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

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 β = 1 2 ω 2 τ n 2 $\beta =\frac 12\omega ^{2}\tau n^{2}$ and ρ = σ 0 2 τ $\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.