Moderate deviations for Euler-Maruyama approximation of Hull-White stochastic volatility model

Abstract

We consider the Euler-Maruyama discretization of stochastic volatility model

{{\displaystyle dS}}_t={{\displaystyle \sigma }}_t{{\displaystyle S}}_t{{\displaystyle dW}}_t,{{\displaystyle d\sigma }}_t=\omega {{\displaystyle \sigma }}_t{{\displaystyle \mathrm{d}Z}}_t,t\in [\mathrm{0},T]

which has been widely used in nancial practice, where {{\displaystyle W}}_t,{{\displaystyle Z}}_t,t\in [\mathrm{0},T] are two uncorrelated standard Brownian motions. Using asymptotic analysis techniques, the moderate deviation principles for log S_n (or log \left|{{\displaystyle S}}_n\right| in case S_n is negative) are obtained as n\to \infty under different discretization schemes for the asset price process S_t and the volatility process {{\displaystyle \sigma }}_t: Numerical simulations are presented to compare the convergence speeds in different schemes.

Key words： Euler-Maruyama discretization; Hull-White stochastic volatility model; moderate deviation principle

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