Fourier Transform

Visit the Mathema Option Pricing System, supporting FX options and structured product pricing and valuation!

Introduction

The Fourier Transform is a powerful mathematical tool used to convert functions from the time or spatial domain to the frequency domain. It is widely applied in signal processing, image analysis, physics, and other fields. In financial mathematics, the Fourier Transform has gained significant attention for its ability to efficiently handle complex pricing problems. Particularly in option pricing, when the distribution of the underlying asset price is complex or no closed-form solution exists, the Fourier Transform provides an effective method to compute option prices. Additionally, the introduction of the Fast Fourier Transform (FFT) has significantly improved computational efficiency, making the Fourier Transform a crucial tool in financial computations.

This article introduces the fundamental principles of the Fourier Transform, the efficient algorithm of the Fast Fourier Transform, and how to use the Fourier Transform for option pricing.

1. Fundamental Principles of the Fourier Transform

1.1 Definition of the Fourier Transform

The Fourier Transform is a transformation that converts a function from one domain (e.g., time domain) to another (e.g., frequency domain). For a function $f(x)$ , its Fourier Transform $\hat{f}(k)$ is defined as:

\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} dx

where:

$f(x)$ : The original function (time or spatial domain).
$\hat{f}(k)$ : The transformed function (frequency domain).
$k$ : The frequency variable.
$e^{-ikx}$ : The complex exponential basis function.

The inverse Fourier Transform converts the frequency domain function $\hat{f}(k)$ back to the time domain:

f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(k) e^{ikx} dk

1.2 Properties of the Fourier Transform

The Fourier Transform has several important properties that make it highly useful in financial computations:

Linearity: $\mathcal{F}\{a f(x) + b g(x)\} = a \hat{f}(k) + b \hat{g}(k)$
Convolution Property: $\mathcal{F}\{f(x) * g(x)\} = \hat{f}(k) \cdot \hat{g}(k)$
Differentiation Property: $\mathcal{F}\left\{\frac{d^n f(x)}{dx^n}\right\} = (ik)^n \hat{f}(k)$
Shift Property: $\mathcal{F}\{f(x - a)\} = e^{-ika} \hat{f}(k)$

These properties make the Fourier Transform highly efficient for solving complex integrals, convolutions, and differential equations.

2. Fast Fourier Transform (FFT)

2.1 Discretization of the Fourier Transform

In practical computations, the continuous Fourier Transform is often discretized. The Discrete Fourier Transform (DFT) is defined as:

\hat{f}_k = \sum_{n=0}^{N-1} f_n e^{-i 2\pi kn / N}, \quad k = 0, 1, \dots, N-1

where:

$N$ : Number of data points.
$f_n$ : Discretized values of the original function.
$\hat{f}_k$ : Transformed values in the frequency domain.

The inverse DFT is:

f_n = \frac{1}{N} \sum_{k=0}^{N-1} \hat{f}_k e^{i 2\pi kn / N}

2.2 Fast Fourier Transform (FFT)

Direct computation of the DFT has a complexity of $O(N^2)$ . The Fast Fourier Transform (FFT) is an efficient algorithm that reduces the complexity to $O(N \log N)$ by recursively decomposing the DFT into smaller DFTs. The core idea of FFT is to exploit periodicity and symmetry to break down the computation.

Advantages of FFT:

Efficiency: FFT significantly reduces computation time, especially for large datasets.
Wide Applications: FFT is the standard implementation of DFT in many financial computations.

3. Applications of the Fourier Transform in Option Pricing

3.1 Relationship Between Fourier Transform and Option Pricing

The core of option pricing is calculating the present value of the expected payoff:

V = e^{-rT} \mathbb{E}[\text{Payoff}]

For a call option, the payoff function is $\max(S_T - K, 0)$ . If the risk-neutral probability density function $f(S_T)$ of the underlying asset price $S_T$ can be described using the Fourier Transform, the option price can be computed efficiently.

3.2 Carr-Madan Method

Carr and Madan (1999) proposed a Fourier Transform-based method for option pricing, transforming the option pricing problem into a frequency domain computation. This method is a classic application of the Fourier Transform in financial computations.

Key Steps:

Modify the Payoff Function:
Direct computation of the Fourier Transform of the payoff function may not converge, so a damping factor $e^{-\alpha k}$ is introduced to transform the payoff function into:
$\tilde{h}(k) = e^{-\alpha k} \max(S_T - K, 0)$
Fourier Transform of the Option Price:
Use the characteristic function $\phi(u)$ (the Fourier Transform of the logarithm of the underlying asset price) to compute the option price:
$\hat{V}(u) = \int_{0}^{\infty} e^{-i u k} \tilde{h}(k) dk$
Inverse Transform:
Use the inverse Fourier Transform to compute the option price:
$V = \frac{e^{-rT}}{2\pi} \int_{-\infty}^{\infty} e^{-i u \ln K} \phi(u - (i\alpha)) du$

Applications:

Advantages: The Carr-Madan method efficiently handles option pricing for complex distributions (e.g., jump-diffusion models, stochastic volatility models).
Disadvantages: Requires selecting an appropriate damping factor $\alpha$ for different models.

3.3 Fourier Transform Method for the Heston Model

The Heston model is a classic stochastic volatility model with an analytical expression for its characteristic function $\phi(u)$ . This makes the Heston model particularly suitable for option pricing using the Fourier Transform.

Characteristic Function of the Heston Model:

Let $\ln(S_T)$ be the logarithm of the underlying asset price. Its characteristic function $\phi(u)$ is:

\phi(u) = \exp\left\{i u (\ln S_0 + (r - q)T) - \frac{\kappa \theta}{\xi^2} \left[(\gamma - d)T - 2\ln\left(\frac{1-g e^{-dT}}{1-g}\right)\right]\right\}

where:

$\gamma = \kappa - \rho \xi i u$
$d = \sqrt{\gamma^2 + \xi^2 (u^2 + iu)}$
$g = \frac{\gamma - d}{\gamma + d}$

Option Price Calculation:

Using the Carr-Madan method combined with the characteristic function of the Heston model, option prices can be computed efficiently.

4. Further Research and Applications

4.1 Jump-Diffusion Models

The Fourier Transform is highly effective in jump-diffusion models (e.g., the Merton model). The characteristic function of jump-diffusion models can often be expressed analytically, enabling fast option pricing using the Fourier Transform.

4.2 High-Dimensional Option Pricing

The Fourier Transform can be extended to high-dimensional problems, such as pricing basket options and options on correlated assets. In high-dimensional problems, using multidimensional Fourier Transforms can significantly reduce computational complexity.

4.3 Fourier Transform and Machine Learning

In recent years, the Fourier Transform has been used to enhance feature engineering in financial machine learning models. For example, the Fourier Transform can extract frequency features from time series data for volatility prediction and optimization of option pricing models.

5. References

Carr, P., & Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance.
- Introduces the Fourier Transform-based method for option pricing, a classic in financial computation.
Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies.
- Proposes the stochastic volatility model and its characteristic function.
Cohen, L., & Walden, A. T. (1995). Fourier Analysis of Time Series. Cambridge University Press.
- Provides a detailed introduction to the theory and applications of the Fourier Transform.

Conclusion

The Fourier Transform provides an efficient tool for option pricing, especially for complex distributions and stochastic volatility models. The use of characteristic functions significantly simplifies computations, and the introduction of the Fast Fourier Transform (FFT) greatly enhances computational efficiency, making the Fourier Transform a key technique in derivative pricing. With the advancement of financial markets and computational technologies, the applications of the Fourier Transform in high-dimensional problems, jump-diffusion models, and machine learning are becoming increasingly promising.