Extracting Endogenous Jumps in Financial Markets Analytically Using Kramers-Moyal Method

Authors
Mehdi Salemi
Hassan khodavaisi
Faculty of Economics and Management, Department of Economics Urmia University
Abstract
Based on the stylized fact, the behavior of price in financial markets is not a continuous process, but we observe jumps in the price that can be endogenous or exogenous. it is claimed that the source of exogenous jumps is news, and the source of endogenous jumps is internal interactions between the market agents. Our goal is to extract these endogenous jumps as a function of the system state variable and time. First, by introducing the Langevin equation as the governing dynamics and linking its parameters with Kramers-Moyal coefficients, we show that these parameters can be extracted based on conditional moments. Next, we use the generalized Langevin equation to model the observed jumps in the data and show that in the new model, the drift coefficient is still equal to the first Kramers-Moyal coefficient, but the diffusion coefficient in this case is lower than the second Kramers-Moyal coefficient. In our model, the jump term consists of two components: jump rate and jump size. We show that these two new components can also be extracted based on Kramers-Moyal coefficients. Also, we introduce a practical criterion based on the fourth and sixth Kramers-Moyal coefficients to choose between the diffusion and jump-diffusion model. Applying the Kramers-Moyal method to extract the generalized Langevin equation shows that this method can accurately reconstruct the process. Tests to evaluate the accuracy of the reconstruction have been used from the information theory. In a practical application, we have extracted the price dynamics of an asset and then shown by simulation that this model is able to answer common statistical questions about stochastic processes with good accuracy. Also, by performing simulations, we show that this model has a good out-of-sample prediction ability. The potential function, which is calculated from the first KM coefficient, is a quadratic parabola for the studied process, and as a result, we have a stable equilibrium at the zero point.
Keywords
Asset Price Dynamics
First time passage distribution
Fokker-Planck equation
Generalized Langevin equation
Jump-diffusion model
Potential Function
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