Option Pricing and the Fourier Transform in Simple Terms

Option pricing is a fundamental issue in financial markets, with the Fourier transform providing a key mathematical tool to address its complexities. This article breaks down these concepts to explain how the Fourier transform helps determine option prices effectively.

What is an Option?

An option is a financial contract that gives the holder the right (but not the obligation) to buy or sell an asset at a specified strike price, \( K \), by a certain expiration date, \( T \). A European call option, for example, allows the buyer to purchase the underlying asset at \( K \) on \( T \).

The challenge is to find the current value of an option, which depends on the expected payoff at maturity. The payoff for a European call is \( \max(S_T - K, 0) \), where \( S_T \) is the asset price at \( T \). The option’s current price is given by:

\[ C = e^{-rT} \mathbb{E}^Q[\max(S_T - K, 0)] \]

where \( r \) is the risk-free interest rate, and \( \mathbb{E}^Q \) denotes the expectation under the risk-neutral measure.

Challenges in Option Pricing

Since future asset prices \( S_T \) follow stochastic processes (e.g., the Black-Scholes model where returns are log-normal), it’s difficult to predict the exact distribution and expected payoff. More complex models may involve jumps and stochastic volatility. Furthermore, the payoff function \( \max(S_T - K, 0) \) is not differentiable at \( S_T = K \), complicating the use of traditional differential methods. Therefore, alternative approaches like the Fourier transform are needed.

The Fourier Transform: A Primer

The Fourier transform is a mathematical tool that decomposes a function into a sum of sinusoids of different frequencies. In this context, it translates a function from the time domain to the frequency domain. Specifically, the transform of a function \( f(t) \) is given by:

\[ F(\xi) = \int_{-\infty}^\infty f(t) e^{-2i\pi\xi t} \, dt \]

where \( \xi \) (the Greek letter "xi") represents frequency. In finance, \( f(t) \) often corresponds to a distribution of returns, and the Fourier transform reveals how different frequencies contribute to this distribution.

Application to Option Pricing

The core idea is to use the Fourier transform to convert the calculation of the expected payoff, complex in the time domain, into a simpler frequency domain calculation. This transformation helps handle the payoff's discontinuity and the complexity of the return distribution.

The Characteristic Function

The characteristic function is crucial in this process. It’s essentially the Fourier transform of the probability distribution of returns, defined as:

\[ \phi(u) = \mathbb{E}^Q\left[\exp(iuX_T)\right], \]

where \( u \) is frequency and \( X_T = \log(S_T) \) is the logarithmic return. The characteristic function encodes all information about \( S_T \) in the frequency domain, representing returns as combinations of sinusoids.

The Carr-Madan Method

A well-known method that utilizes this approach is the Carr-Madan method. By leveraging the characteristic function, this method efficiently computes option prices by working in the frequency domain, bypassing the challenges of non-differentiability and complex distributions in the time domain.