TY - JOUR
T1 - Discrete sums of geometric Brownian motions, annuities and Asian options
AU - Pirjol, Dan
AU - Zhu, Lingjiong
N1 - Publisher Copyright:
© 2016 Elsevier B.V.
PY - 2016/9/1
Y1 - 2016/9/1
N2 - The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of geometric Brownian motions with geometric stopping time, and the finite sum of the geometric Brownian motions. These results are extended to the discrete sum of the exponential Lévy process. We derive tail asymptotics and compute numerically the asymptotic distribution function. We compare the results against the known results for the continuous time integral of the geometric Brownian motion up to an exponentially distributed time. The results are illustrated with numerical examples for life annuities with discrete payments, and Asian options.
AB - The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of geometric Brownian motions with geometric stopping time, and the finite sum of the geometric Brownian motions. These results are extended to the discrete sum of the exponential Lévy process. We derive tail asymptotics and compute numerically the asymptotic distribution function. We compare the results against the known results for the continuous time integral of the geometric Brownian motion up to an exponentially distributed time. The results are illustrated with numerical examples for life annuities with discrete payments, and Asian options.
KW - Annuities
KW - Asian options
KW - Exponential Lévy processes
KW - Geometric stopping
KW - Stochastic recurrence equations
KW - Sum of geometric Brownian motions
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U2 - 10.1016/j.insmatheco.2016.05.020
DO - 10.1016/j.insmatheco.2016.05.020
M3 - Article
AN - SCOPUS:84976342107
SN - 0167-6687
VL - 70
SP - 19
EP - 37
JO - Insurance: Mathematics and Economics
JF - Insurance: Mathematics and Economics
ER -