论文标题
在时间依赖的赫斯顿模型中的屏障选项的半分析定价
Semi-analytical pricing of barrier options in the time-dependent Heston model
论文作者
论文摘要
我们开发了一般的积分变换（GIT）方法，用于时间依赖性的Heston模型（也具有时间相关的屏障）中的定价屏障选项，其中期权价格以半分析形式表示为二维积分。该积分取决于未知函数$φ（t，v）$，它是移动边界$ s = l（t）$的解决方案的梯度，并求解了第二种也得出的线性混合伏特拉 - 弗雷姆方程。因此，我们将在数学金融中（WS，2021年）和相应论文中开发的一维GIT方法（Itkin，Lipton，Muravey，广义积分转换）开发到二维情况。换句话说，我们表明GIT方法可以扩展到随机波动率模型（两个具有不均匀相关性的驱动程序）。因此，这种2D方法自然继承了相应1D方法的所有优点，尤其是它们的速度和准确性。这个结果是新的，不仅在金融中，而且在物理学中都有各种应用。数值示例说明了与有限差异方法相比，该方法的高速和准确性。
We develop the general integral transforms (GIT) method for pricing barrier options in the time-dependent Heston model (also with a time-dependent barrier) where the option price is represented in a semi-analytical form as a two-dimensional integral. This integral depends on yet unknown function $Φ(t,v)$ which is the gradient of the solution at the moving boundary $S = L(t)$ and solves a linear mixed Volterra-Fredholm equation of the second kind also derived in the paper. Thus, we generalize the one-dimensional GIT method, developed in (Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, WS, 2021) and the corresponding papers, to the two-dimensional case. In other words, we show that the GIT method can be extended to stochastic volatility models (two drivers with inhomogeneous correlation). As such, this 2D approach naturally inherits all advantages of the corresponding 1D methods, in particular, their speed and accuracy. This result is new and has various applications not just in finance but also in physics. Numerical examples illustrate high speed and accuracy of the method as compared with the finite-difference approach.
