Application of microlocal analysis to an inverse problem arising from financial markets

Abstract

One of the most interesting problems discerned when applying the Black-Scholes model to financial derivatives, is reconciling the deviation between expected and observed values. In our recent work, we derived a new model based on the Black-Scholes model and formulated a new mathematical approach to an inverse problem in financial markets. In this paper, we apply microlocal analysis to prove a uniqueness of the solution to our inverse problem. While microlocal analysis is used for various models in physics and engineering, this is the first attempt to apply it to a model in financial markets. First, we explain our model, which is a type of arbitrage model and illustrate our new mathematically applying microlocal analysis to the integral equation, we prove our uniqueness of the solution to our new mathematical model in financial markets. Finally we propose and test the numerical algorithm for our model.