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- Title
Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model.
- Authors
Pirjol, Dan; Zhu, Lingjiong
- Abstract
We consider the stochastic volatility model d S = σ S d W , d σ = ω σ d Z , with ( W , Z ) 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 $\beta =\frac 12\omega ^{2}\tau n^{2}$ and $\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 . Under the Euler-Maruyama discretization for ( S ,log σ ), 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.
- Subjects
STOCHASTIC processes; WIENER processes; HULL-White model; MARKET volatility; ASYMPTOTIC expansions; EULER method; DISCRETIZATION methods
- Publication
Methodology & Computing in Applied Probability, 2018, Vol 20, Issue 1, p289
- ISSN
1387-5841
- Publication type
Article
- DOI
10.1007/s11009-017-9548-5