分数布朗运动环境下带红利支付的脆弱期权定价模型

The Pricing Model of Vulnerable Options with Dividend Payment under Fractional Brownian Motion

考虑到金融资产的长期依赖性,以及具有违约风险的可能性,假设期权标的资产价格和承约方资产的市场价值均服从分数布朗运动过程,研究了红利支付的脆弱期权定价问题。基于Delta对冲策略得到脆弱看涨期权价值满足的偏微分方程模型,运用双Mellin变换技巧将该模型转化为形式简单的常微分方程,推导出期权价格闭形式的解析公式,简化了模型的计算,解决了求解偏微分方程的复杂性问题。通过数值算例,将定价公式得到的期权价值与Monte-Carlo模拟值对比,验证了定价公式的正确性和有效性,并分析了模型中的各参数变化对欧式脆弱看涨期权价值的影响。相较于以往定价模型,该模型同时考虑了标的资产的长期依赖性和红利支付,因此可将其应用于标的资产具有这两种情形之一,且具有违约风险的债券定价中。

Considering the long-term dependence of financial assets and the possibility of default risk, this research explores the pricing problem of vulnerable option with dividend payment by assuming both the price of the underlying asset and the market value of the counterparty’s asset follow fractional Brownian motion process. Based on Delta-hedging strategy, the partial differential equation model for vulnerable call option value is obtained. Then the model is transformed into ordinary differential equation in simple form by using double Mellin transforms. The closed-form pricing formula is derived, which simplifies the calculation of the model and overcomes the complexity of solving partial differential equation. Through a numerical example, this research compares the option value obtained by the pricing formula with the Monte Carlo simulation value to verify its correctness and effectiveness, and analyzes the influence of the parameters in the model on the value of European vulnerable call options. Compared with previous models, this model involves both the long-term dependence and the dividend payment of the underlying asset. Therefore, it can be applied to the bond pricing with default risk for one of these two situations.