Perturbative Approach on Financial Markets

TL;DR

This paper introduces the Perturbed Black Scholes (PBS) model, using Dirichlet Forms to analyze the transition between complete and incomplete markets, capturing market features like smile effects and bid-ask spreads.

Contribution

It develops a novel perturbative approach with Dirichlet Forms to model market imperfections and links error theory with utility functions.

Findings

PBS model reproduces smile effects and bid-ask spreads

Volatility function derived for local-volatility model equivalent to PBS

Connection established between Dirichlet Forms error theory and utility functions

Abstract

We study the point of transition between complete and incomplete financial models thanks to Dirichlet Forms methods. We apply recent techniques, developped by Bouleau, to hedging procedures in order to perturbate parameters and stochastic processes, in the case of a volatility parameter fixed but uncertain for traders; we call this model Perturbed Black Scholes (PBS) Model. We show that this model can reproduce at the same time a smile effect and a bid-ask spread; we exhibit the volatility function associated to the local-volatility model equivalent to PBS model when vanilla options are concerned. Lastly, we present a connection between Error Theory using Dirichlet Forms and Utility Function Theory.