The fractional Cox-Ingersoll-Ross process of the form X t = Z 2 t , where the process (Z t ) t≥0 is given by dZt≥0 is a fractional Brownian motion with Hurst parameter H ∈ (0, 1) (which is an extension of the classical Cox-Ingersoll-Ross process) was recently studied by Mishura and Yurchenko-Tytarenko [Fractional Cox-Ingerson-Ross process with non-zero "mean", Modern Stochastics: Theory and Applications 5(1), (2018) 99-111]. They proved that (X t ) t≥0 satisfies the equation dX t = (k − θX t )dt + σ √ X t • dW H t where • refers to the Stratonovich integral. Moreover, for H > 1/2, (X t ) t≥0 never hits zero and for H < 1/2, the probability of hitting zero tends to 0 provided the drift coefficient k increases to +∞. In this paper, we extend these results to the general process defined by+ under mild conditions weaker than previously considered in the literature. In the case where H < 1/2, we consider a sequence of increasing functions (f n ) and we prove that the probability of hitting zero tends to 0 as n → ∞. These results are illustrated with some simulations using the generalisation of the extended Cox-Ingersoll-Ross process.
This paper defines fractional Heston-type (fHt) model as an arbitrage-free financial market model with the infinitesimal return volatility described by the square of a single stochastic equation with respect to fractional Brownian motion with Hurst parameter H ∈ (0, 1). We extend the idea of Alos and Ewald (2008) [Alos, E., & Ewald, C. O. (2008). Malliavin differentiability of the Heston volatility and applications to option pricing. Advances in Applied Probability, 40(1), 144-162.] to prove that fHt model is Malliavin differentiable and deduce an expression of expected payoff function having discontinuity of any kind. Some simulations of stock price process and option prices are performed.
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