Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

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.

Diffusion–Advection Equations on a Comb: Resetting and Random Search

“…It results in turbulent diffusion, which is characterized by the log-normal distribution and exponential growth of the MSD in time [29]. This behavior is analogous to one-dimensional geometric Brownian motion [29], which is used in the Black-Scholes model for option pricing [59,60]. The FATD is the Lévy-Smirnov distribution, and the process is suitable for searching for long-distance targets.…”

Statistical Dynamics of Wealth Inequality in Stochastic Models of Growth

“…To answer this question, we consider the solution to the Fokker-Planck equation (FPE) for the time dependent probability of resources r for each agent. The solution with initial conditions r(0) = r 0 (a delta function at the individual level) is a lognormal distribution [25] with time dependent mean and variance. This can be re-written as a Gaussian distribution of ln r as…”

“…In such dynamical picture, initial correlations will also take on dynamical properties in ways that we stated anticipating here. In recognizing recent research on the effects of heterogeneous and dynamical growth on distributions of wealth [25,[37][38][39][40], we seek a theoretical framework for wealth dynamics that both complements these phenomenological approaches and incorporates strategic agent behavior in statistical environments. These topics will be presented in future work.…”

Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity

“…One can use the renewal approach as described above (also see [7]) to show that that the probability density function (PDF) has a stationary solution. In particular, the PDF with resetting (r > 0) can be written as P r (x,t|x 0 ) = e −rt P 0 (x,t|x 0 ) + r t 0 e −ru P 0 (x, u|x 0 ) du, (5) where P 0 (x,t|x 0 ) is the PDF of the reset-free (r = 0) income dynamics [24,25] 5), and by using the limit s → 0, we find the steady state, P ss r (x|x 0 ) = lim t→∞ P r (x,t|x 0 ) = lim s→0 s Pr (x,t|x 0 ) = r P0 (x, r|x 0 ). Following this, it can be shown that the stationary distribution follows the power law…”