Heston和Black-Scholes随机利率模型下GMWB的定价与套期保值
Pricing and Hedging GMWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models
摘要：
保证最低提取收益（Guaranteed Minimum Withdrawal Benefit，简称GMWB）的评估已经引起了学术界和现实世界金融市场的极大关注。正如Yang和Dai所说，Black和Scholes框架似乎不适合如此长的成熟度产品。Chen Vetzal和Forsyth也表明，这些产品的价格对利率和波动率参数非常敏感。本文提出了一个随机波动率模型（Heston模型）和一个带有随机利率的布莱克-斯科尔斯模型（Hull White模型）。为此，我们提出了四个数值方法定价GMWB变量年金：混合树有限差分法和混合蒙特卡罗方法，ADI有限差分格式，和标准蒙特卡罗方法。这些方法用于确定最流行的GMWB合约版本的无套利费用，并计算套期保值中使用的希腊人。同时考虑了常数提款、最优退保和最优提款策略。数值结果表明，经济，合同和长寿假设的无套利费用的敏感性。
英文摘要：
Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Yang and Dai, the Black and Scholes framework seems to be inappropriate for such a long maturity products. Also Chen Vetzal and Forsyth in showed that the price of these products is very sensitive to interest rate and volatility parameters. We propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GMWB variables annuities: a hybrid tree-finite difference method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a Standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GMWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal, optimal surrender and optimal withdrawal strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.
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