Discrete sums of geometric Brownian motions, annuities and Asian options

Dan Pirjol, Lingjiong Zhu

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)19-37
Number of pages19
JournalInsurance: Mathematics and Economics
Volume70
DOIs
StatePublished - 1 Sep 2016

Keywords

  • Annuities
  • Asian options
  • Exponential Lévy processes
  • Geometric stopping
  • Stochastic recurrence equations
  • Sum of geometric Brownian motions

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