Interpreting Complex FFT Outputs in Carr–Madan Option Pricing

Summary

The discussion addresses why a Carr–Madan fast Fourier transform calculation can return complex-valued quantities when pricing caplets. The accepted answer says the complex form is expected in the method and that the option price is obtained from the real part of the computed result, rather than its modulus. It points to financial modelling references that helped clarify this interpretation and to a separate outline of a straightforward FFT implementation.

The excerpt offers a practical interpretation of the output, but it does not derive the Carr–Madan transform or explain the numerical conditions under which the imaginary component should be small. It also does not diagnose whether a particular implementation has sign, scaling, discretization, or convention errors. Thus, taking the real part is the stated answer to the conceptual question, while validation against known prices and careful implementation checks remain outside the discussion.

Key ideas

• Carr–Madan FFT calculations can produce complex-valued intermediate or final outputs.
• The stated option-price result is the real part, not the complex modulus.
• The response cites financial modelling references for interpreting the method.
• The excerpt does not provide implementation diagnostics or explain numerical error sources.

Tags

# Understanding FFT's complex number result on option pricing


# Understanding FFT's complex number result on option pricing












I have been using the Carr-Madan method to price caplets using the FFT. I have followed every step closely and (i believe) successfully. I understand the procedure theoretically but I cannot interpret the complex nature of my results. Is the modulus my answer? Or is it the real part? Am I doing something wrong to have a complex number as my result?

https://wwwf.imperial.ac.uk/~ajacquie/IC_Num_Methods/IC_Num_Methods_Docs/Literature/CarrMadan.pdf

Thanks in advance

## Answer by Sotiris Zampelis (score 1, accepted)

https://quant.stackexchange.com/a/35238

As I figured after reading Stephane Crepey's Financial Modelling book the result is meant to be complex, we should only include the real part as the answer something not 100% clear (at least to me) from Carr-Madan's paper.

I also found a useful outline of the algorithm for a naive version of FFT at F.Rouah's book on the Heston Model.

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