Fourier Inversion in Finance

Fourier inversion is a powerful mathematical technique with significant applications in financial modeling, particularly in option pricing. It provides an efficient method to calculate option prices when analytical solutions are unavailable, often encountered with complex models that incorporate realistic market dynamics.

At its core, the Fourier transform decomposes a function (in finance, typically the probability density function of an asset’s return) into its constituent frequencies. The characteristic function, which is the Fourier transform of the probability density function, is often easier to obtain analytically, especially for models with stochastic volatility, jumps, or other complex features. These features are challenging to handle directly with traditional pricing techniques.

The option pricing problem then transforms into finding the probability density function of the underlying asset price at maturity. Using the characteristic function, the Fourier inversion theorem allows us to retrieve this probability density function. This involves an integral that can often be efficiently evaluated numerically, enabling the computation of the option price.

Here’s a simplified outline of the process:

1. Model Specification: Define the model for the underlying asset’s price dynamics. This could be a Black-Scholes model, a Heston model (stochastic volatility), or a Merton jump-diffusion model, among others.
2. Characteristic Function Derivation: Determine the characteristic function of the log-asset price at maturity, typically denoted as φ(u,T). This is often model-specific and can be derived analytically or using numerical methods.
3. Fourier Inversion: Apply the Fourier inversion theorem to obtain the probability density function f(x) of the log-asset price. This involves the integral: f(x) = (1 / 2π) ∫ exp(-i * u * x) * φ(u,T) du where the integral is taken over the real line, and ‘i’ is the imaginary unit.
4. Option Price Calculation: Once the probability density function is known, the option price can be calculated by integrating the payoff function multiplied by the density function. For example, the price of a call option with strike K is: C = exp(-rT) ∫ max(exp(x) – K, 0) * f(x) dx where the integral is taken over all possible values of x, ‘r’ is the risk-free rate, and ‘T’ is the time to maturity.

The main advantage of using Fourier inversion is its speed and efficiency, especially for complex models where closed-form solutions are unavailable. Furthermore, it provides a versatile framework applicable to various option types, including European, Asian, and barrier options. However, careful selection of the integration range and numerical integration techniques is crucial to ensure accuracy and avoid spurious oscillations in the resulting density function.

In summary, Fourier inversion provides a valuable tool for financial engineers and researchers, allowing them to accurately price options under complex and realistic market conditions.

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