FFT and Carr-Madan Pricing

Setup

Motivation

For many models of practical importance — Heston, Variance Gamma, CGMY, NIG — the dynamics of $\ln S_T$ under the risk-neutral measure are not log-normal, and closed-form option prices do not exist. However, the characteristic function of the log-price is often available in closed form.

The Carr-Madan (1999) method prices European options by Fourier-inverting the characteristic function. Its efficiency comes from evaluating option prices at all strikes simultaneously using a single Fast Fourier Transform (FFT) pass — pricing at $N$ strikes for the cost of a single $O(N \log N)$ FFT, compared to $O(N)$ separate numerical integrations.

Characteristic Function

Let $x_T = \ln(S_T / S_0)$ be the log-return. Under the risk-neutral measure $\mathbb{Q}$:

$\varphi_T(u) = \mathbb{E}^{\mathbb{Q}}\!\left[e^{iu x_T}\right], \qquad u \in \mathbb{R}.$

For GBM: $\varphi_T(u) = \exp\!\left(iu(r - \frac{1}{2}\sigma^2)T - \frac{1}{2}u^2\sigma^2 T\right)$ (a known Gaussian characteristic function). For Heston: see the preceding module. For Lévy models (VG, NIG, CGMY): the characteristic function is an exponential of the Lévy exponent, available analytically.

Key property. $\varphi_T(u)$ is a bounded, uniformly continuous function of $u$. It decays as $|u| \to \infty$ whenever $S_T$ has a sufficiently smooth density.

Notation

• $k = \ln K$: log-strike.
• $F = S_0 e^{rT}$: forward price.
• $C_T(k) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T - e^k)^+]$: call price as a function of log-strike.
• $\varphi_T(u)$: characteristic function of $x_T = \ln(S_T/S_0)$, so $\mathbb{E}^{\mathbb{Q}}[e^{iux_T}] = \varphi_T(u)$.

The Carr-Madan Formula

Problem with the Naive Approach

The undiscounted call payoff $(S_T - K)^+ = (e^{x_T + \ln S_0} - e^k)^+$ is not square-integrable in $k$: as $k \to -\infty$, $C_T(k) \to S_0 - e^{k-rT} \to S_0$ (a constant), so $C_T \notin L^2(\mathbb{R})$. The Fourier transform does not exist in the classical sense.

Dampening

Multiply $C_T(k)$ by an exponential damping factor $e^{\alpha k}$ for some $\alpha > 0$:

$z_T(k) = e^{\alpha k} C_T(k).$

For $\alpha > 0$ and sufficiently small (specifically $\alpha < \bar{\alpha}$ where $\mathbb{E}[S_T^{1+\alpha}] < \infty$): as $k \to -\infty$, $z_T(k) = e^{\alpha k} \cdot (S_0 - e^{k-rT} + O(e^{2k})) \to 0$. As $k \to +\infty$, $z_T(k) = e^{\alpha k} \cdot O(e^{-k}) = O(e^{(\alpha-1)k}) \to 0$ for $\alpha < 1$. So $z_T \in L^2(\mathbb{R})$ and its Fourier transform exists.

Fourier Transform of the Modified Price

Define $\Psi_T(v) = \int_{-\infty}^{\infty} e^{ivk} z_T(k)\, dk$. Substituting $z_T(k) = e^{\alpha k + ivk} e^{-rT}\mathbb{E}^{\mathbb{Q}}[(e^{x_T + \ln S_0} - e^k)^+]$ and exchanging expectation and integration (Fubini, justified under the moment condition):

$\Psi_T(v) = e^{-rT} \int_{-\infty}^{\infty} e^{(\alpha + iv)k} \mathbb{E}^{\mathbb{Q}}\!\left[(S_0 e^{x_T} - e^k)^+\right] dk.$

Switching the order of integration (integrating over $k$ from $-\infty$ to $x_T + \ln S_0$ where the payoff is positive):

$\Psi_T(v) = \frac{e^{-rT} \varphi_T(v - (\alpha + 1)i)}{\alpha^2 + \alpha - v^2 + i(2\alpha + 1)v}.$

Derivation. Inside the expectation, $\int_{-\infty}^{x_T + \ln S_0} e^{(\alpha+iv)k}(S_0 e^{x_T} - e^k)\,dk$ evaluates to a ratio involving $e^{(\alpha+1+iv)(x_T + \ln S_0)}$. Taking the expectation introduces $\mathbb{E}[e^{(\alpha+1+iv)x_T}] = \varphi_T(-i(\alpha+1) + v)$. Collecting terms yields the formula above.

Inversion

The call price is recovered by Fourier inversion:

$C_T(k) = e^{-\alpha k} z_T(k) = \frac{e^{-\alpha k}}{\pi} \int_0^\infty \mathrm{Re}\!\left[e^{-ivk} \Psi_T(v)\right] dv,$

using the symmetry of the real part of the integrand (since the imaginary part integrates to zero for real $C_T$):

$\boxed{C_T(k) = \frac{e^{-\alpha k}}{\pi} \int_0^{\infty} e^{-ivk} \Psi_T(v)\, dv \quad \text{(real part implied)}.}$

This is exact for any model with a known characteristic function $\varphi_T$, provided the dampening condition $\mathbb{E}^{\mathbb{Q}}[S_T^{1+\alpha}] < \infty$ holds.

FFT Implementation

Discretisation of the Integral

Approximate the integral by a truncated sum over frequencies $v_j = \eta j$ for $j = 0, 1, \ldots, N-1$, with spacing $\eta > 0$ and upper limit $v_{\max} = \eta(N-1)$:

$C_T(k) \approx \frac{e^{-\alpha k}}{\pi} \sum_{j=0}^{N-1} e^{-iv_j k} \Psi_T(v_j)\, \eta \cdot w_j,$