LAUSR

We consider the stochastic volatility model dSt = σtStdWt,dσt = ωσtdZt, with (Wt,Zt) 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$\beta =\frac 12\omega ^{2}\tau n^{2}$ and ρ=σ02τ$\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 St. Under the Euler-Maruyama discretization for (St,logσt), 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.