Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model

Abstract

We consider the stochastic volatility model d S _t = σ _t S _t d W _t ,d σ _t = ω σ _t d Z _t , with (W _t ,Z _t ) uncorrelated standard Brownian motions. This is a special case of the Hull-White and the β=1 (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the n→∞ limit of a very large number of time steps of size τ, at fixed $\beta =\frac 12\omega ^{2}\tau n^{2}$ and $\rho ={\sigma _{0}^{2}}\tau$, and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of S _t . Under the Euler-Maruyama discretization for (S _t ,logσ _t ), the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.