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

Dan Pirjol () and Lingjiong Zhu ()
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Lingjiong Zhu: Florida State University

Methodology and Computing in Applied Probability, 2018, vol. 20, issue 1, 289-331

Abstract: Abstract 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.

Keywords: Linear stochastic recursion; Lyapunov exponent; Phase transitions; Critical exponent; Large deviations; Central limit theorems; 60G99; 60K99; 82B26; 60F10; 60F05 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s11009-017-9548-5

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