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Fourier Analysis Methods in Finance

Alfred Ricker

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What is Fourier Analysis?

General Functions —> Trigonometric Functions

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Fourier Series - Represent a function as an infinite sum

of sines and cosines

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Fourier Transform - Operation that decomposes a function

into its “frequency domain.” Has a variety of applications,

including signal processing and solving

differential equations.

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DFT

-Maps a discrete set of data to its “frequency domain”

-May be possible to extract periodicity from (seemingly) Brownian market motion by looking at the power spectrum of the DFT

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FFT

-Clever algorithm that changes the number of operations of the DFT from N*N → NlogN

-Useful for applying the DFT to a large set of market data points.

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Freq. Domain of S&P 500

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Freq. Domain of GOOGLE

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What to make of this?

There doesn’t appear to be useful frequencies that could be applied to short term trading strategies.

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Patterns of market motion aren’t usefully obtained through the FFT

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Fourier Transform for Pricing Financial Derivatives

In the figure, K is the strike price, and x = lnS(t) where S is the asset price at time t.

A. Lewis method: transform the payoff functions to Fourier Space. The valuation of the option is then determined by integrating price density times the Fourier payoff using the convolution theorem and Parseval’s identity.

Divergent payoff Fourier transforms are normalized using an exponential damping factor.

One approach is to use the Fourier Transform as a method to solve the Black Scholes PDE

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Works Cited

Negrea, B. (2002). Option pricing with stochastic volatility: A closed-form solution using the Fourier transform. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.314406

Schmelzle, M. (2010). Option Pricing Formulae using Fourier Transform: Theory and Application.

Stádník, Bohumil & Raudeliuniene, Jurgita & Davidavičienė, Vida. (2016). Fourier Analysis for Stock Price Forecasting: Assumption and Evidence. Journal of Business Economics and Management. 17. 365-380. 10.3846/16111699.2016.1184180.

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