TY - JOUR
T1 - Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model
AU - Pirjol, Dan
AU - Zhu, Lingjiong
N1 - Publisher Copyright:
© 2017, Springer Science+Business Media New York.
PY - 2018/3/1
Y1 - 2018/3/1
N2 - 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.
AB - 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.
KW - Central limit theorems
KW - Critical exponent
KW - Large deviations
KW - Linear stochastic recursion
KW - Lyapunov exponent
KW - Phase transitions
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U2 - 10.1007/s11009-017-9548-5
DO - 10.1007/s11009-017-9548-5
M3 - Article
AN - SCOPUS:85014176279
SN - 1387-5841
VL - 20
SP - 289
EP - 331
JO - Methodology and Computing in Applied Probability
JF - Methodology and Computing in Applied Probability
IS - 1
ER -