Fourier Transform
1 Fourier Transform
The Fourier transform decomposes a function into the frequencies that make it up. It turns convolution into multiplication and differentiation into multiplication by a polynomial, so it converts many hard problems (linear PDEs, sums of independent random variables) into algebra. It is the tool behind the PDE approach to the Heston Model, the Heat Equation and characteristic functions of the Normal Distribution. It is closely related to the Laplace Transform.
Throughout, we use the convention below. Other texts move the factors of \(2\pi\) and the sign of the exponent around, so check the convention before quoting a formula.
2 Definition
For \(f\in L^1(\mathbb{R})\) the Fourier transform of \(f\) is
\[ \hat{f}(\xi):=\int_{-\infty}^\infty f(x)e^{-i\xi x}\,dx,\qquad \xi\in\mathbb{R}. \]
The function \(\hat{f}\) is bounded and continuous, and \(\hat{f}(\xi)\to 0\) as \(|\xi|\to\infty\) (the Riemann-Lebesgue lemma).
3 Properties
Let \(f,g\in L^1(\mathbb{R})\), \(a\in\mathbb{R}\) and \(c\neq 0\). The derivative and multiplication rules assume \(f\) is sufficiently smooth and decaying.
| Operation | Transform |
|---|---|
| Linearity: \(\alpha f+\beta g\) | \(\alpha\hat{f}(\xi)+\beta\hat{g}(\xi)\) |
| Derivative: \(f'(x)\) | \(i\xi\hat{f}(\xi)\) |
| Multiplication: \(xf(x)\) | \(i\hat{f}'(\xi)\) |
| Translation: \(f(x-a)\) | \(e^{-ia\xi}\hat{f}(\xi)\) |
| Modulation: \(e^{iax}f(x)\) | \(\hat{f}(\xi-a)\) |
| Scaling: \(f(cx)\) | \(\frac{1}{|c|}\hat{f}(\xi/c)\) |
| Convolution: \(f*g\) | \(\hat{f}(\xi)\hat{g}(\xi)\) |
The convolution line refers to Convolution. The derivative rule follows by integrating by parts, since the boundary terms vanish.
4 Results
Let \(f\in L^1(\mathbb{R})\) with \(\hat{f}\in L^1(\mathbb{R})\). Then, for almost every \(x\),
\[ f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty\hat{f}(\xi)e^{i\xi x}\,d\xi, \]
with equality for every \(x\) if \(f\) is continuous.
If \(f\in L^1\cap L^2(\mathbb{R})\) then \(\hat{f}\in L^2(\mathbb{R})\) and
\[ \int_{-\infty}^\infty|f(x)|^2\,dx=\frac{1}{2\pi}\int_{-\infty}^\infty |\hat{f}(\xi)|^2\,d\xi. \]
The transform therefore extends (up to the factor \(2\pi\)) to a unitary operator on \(L^2(\mathbb{R})\).
For \(f,g\in L^1(\mathbb{R})\),
\[ \widehat{f*g}(\xi)=\hat{f}(\xi)\hat{g}(\xi). \]
Proof. By Fubini’s theorem,
\[ \widehat{f*g}(\xi)=\int\!\!\int f(s)g(x-s)e^{-i\xi x}\,ds\,dx =\int f(s)e^{-i\xi s}\left(\int g(x-s)e^{-i\xi(x-s)}\,dx\right)ds =\hat{f}(\xi)\hat{g}(\xi). \]
5 Examples
Let \(f(x)=e^{-x^2/2}\). Then \(\hat{f}(\xi)=\sqrt{2\pi}\,e^{-\xi^2/2}\), so up to a constant the Gaussian is its own Fourier transform.
Since \(f'(x)=-xf(x)\), the derivative and multiplication rules give
\[ i\xi\hat{f}(\xi)=\widehat{f'}(\xi)=-\widehat{xf}(\xi)=-i\hat{f}'(\xi), \]
so \(\hat{f}'(\xi)=-\xi\hat{f}(\xi)\). This ODE has solution \(\hat{f}(\xi)=\hat{f}(0)e^{-\xi^2/2}\), and \(\hat{f}(0)=\int e^{-x^2/2}\,dx=\sqrt{2\pi}\) by the Gaussian Integral.
Consider \(u_t=u_{xx}\) on \(\mathbb{R}\) with \(u(x,0)=f(x)\). Transforming in \(x\) gives the ODE \(\hat{u}_t=-\xi^2\hat{u}\) in \(t\), so
\[ \hat{u}(\xi,t)=e^{-\xi^2t}\hat{f}(\xi). \]
The inverse transform of \(e^{-\xi^2t}\) is the Gaussian kernel \(K_t(x)=\frac{1}{\sqrt{4\pi t}}e^{-x^2/(4t)}\), so by the convolution theorem
\[ u(x,t)=(K_t*f)(x). \]
See the Heat Equation.
6 Characteristic Functions
For a random variable \(X\) with density \(p_X\) the characteristic function is
\[ \varphi_X(\xi):=\mathbb{E}\left[e^{i\xi X}\right]=\hat{p}_X(-\xi), \]
so it is the Fourier transform of the density, up to the sign of the argument. It always exists and determines the distribution of \(X\) uniquely. Fourier inversion recovers the density from \(\varphi_X\), and the convolution theorem shows that \(\varphi_{X+Y}=\varphi_X\varphi_Y\) for independent \(X,Y\), since the density of \(X+Y\) is \(p_X*p_Y\). For \(X\sim N(\mu,\sigma^2)\) the Gaussian example above, after shifting and scaling, gives \(\varphi_X(\xi)=e^{i\mu\xi-\sigma^2\xi^2/2}\).
See Characteristic Functions for the full treatment, including the uniqueness, inversion and continuity theorems.
7 Application to Option Pricing
In models such as the Heston Model the density of \(\log S_T\) is not available in closed form, but its characteristic function is, because the dynamics are affine in the variance process \(\{Y_{t}\}\). The risk-neutral price of a European option is then computed by Fourier inversion of the characteristic function, replacing a PDE solve with a one-dimensional numerical integral. See Stochastic Volatility Models.
Note: The conditions matter. Inversion requires \(\hat{f}\in L^1\), and the derivative rule requires \(f\) to decay at infinity. The Fourier transform of a call payoff does not exist as is, because the payoff is not integrable, and pricing methods damp it by an exponential factor first.
9 References
- Folland, Real Analysis, Section 8.3.
- Stein and Shakarchi, Fourier Analysis: An Introduction.
- Carr and Madan (1999), “Option valuation using the fast Fourier transform”, Journal of Computational Finance.