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

We consider the stochastic volatility model dS(t) = sigma(t)S(t)dW(t), d sigma(t) = omega sigma(t)dZ(t), with (W-t, Z(t)) uncorrelated standard Brownian motions. This is a special case of the Hull-White and the ss = 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 -> infinity limit of a very large number of time steps of size tau, at fixed ss = 1/2 omega(2)tau n(2) and rho = sigma(2)(0)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 St. Under the Euler-Maruyama discretization for (St, log st), 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.