Calibration of ARIMA-GARCH-Model of Basic Asset Price Based on Market Option Quotes

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Abstract

The paper is devoted to the study of the possibility of calibrating the time series model of the underlying asset on the basis of market quotes of options on this asset. Market prices of options reflect the expectations of traders on the future dynamics of the underlying asset. At the beginning of the paper we present the general form of the ARIMA-GARCH time series model, as well as the form of the ARMA-GARCH model corresponding to the martingale risk-neutral probability measure. Next, the paper presents the formulation of the optimization problem of calibrating the risk-neutral model of logarithmic returns of the underlying asset based on the market prices of European options using the Monte Carlo method. The method of stochastic gradient approximation projection is applied to solve the problem. It is then shown how the form of the model changes when switching from a risk-neutral to a risk-averse probability measure under the assumption of the agent’s power utility function. The paper presents the results of calibrating the models on historical data on the quoted prices of the S&P500 index and the prices of European options on this index over the period from 2019 to 2023. Finally, the statistical test of Crnkovic–Drachman is performed to assess the accuracy of the calibrated models of the underlying asset returns for different future points in time.

About the authors

P. A. Arbuzov

Lomonosov Moscow State University

Email: arbuzov.parb@gmail.com
Moscow, Russia

D. Y. Golembiovskiy

Lomonosov Moscow State University

Email: dgolembiovskiy@yandex.ru
Moscow, Russia

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