Quadrature

Article Quant Q&A · Author: ewofeo
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Summary

The document explains why the Carr–Madan Fourier option-pricing method uses the fast Fourier transform: after discretizing the pricing integral with the trapezoid rule, one FFT can produce call prices for many strikes at a computational cost that scales as O(N log N). This is useful when prices across a regular grid of strikes are needed, but the grid may not align with a trader’s desired strikes and can spend computation on irrelevant values.

It contrasts that approach with evaluating the integral directly for selected strikes, a method whose basic cost is described as O(NM) for N integration points and M strikes. The cited numerical study compares the traditional FFT with a strike-optimized Carr–Madan implementation and reports that the optimized version was faster and more accurate in its tests, with more flexibility over strikes, integration, and interpolation. These are reported findings from a paper, not a universal guarantee; the document notes that further analysis could be needed to confirm the conclusion.

Key ideas

  • The Carr–Madan FFT applies a discretized pricing integral to produce option prices across a strike grid.
  • The FFT’s computational cost is stated as O(N log N), but its strike grid may not suit a particular pricing task.
  • Direct quadrature can target selected strikes, with a basic cost stated as O(NM).
  • A cited numerical comparison found its strike-optimized method faster and more accurate in the tested settings.
  • The comparison does not establish that FFT is inferior in every application.

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Full text
# In Carr-Madans option pricing method, why do they use FFT?


# In Carr-Madans option pricing method, why do they use FFT?












In the famous fourier option pricing method by Carr-Madan, (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.4044&rep=rep1&type=pdf), the crucial formula is

They evaluate this by using the trapezoid rule to write it in a form to which FFT can be applied. The output of the FFT will give us the call prices for multiple strikes. The complexity is $O(N \log N)$.

But the problem is that we have little control over the strikes. In particular if $N$ is large, the strikes become spread out, and many of the values are completely useless to us. Really, we are only interested in strikes close to $S_0$, the asset price.

So would it not be better to just calculate the integral above manually using quadrature rules, and then repeating that process for any given strike? This would be a $O(NM)$ algorithm with $M$ being the number of strikes, but in practice it would be very fast since we can re-use a lot of the computations (notice that $k = \log K$ is independent of $\psi(v)$ in the integral).

## Answer by sets (score 8, accepted)

https://quant.stackexchange.com/a/50322

Indeed, the FFT was a notable improvement in computational option pricing in 1999, but further investigation has shown that it can be easily optimized both in terms of speed and accuracy.

For instance, this paper compares the traditional FFT with a strike optimized version of the Carr-Madan formula (CM-OPT), concluding that the CM-OPT is simultaneously faster and more accurate that the FFT.

> Finally, the comparison between the FFT and the CM-OPT deserves a special mention. While both are based on the same pricing approach, the CM-OPT's flexibility allows (i) pricing any required strikes, (ii) choosing any integration size and technique and (iii) avoiding interpolation biases. As a result, the CM-OPT is both faster and more accurate than the FFT, thus rendering this method inefficient. Based on our results, we see no reason to employ the FFT over the CM-OPT, but further analysis may be needed in order to confirm this hypothesis.

Source: Crisóstomo, R (2018): Speed and biases of Fourier-based pricing choices: a numerical analysis, International Journal of Computer Mathematics, 95:8, 1565-1582, DOI: 10.1080/00207160.2017.1322691

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.