论文标题
广义的几何布朗运动：理论和适用于期权定价的应用
Generalised geometric Brownian motion: Theory and applications to option pricing
论文作者
论文摘要
经典期权定价方案假定金融资产的价值遵循几何布朗尼运动（GBM）。然而，越来越多的研究表明，由于将其特性与经验分布进行比较时发现的不规则性，因此简单的GBM轨迹并不是资产动态的足够代表。作为解决方案，我们开发了GBM的概括，其中记忆内核的引入批判性地决定了随机过程的行为。我们找到了矩，对数时刻和周期日对数返回的期望的一般表达式，并使用从属方法获得相应的概率密度函数。特别是，我们考虑使用SGBM的二次GBM（SGBM），GBM和SGBM的混合物以及SGBM的混合物。我们利用由此产生的广义GBM（GGBM）来检查欧洲呼叫选项的定价中选定的一组内核的经验性能。我们的结果表明，内核的性能最终取决于选项的成熟度及其金钱。
Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics due to irregularities found when comparing its properties with empirical distributions. As a solution, we develop a generalisation of GBM where the introduction of a memory kernel critically determines the behavior of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and obtain the corresponding probability density functions by using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.
