Journal of Mathematical Finance, 2017, 7, 633-656 http://www.scirp.org/journal/jmf ISSN Online: 2162-2442 ISSN Print: 2162-2434 DOI: 10.4236/jmf.2017.73033 July 18, 2017 Theories on the Relationship between Price Process and Stochastic Volatility Matrix with Compensated Poisson Jump Using Fourier Transforms Perpetual Saah Andam 1 , Joseph Ackora-Prah 1 , Sure Mataramvura 2 1 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana 2 Department of Actuarial Science, University of Cape Town, Cape Town, South Africa Abstract Investors find it difficult to determine the movement of prices of stock due to volatility. Empirical evidence has shown that volatility is stochastic which contradicts the Black-Scholes framework of assuming it to be constant. In this paper, stochastic volatility is estimated theoretically in a model-free way without assuming its functional form. We show proof of an identity estab- lishing an exact expression for the volatility in terms of the price process. This theoretical presentation for estimating stochastic volatility with the presence of a compensated Poisson jump is achieved by using Fourier Transform with Bohr’s convolution and quadratic variation. Our method establishes the addi- tion of a compensated Poisson jump to a stochastic differential equation using Fourier Transforms around a small time window from the observation of a single market evolution. Keywords Stochastic Differential Equation, Fourier Transform, Compensated Poisson Jump 1. Introduction Volatility measures uncertainty of returns which plays a major role in cash flows from selling assets at a precise future date. It is very essential in financial markets due to price fluctuations, prediction of stock prices, option pricing, portfolio management and hedging. Decision and policy makers depend on volatility to determine the bullish and bearish nature of the market to avoid loss. The varying How to cite this paper: Andam, P.S., Ackora-Prah, J. and Mataramvura, S. (2017) Theories on the Relationship between Price Process and Stochastic Volatility Matrix with Compensated Poisson Jump Using Fourier Transforms. Journal of Mathema- tical Finance, 7, 633-656. https://doi.org/10.4236/jmf.2017.73033 Received: May 23, 2017 Accepted: July 15, 2017 Published: July 18, 2017 Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access