Sabr Type Stochastic Volatility Operator in Hilbert Space

A Stochastic Volatility Alternative to SABR

We present two new stochastic–volatility models in which option prices for European plain vanilla options have closed–form expressions. The models are motivated by the wellknown SABR model but use modified dynamics of the underlying asset. The asset process is modelled as a product of functions of two independent stochastic processes: a Cox– Ingersoll–Ross process and a geometric Brownian motio...

Operator Preconditioning in Hilbert Space

2010 1 Introduction The numerical solution of linear elliptic partial differential equations consists of two main steps: discretization and iteration, where generally some conjugate gradient method is used for solving the finite element discretization of the problem. However, when for elliptic problems the dis-cretization parameter tends to zero, the required number of iterations for a prescrib...

Efficient unbiased simulation scheme for the SABR stochastic volatility model

The stochastic Alpha Beta Rho stochastic volatility (SABR-SV) model is widely used in the financial industry for the pricing of fixed income instruments. In this paper we develop an unbiased simulation scheme for the SABR-SV model, which deals efficiently with (undesired) possible negative values, the martingale property of the discrete scheme and the discretization bias of commonly used Euler ...

Operator Extensions on Hilbert Space

Stochastic differential inclusions of semimonotone type in Hilbert spaces

In this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$ is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition. We define the It^{o} stochastic integral of operator set-valued stochastic pr...