Developing Fractional Stochastic Differential Equations with Stochastic Interest Rate Following the CIR Model

In the light of present-day research, as we all know, the financial market owns the characteristics of self-similarity and long-range dependence and Fractional Brownian motion has these properties. From that point, The model with fractional stochastic volatility could be more businesslike than the model with standard Brownian motion. Furthermore, for more hardheaded models we can use hybrid models by incorporating stochastic interest rate into stochastic volatility. In this paper, we bridge the gap in the financial literature by proposing a new mathematical model, more general and realistic model based on concocting the stochastic interest rate following the CIR process together with fractional stochastic volatility. By using the replication technique, Itos lemma, and Malliavin calculus, we derive a partial differential equation for valuing European options, also we provide many numerical examples in order to illustrate the model solutions