Carr-Madan FFT
Carr-Madan Fast Fourier Transform Option Pricing · Also known as: FFT Pricing, Characteristic Function Method
The Carr-Madan Fast Fourier Transform (1999) is a highly efficient method for computing option prices across a range of strikes using characteristic functions and FFT. It enables rapid pricing of European options under any model with a known characteristic function (Heston, Merton jumps, Variance Gamma), with computational complexity that scales logarithmically in the number of strikes.
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When to use it
Use FFT when you need to price options across many strikes simultaneously (swaption matrices, volatility surfaces). Essential for Heston and jump models where no closed-form formula exists. Less useful if pricing single strikes or very few strikes (overhead dominates). Requires characteristic function, limiting applicability.
Strengths & limitations
- Speed: O(N log N) complexity for pricing N strikes simultaneously, much faster than individual pricing
- Many strikes: compute entire volatility surface in seconds, ideal for calibration
- General: applies to any model with known characteristic function (Heston, jumps, Levy)
- Accuracy: smooth integrand enables high accuracy with modest grid sizes
- Characteristic function requirement: must have closed form (excludes many exotic models)
- Grid points required: number of strikes must be power of 2; interpolation needed for other strikes
- European only: FFT method is for European options; American options require other approaches
- Damping parameter selection: parameter alpha must be chosen carefully to avoid numerical issues
Frequently asked
How do I choose the damping parameter alpha?
Alpha controls the damping of the integrand. For puts, typical alpha ≈ 1.5; for calls, alpha ≈ 1.5 works. Test: compute a few strikes analytically, then vary alpha until FFT matches. Too large alpha causes damping-induced bias; too small causes integration error.
Can I use FFT for American options?
Not directly. FFT is for European options. For American options under Heston, you must combine FFT with early exercise algorithms (LSM or tree). Alternatively, use pure PDE or tree methods.
What if my model doesn't have a closed-form characteristic function?
You can still use FFT if you can compute phi(u) numerically (e.g., via Monte Carlo over log-returns). However, this adds noise and defeats the efficiency advantage. In such cases, use pure Monte Carlo or PDE methods.
How do I compute puts?
Compute calls via FFT. Use put-call parity: P = C - S + K*exp(-rT). Alternatively, adapt the Carr-Madan formula to price puts directly by modifying the integrand.
Sources
- Carr, P., & Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2(4), 61-73. DOI: 10.21314/JCF.1999.043 ↗
- Lee, R. W. (2004). Option pricing by transform methods: extensions, unification, and error analysis. Journal of Computational Finance, 7(3), 51-102. link ↗
How to cite this page
ScholarGate. (2026, June 3). Carr-Madan Fast Fourier Transform Option Pricing. ScholarGate. https://scholargate.app/en/quantitative-finance/carr-madan-fft
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