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

Abstract

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.

Approximating Expected Value of an Option With Non-Lipschitz Payoff in Fractional Heston-Type Model

“…Arbitrage-free property and incompleteness. For the market (1)-( 3), we can obtain the following result which is similar to the one in [6], Theorem 4. (i) It is arbitrage-free and incomplete.…”

“…They allow to reflect the so-called "memory phenomenon" of the market (for more details on market models with memory see, for instance, [3,13,35]). In this context, we should also mention [7,9,10] and [6].…”

“…An analogue of the model ( 2), (3) was considered in [6] with fractional Ornstein-Uhlenbeck process instead of Y . However, Ornstein-Uhlenbeck process can take negative values with positive probability which is a notable drawback for a stochastic volatility model.…”

“…In Section 3, several important properties of both price and volatility processes are presented and the arbitrage-free property is discussed. In Section 4 we apply the Malliavin calculus techniques, following [1] and [6], to obtain the formula for Ef (S T ) that does not contain discontinuities (which are allowed for the payoff function f ). In Section 5, we study the rate of convergence Ef (Ŝ n T ) → Ef (S T ), whereŜ n is based on inverse Euler approximation scheme for fractional CIR process presented in [18].…”

General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory

“…As for stochastic volatility with memory, it is not considered in the present paper, but we can refer the reader to Gatheral et al (2018) and Bezborodov et al (2019), among many others.…”