Fourier transform is a mathematical technique used to express a function as a sum of harmonic functions. In quantitative finance, Fourier transform had been widely used in option pricing, risk management, volatility estimation and extraction, and signal processing.
One of the most popular applications of Fourier transform in quantitative finance is in option pricing. The core of option pricing is evaluating expected discounted payoffs of the option under different underlying asset prices. Fourier transform can simplify the computation of these expected payoffs by transforming the option pricing equation from the time domain to the frequency domain. This approach is called the Fourier transform method for option pricing.
The basic idea behind the Fourier transform method is to first transform the option pricing equation from the time domain to the frequency domain. Then use the fast Fourier transform algorithm (FFT) to numerically evaluate the transformed equation. Finally, transform the results back to the time domain.
Another application of Fourier transform in quantitative finance is in volatility estimation and extraction. Volatility is a key factor that determines the pricing of many financial instruments. However, volatility is not directly observable, and often needs to be estimated or extracted from data. Fourier transform provides a powerful tool to estimate and extract volatility from financial time series.
One popular method is the Fourier-based estimator proposed by Andersen and Bollerslev (1998). The basic idea is to decompose the financial time series into different frequency components using Fourier transform. Then, estimate the volatility for each frequency component separately. This approach can capture the frequency-dependent property of volatility and improve the accuracy of volatility estimation.
To summarize, Fourier transform is a versatile tool in quantitative finance that can be used for option pricing, risk management, volatility estimation, and signal processing. Its applications are not limited to the ones mentioned above and are continually evolving.