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

We consider the stochastic volatility model dS_t = σ_tS_tdW_t,dσ_t = ωσ_tdZ_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 β=12ω2τn2 and ρ=σ02τ, 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.