Analytic Formulae for Valuing Guaranteed Minimum Withdrawal Benefits in a Multi-Asset Framework

Abstract

本文試圖找到於多資產架構下保證最低提領給付（GMWB）之解析解公式。假設每個資產過程皆依據幾何布朗運動，藉運用反咖瑪與對數常態分配之平均值概念近似有相關性的對數常態隨機變數之和，我們得到GMWB之解析解公式。其數值顯示平均值概念之解析解公式表現精確，用於長期保單時亦然。評價GMWB時，此解析解公式具有顯著優勢與效率。

This study attempts to find the analytic formulae for valuing guaranteed minimum withdrawal benefits (GMWB) in a multi-asset framework. With the assumption that each equity process follows a geometric Brownian motion, we obtain the analytic formulae for GMWB by approximating the sum of the correlated lognormal variables using the averaging concept of reciprocal gamma and lognormal distributions. Numerical experiments show that analytic formulae of the averaging concept are accuracy, even for long-duration policies. Thus, the analytic formulae provide the significant advantage and efficiency in valuing GMWB.