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

We consider the pricing problem related to payoffs that can have discontinuities of polynomial growth. The asset price dynamic is modeled within the Black and 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, the latter one being impossible to be evaluated analytically. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Then we derive a closed formula which involves only integral functional depending on the fBm trajectory, to evaluate the price; finally we discretize the fBm and estimate the rate of convergence of the associated numerical scheme to the option price. In both cases the rate of convergence is the same and equals $n^{-rH}$, where $n$ is a number of the points of discretization, $H$ is the Hurst index of fBm, and $r$ is the Hölder exponent of volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus also providing the option price in terms of the associated probability density.