[HTML] from mdpi.com Full View Generalised geometric Brownian motion: Theory and applications to option pricing
Journal
Entropy
Date Issued
2020-12-18
Author(s)
Stojkoski, Viktor
Sandev, Trifce
Kocarev, Ljupco
Metzler, Ralf
Abstract
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.
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.
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