Stojkoski Viktor, Sandev Trifce, Basnarkov Lasko, Kocarev Ljupco, Metzler Ralf
Faculty of Economics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia.
Research Centre for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia.
Entropy (Basel). 2020 Dec 18;22(12):1432. doi: 10.3390/e22121432.
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 investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions 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) in order 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.
经典期权定价方案假设金融资产的价值遵循几何布朗运动（GBM）。然而，越来越多的研究表明，简单的GBM轨迹不足以代表资产动态，因为将其性质与经验分布进行比较时发现了不规则性。作为一种解决方案，我们研究了GBM的一种推广形式，其中引入记忆核关键地决定了随机过程的行为。我们找到了矩、对数矩以及周期对数收益率期望的一般表达式，然后使用从属方法得到相应的概率密度函数。特别地，我们考虑亚扩散GBM（sGBM）、缓和sGBM、GBM和sGBM的混合形式以及sGBM的混合形式。我们利用所得的广义GBM（gGBM）来检验一组选定核在欧式看涨期权定价中的实证表现。我们的结果表明，核的表现最终取决于期权的到期日及其实虚值状态。
