英文标题：
《Valuation of Variable Annuities with Guaranteed Minimum Withdrawal
  Benefit under Stochastic Interest Rate》
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作者：
Pavel V. Shevchenko and Xiaolin Luo
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最新提交年份：
2017
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英文摘要：
  A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB) promises to return the entire initial investment through cash withdrawals during the contract plus the remaining account balance at maturity, regardless of the portfolio performance. Under the optimal(dynamic) withdrawal strategy of a policyholder, GMWB pricing becomes an optimal stochastic control problem that can be solved by backward recursion of Bellman equation. In this paper we develop a very efficient new algorithm for pricing these contracts in the case of stochastic interest rate not considered previously in the literature. Presently our method is applied to the Vasicek interest rate model, but it is generally applicable to any model when transition density or moments of the underlying asset and interest rate are known in closed form or can be evaluated efficiently. Using bond price as a numeraire the required expectations in the backward recursion are reduced to two-dimensional integrals calculated through a high order Gauss-Hermite quadrature applied on a two-dimensional cubic spline interpolation. Numerical results from the new algorithm for a series of GMWB contracts for both static and optimal cases are presented. As a validation, results of the algorithm are compared with the closed form solutions for simple vanilla options, and with Monte Carlo and finite difference results for the static GMWB. The comparison demonstrates that the new algorithm is significantly faster than finite difference or Monte Carlo for all the two-dimensional problems tested so far. For dynamic GMWB pricing, we found that for positive correlation between the underlying asset and interest rate, the GMWB price under the stochastic interest rate is significantly higher compared to the case of deterministic interest rate, while for negative correlation the difference is less but still significant.
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中文摘要：
具有最低提款保障福利（GMWB）的可变年金合同承诺，无论投资组合表现如何，都将通过合同期间的现金提款以及到期时的剩余账户余额来返还全部初始投资。在投保人的最优（动态）退出策略下，GMWB定价成为一个最优随机控制问题，可通过Bellman方程的反向递归求解。在本文中，我们开发了一个非常有效的新算法，用于在文献中未考虑的随机利率情况下对这些合同进行定价。目前，我们的方法适用于Vasicek利率模型，但当标的资产和利率的转移密度或矩以封闭形式已知或可以有效评估时，它通常适用于任何模型。以债券价格为基准，将反向递归中所需的期望值简化为二维积分，通过应用于二维三次样条插值的高阶高斯-埃尔米特求积计算。给出了静态和最优情况下一系列GMWB契约的新算法的数值结果。作为验证，将该算法的结果与简单香草期权的闭式解进行了比较，并与静态GMWB的蒙特卡罗和有限差分结果进行了比较。比较表明，对于迄今为止测试的所有二维问题，新算法明显快于有限差分法或蒙特卡罗法。对于动态GMWB定价，我们发现，对于标的资产和利率之间的正相关，随机利率下的GMWB价格显著高于确定性利率下的GMWB价格，而对于负相关，差异较小但仍然显著。
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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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