TY - JOUR
T1 - On the distribution of the time-integral of the geometric Brownian motion
AU - Nándori, Péter
AU - Pirjol, Dan
N1 - Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2022/3/1
Y1 - 2022/3/1
N2 - We study the numerical evaluation of several functions appearing in the small time expansion of the distribution of the time-integral of the geometric Brownian motion as well as its joint distribution with the terminal value of the underlying Brownian motion. A precise evaluation of these distributions is relevant for the simulation of stochastic volatility models with log-normally distributed volatility, and Asian option pricing in the Black–Scholes model. We derive series expansions for these distributions, which can be used for numerical evaluations. Using tools from complex analysis, we determine the convergence radius and large order asymptotics of the coefficients in these expansions. We construct an efficient numerical approximation of the joint distribution of the time-integral of the gBM and its terminal value, and illustrate its application to Asian option pricing in the Black–Scholes model.
AB - We study the numerical evaluation of several functions appearing in the small time expansion of the distribution of the time-integral of the geometric Brownian motion as well as its joint distribution with the terminal value of the underlying Brownian motion. A precise evaluation of these distributions is relevant for the simulation of stochastic volatility models with log-normally distributed volatility, and Asian option pricing in the Black–Scholes model. We derive series expansions for these distributions, which can be used for numerical evaluations. Using tools from complex analysis, we determine the convergence radius and large order asymptotics of the coefficients in these expansions. We construct an efficient numerical approximation of the joint distribution of the time-integral of the gBM and its terminal value, and illustrate its application to Asian option pricing in the Black–Scholes model.
KW - Asymptotic expansions
KW - Complex analysis
KW - Numerical approximation
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U2 - 10.1016/j.cam.2021.113818
DO - 10.1016/j.cam.2021.113818
M3 - Article
AN - SCOPUS:85116030871
SN - 0377-0427
VL - 402
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
M1 - 113818
ER -