Based on the theoretical framework of Black–Scholes model, convergence inverse volatility problem based degenerate parabolic equation is studied. Being different from other problems in classical equations, we introduce some variable substitutions to convert original into an principal coefficient a bounded area, which unknown can be recovered and deficiencies caused by artificial truncation solv...

In this paper we introduce a new numerical method for the linear complementarity problems (LCPs) arising from two-asset Black–Scholes and Heston's stochastic volatility American options pricing. Based on barycenter dual mesh, class of finite volume (FVM) is proposed spatial discretization, coupled with backward Euler Crank–Nicolson schemes are employed time stepping partial differential equatio...

The Black-Scholes (1973) option pricing model provides the foundation for the modern theory of options valuation. In actual applications, however, the model has certain well-known deficiencies. For example, when calibrated to accurately price at-the-money options the Black-Scholes (1973) model often misprices deep in-the-money and deep out-of-themoney options. This model-anomalous behavior give...

We study the eeect of stochastic volatility on option prices. In the fast-mean reversion model for stochastic volatility of 5], we show that there is a full asymptotic expansion for the option price, centered at the Black-Scholes price. We show, however, that this price does not converge in a strong sense to Black-Scholes as the mean-reversion rate increases. We also introduce a general (possib...

In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial (PDEs), which is related to FBSDEs from Pardoux-Peng theory. The rely on a process by minimizing path-wise difference between two discrete processes, are defined time discretization DNN r...

In this paper, as far the authors know, for first time, a one-dimensional partial differential model is generalized using fractional operators and same principle that provides dimensional invariance of radial basis functions methodology, resulting in multidimensional can be solved numerical scheme functions. A proposed to solve numerically, on different node configurations, equations, both spac...

We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. ...