Jump-Diffusion Model for Excess Volatility in Asset Prices: Generalized Langevin Equation Approach

Subject Areas : Financial and Economic Modelling

Mahdi Salemi ¹ , Hassan Khodavaisi ²

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Received: 2024-10-29 Accepted : 2025-03-02 Published : 2025-01-17

Keywords: Excess Volatility , First Time Passage Problem, Generalized Langevin Equation (GLE) , Kramers-Moyal coefficients , Potential Function,

Abstract :

The excess volatility puzzle refers to the observation of returns that cannot be explained only by fundamentals, and this research attempts to explain such volatilities using the concept of endogenous jumps and modelling them based on the generalized Langevin equation. Based on stylized facts, price behaviour in financial markets is not simply a continuous process, but rather jumps are observed in asset prices that may be exogenous or endogenous. It is claimed that the source of exogenous jumps is news, and the source of endogenous jumps is internal interactions between the agents. The goal is to extract these endogenous jumps as a function of the state variable and time. For this purpose, the generalized Langevin equation is introduced and it is shown that the parameters of this model can be extracted based on the Kramers-Moyal coefficients. The results of self-consistency tests to evaluate the accuracy of the Kramers-Moyal method in extracting the generalized Langevin equation show that this method has good accuracy. In a practical application of the aforementioned method, Ethereum cryptocurrency price data was used between October 2017 and February 2024 with a sampling rate of one minute. By simulating the extracted dynamics, the probability distribution of the first time passage of this cryptocurrency from a specific level was calculated, and an examination of the price behavior of this asset shows that the aforementioned distribution was extracted with good accuracy. The potential function, which is calculated from the first KM coefficient, will be a quadratic parabola for the studied process, and as a result, we have a stable equilibrium at the zero point. Also using the extracted dynamics we show that this model has good out-of-sample prediction ability.

 

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