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Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth

  • Viktor Bezborodov
  • Luca Di Persio
  • Yuliya Mishura
Article

Abstract

We consider the pricing problem related to payoffs of polynomial growth that can have discontinuities of the 1st kind. The asset price dynamic is modeled within the Black-Scholes framework characterized by a stochastic volatility term driven by a fractional Ornstein-Uhlenbeck process. In order to solve the aforementioned problem, we consider three approaches. The first one consists in a suitable transformation of the initial value of the asset price, in order to eliminate possible discontinuities. Then we discretize both the Wiener process and the fractional Brownian motion and estimate the rate of convergence of the related discretized price to its real value whose closed-form analytical expression is usually difficult to obtain. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Here we derive a presentation for the option price which involves only an integral functional depending on the fBm trajectory, and then discretize the fBm and estimate the rate of convergence of the associated numerical scheme. In both cases the rate of convergence is the same and equals nrH, where n is the partition size, H is the Hurst index of the fBm, and r is the Hölder exponent of the volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus and providing the option price in terms of the associated probability density.

Keywords

Option pricing Stochastic volatility Black–Scholes model Wiener process Discontinuous payoff function Polynomial growth Rate of convergence Discretization Conditioning Malliavin calculus Stochastic derivative Skorokhod integral 

Mathematics Subject Classification (2010)

91G20 91B24 91B25 91G60 60G22 60H07 

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Authors and Affiliations

  1. 1.The Department of Computer Science Universita degli Studi di Verona - Cà Vignal 2The University of VeronaVeronaItaly
  2. 2.The Faculty of Mechanics and Mathematics Volodymyrska 64Taras Shevchenko National University of KyivKyivUkraine

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