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Contreras, M., Echeverria, J., Pena, J. P., & Villena, M. (2020). Resonance phenomena in option pricing with arbitrage. Physica A, 540, 21 pp.
Abstract: In this paper, we want to report an interesting resonance phenomena that appears in option pricing, when the presence of arbitrage is incorporated explicitly into the Black-Scholes model. In Contreras et al. (2010), the authors after analyse empirical financial data, determines that the mispricing between the empirical and the Black-Scholes prices can be described by Heaviside type function (called an arbitrage bubble there). These bubbles are characterised by a finite time span and an amplitude which measures the price deviation from the Black-Scholes model. After that, in Contreras et al. (2010), the Black-Scholes equation is generalised to incorporates explicitly these arbitrage bubbles, which generates an interaction potential that changes the usual Black-Scholes free dynamics completely. However, an interesting phenomena appears when the amplitude of the arbitrage bubble is equal to the volatility parameter of the Black-Scholes model: in that case, the potential becomes infinite, and option pricing decrease abruptly to zero. We analyse this limit behaviour for two situations: a European and a barrier option. Also, we perform an analytic study of the propagator in each case, to understand the cause of the resonance. We think that it resonance phenomena could to help to understand the origin of certain financial crisis in the option pricing area. (C) 2019 Elsevier B.V. All rights reserved.