We are interested in problems related to forward starting options for Hull–White stochastic volatility model. Our objective is to obtain analytical, semi-analytical, or approximated expressions of its price for simulation. To obtain an analytical representation of the price, we use Yor’s formula. However, the analytical formula is difficult to implement. Next we consider semi-analytical expressions for the price. In order to have them, we use the tower property for conditional expectations with a certain filtration and explicitly calculate it. Then, we consider an expansion expression for the price using the semi-analytical expression to have a simple expression. The semi-analytical expressions and the expansion expression can reduce computational costs and standard errors when the Monte Carlo method is used. Finally, some numerical results are given to show their accuracy and efficiency.

我们对与 Hull-White 随机波动率模型的前向启动选项相关的问题感兴趣。我们的目标是获得模拟价格的分析、半分析或近似表达式。为了获得价格的解析表示，我们使用 Yor 公式。但是，解析公式很难实现。接下来我们考虑价格的半解析表达式。为了拥有它们，我们将塔属性用于具有特定过滤的条件期望并明确计算它。然后，我们考虑使用半解析表达式的价格扩展表达式以获得简单的表达式。使用蒙特卡罗方法时，半解析表达式和扩展表达式可以降低计算成本和标准误差。最后，