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

Dan Pirjol, Lingjiong Zhu

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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 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 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.

Original languageEnglish
Pages (from-to)289-331
Number of pages43
JournalMethodology and Computing in Applied Probability
Volume20
Issue number1
DOIs
StatePublished - 1 Mar 2018

Keywords

  • Central limit theorems
  • Critical exponent
  • Large deviations
  • Linear stochastic recursion
  • Lyapunov exponent
  • Phase transitions

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