Fourier Pricing in Exponential Lévy Models — The Carr-Madan FFT Approach

For any exponential Lévy model $S_t = S_0 e^{X_t}$, the characteristic function of the log-return $X_T$ is given by the Lévy-Khintchine formula in closed form. The Carr-Madan FFT approach (1999) transforms the call price into Fourier space, evaluates it using the characteristic function, and inverts via the Fast Fourier Transform — obtaining an entire grid of European call prices in $O(N\log N)$ time.

This is the workhorse pricing algorithm for all exponential Lévy models: VG, NIG, CGMY, and the Bates SVJ model studied in the next lesson. It generalises the Heston Gil-Pelaez inversion (seen in the Heston Fourier pricing lesson) to arbitrary Lévy characteristic exponents.

Learning objectives. By the end of this note you should be able to: