英文标题：
《Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options》
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作者：
Dan Pirjol, Lingjiong Zhu
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最新提交年份：
2016
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英文摘要：
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 L\\\'evy 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.
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中文摘要：
几何布朗运动的离散和在保险随机年金建模中起着重要作用。它在数学金融学中的亚洲期权定价中也起着关键作用。本文研究了几何布朗运动的无穷和、几何布朗运动的几何停止时间和几何布朗运动的有限和的概率分布。这些结果被推广到指数Levy过程的离散和。我们推导了尾部渐近，并数值计算了渐近分布函数。我们将这些结果与已知的几何布朗运动在指数分布时间内的连续时间积分的结果进行了比较。结果以离散支付的终身年金和亚洲期权的数值例子进行了说明。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Pricing of Securities 证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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