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Dynamic Option Pricing with Endogenous Stochastic Arbitrage

Mauricio Contreras, Rodrigo Montalva, Rely Pellicer and Marcelo Villena

posted on 16 May 2016

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Only few efforts have been made in order to relax one of the key assumptions of the Black-Scholes model: the no-arbitrage assumption. This despite the fact that arbitrage processes usually exist in the real world, even thought they tend to be short-lived. The purpose of this paper is to develop an option pricing model with endogenous stochastic arbitrage, capable of modelling in a general fashion any future and underlying asset that deviate itself from its market equilibrium. Thus, this investigation calibrates empirically the arbitrage on the futures on the S&P 500 index using transaction data from September 1997 to June 2009, from here a specic type of arbitrage called arbitrage bubble, based on a t-step function, is identied and hence used in our model. The theoretical results obtained for Binary and European call options, for this kind of arbitrage, show that an investment strategy that take advantage of the identied arbitrage possibility can be dened, whenever it is possible to anticipate in relative terms the amplitude and timespan of the process. Finally, the new trajectory of the stock price is analytically estimated for a specific case of arbitrage and some numerical illustrations are developed. We find that the consequences of a finite and small endogenous arbitrage, not only change the trajectory of the asset price during the period when it started, but also after the arbitrage bubble is already gone. In this context, our model will allow us to calibrate the B-S model to that new trajectory even when the arbitrage already started.