Nonlinear Parabolic Stochastic Partial Differential Equation with Application to Finance

Abstract

We study a nonlinear parabolic stochastic partial differential equation (SPDE) with multiplicative space–time white noise. The noise coecient is the square root of the unknown, with respect to which such nonlinearity is H ̈older but not Lipschitz. The SPDE’s deterministic part includes also a first order quadratic nonlinearity. Mathematically speaking this space–time SPDE reduces to one of two stochastic differential equations (SDEs) appearing in the celebrated Heston and Cox–Ingersoll–Ross models describing the stock price volatility evolution; we therefore propose and test its use as a possible tool in understanding the so-called “volatility smile” observed in the implied volatility upon inverting the Black–Scholes–Merton model against option market data [Rouah, 2013][Gatheral and Taleb, 2011]. Similar equations arise in other fields as well, for example, in surface growth modelling with or without a random forcing term. A typical example is the nonlinear stochastic Kardar–Parisi–Zhang equation (KPZ) whose unique solvability re- quired the establishment of regularity structure methods [Hairer, 2013]. In principle, the model we herein propose lends itself to analysis via the regularity structures approach, but it exhibits better stability properties than KPZ thanks to a favourable sign in the first order quadratic nonlinearity and a resulting crucial energy identity; owing to this we are thus able to take a more straight-forward approach using energy methods and Galerkin approximations and show the SPDE is well posed in one spatial dimension, which is the relevant case in financial model- ling. In line with the Galerkin approximation idea, we introduce an Euler–Maruyama numerical scheme to approximate the solution [Lord et al., 2014] which we use to close our work by looking at possible applications of the extended Heston model we propose. This extended Heston model includes a new independent variable (which acts as “space” in “space–time”) which signifies the option’s strike price on which the implied volatility depends