Correlated Log-Normal Random Variables under a Multiscale Volatility Model

Yong-Ki Ma,Maria L Gandarias

doi:10.1155/2021/5916312

Abstract

The study focuses on extending the fast mean-reversion volatility, which was developed by the author in a previous work, to the multiscale volatility model so that it can express a well-separated time scale. The leading-order term and first-order correction terms are analytically computed using the perturbation theory based on the Lie–Trotter operator splitting method. Finally, the study is concluded by deriving the numerical results that further validate the effectiveness of the model.

Highlights

A stochastic process provides a useful tool to analyze time series data and wide applications in many fields such as physics, finance, biotechnology, and telecommunication studies
The transition density function of a continuous-time process plays an important role in understanding and explaining the dynamics of the process
Finding analytical approximations to them is important as numerical methods, such as finite-difference method, Monte Carlo simulation, and Fourier inversion, because of much faster and precise at least under certain model parameter regime

Summary

Introduction

A stochastic process provides a useful tool to analyze time series data and wide applications in many fields such as physics, finance, biotechnology, and telecommunication studies. It is known that high volatility leads to high volatility and low volatility leads to low volatility Such phenomena are observed on different time scales and are characterized by the tendency of short-run volatility clustering (or fast mean-reversion) and long-run volatility clustering. These phenomena can only be delineated by multiscale models and not by single-scale models. This study focuses on extending the fast meanreversion volatility (FMRV) model, which is a previously proposed model by the author, to multiscale volatility (MSV) model This extension would aid the researchers and practitioners, such as Adrian and Rosenberg [9], Chernov et al [10], Fouque et al [11], and Gallant et al [12], in obtaining financial derivatives, especially those with derived empirical evidence. Approximate transition density under the MSV model is derived by the perturbation theory based on the Lie–Trotter operator splitting method

Asymptotic Analysis

Perturbation Theory Based on Lie–Trotter Operator Splitting Method

Numerical Experiment

Final Remarks