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We consider the pricing problem related to payoffs that can havediscontinuities of polynomial growth. The asset price dynamic is modeled withinthe Black and Scholes framework characterized by a stochastic volatility termdriven by a fractional Ornstein-Uhlenbeck process. In order to solve theaforementioned problem, we consider three approaches. The first one consists ina suitable transformation of the initial value of the asset price, in order toeliminate possible discontinuities. Then we discretize both the Wiener processand the fractional Brownian motion and estimate the rate of convergence of therelated discretized price to its real value, the latter one being impossible tobe evaluated analytically. The second approach consists in considering theconditional expectation with respect to the entire trajectory of the fractionalBrownian motion (fBm). Then we derive a closed formula which involves onlyintegral functional depending on the fBm trajectory, to evaluate the price;finally we discretize the fBm and estimate the rate of convergence of theassociated numerical scheme to the option price. In both cases the rate ofconvergence is the same and equals $n^{-rH}$, where $n$ is a number of thepoints of discretization, $H$ is the Hurst index of fBm, and $r$ is theH\"{o}lder exponent of volatility function. The third method consists incalculating the density of the integral functional depending on the trajectoryof the fBm via Malliavin calculus also providing the option price in terms ofthe associated probability density.

Option pricing with fractional stochastic volatility and discontinuous payoff function of polynomial growth