论文标题
与时间相关的CEV和CIR模型中的障碍选项的半关闭形式价格
Semi-closed form prices of barrier options in the time-dependent CEV and CIR models
论文作者
论文摘要
我们继续进行一系列论文，其中在基础上写的障碍选项的价格（动态遵循一个因时间依赖的系数和屏障）以半锁定形式获得的一个因素随机模型，请参见（Carr and Itkin，2020年，Itkin，Itkin，Itkin和Muravey，2020）。本文将此方法扩展到零企业键的CIR模型，并将其用于股票的CEV模型，该股票用作屏障选项的相应基础。我们描述了两种方法。其中一种是将热方程的热能方法的概括为贝塞尔过程，因此我们称其为贝塞尔电位的方法。我们还提出了一个通用方案，如何使用无时间的系数构建任何线性差分运算符的潜在方法。第二个是广义积分变换的方法，它也扩展到贝塞尔过程。在所有情况下，半锁定的解决方案都意味着首先，我们需要在数值上求解第二种的线性伏特拉方程，然后将期权价格表示为一维积分。我们证明，计算方法比向后和前进有限差异方法更有效，同时提供了更好的准确性和稳定性。同样，这表明这两种方法都不重复，而是相互补充，因为一个方法在较小的成熟度中提供了非常准确的结果，而另一种则在高到期的情况下提供了非常准确的结果。
We continue a series of papers where prices of the barrier options written on the underlying, which dynamics follows some one factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form, see (Carr and Itkin, 2020, Itkin and Muravey, 2020). This paper extends this methodology to the CIR model for zero-coupon bonds, and to the CEV model for stocks which are used as the corresponding underlying for the barrier options. We describe two approaches. One is generalization of the method of heat potentials for the heat equation to the Bessel process, so we call it the method of Bessel potentials. We also propose a general scheme how to construct the potential method for any linear differential operator with time-independent coefficients. The second one is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to solve numerically a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. We demonstrate that computationally our method is more efficient than both the backward and forward finite difference methods while providing better accuracy and stability. Also, it is shown that both method don't duplicate but rather compliment each other, as one provides very accurate results at small maturities, and the other one - at high maturities.
