ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ ЦІНОВОЇ  ДИНАМІКИ В РОЗШИРЕНІЙ МОДЕЛІ ГЕСТОНА

Vasyl Yanishevskyi; Olha Pelekh

doi:10.32782/2524-0072/2024-69-10

Vasyl Yanishevskyi Lviv Polytechnic National University https://orcid.org/0000-0002-0449-6686
Olha Pelekh Lviv Polytechnic National University https://orcid.org/0009-0006-5827-6829

DOI: https://doi.org/10.32782/2524-0072/2024-69-10

Keywords: stochastic equations, Brownian motion, option pricing, Black-Scholes model, extended Heston model, Cox-Ingersoll-Ross stochastic process, Euler and Milstein schemes

Abstract

The paper presents a numerical study of the extended Heston model for asset and derivative pricing in financial engineering. Building on the foundational one-dimensional Black-Scholes model, the classical Heston model introduces stochastic volatility through two stochastic differential equations: one describing the dynamics of asset prices and the other governing volatility. The extended Heston model further incorporates the CIR stochastic process to represent the dynamics of the interest rate. The development of multi-dimensional models, such as the extended Heston model, has necessitated the broader application of numerical methods for analyzing stochastic pricing models. Analytical solutions for models involving multiple stochastic equations are often difficult to obtain or may not exist. In the extended Heston model, analytical solutions are known only in the absence of correlation in the Wiener process for the interest rate. These solutions for the conditional probability density of the variable x(t)=ln((S(t))/S_0 ) are used to validate numerical solutions of the model's stochastic equations. The paper employs the Euler and Milstein schemes to numerically analyze the extended Heston model, generating arrays of values for asset prices, volatility, and interest rates. Interpolation methods, based on histogram data, are used to estimate the conditional probability densities of the asset price x(t) under different parameter configurations. Due to the high dimensionality and numerous parameters of the extended Heston model, exploring the full parameter space to study asset price dynamics is a complex task. This study focuses on the influence of the correlation coefficient (ρ_r) of the Wiener process in the interest rate equation on the conditional probability density of asset prices. The findings reveal that, for both positive and negative values of ρ_r, the effect is negligible, resulting in minimal changes to the conditional probability density curve. Consequently, the impact of the stochastic dynamics of interest rates on option pricing is primarily determined by the discount factor, which defines the temporal structure of interest rates within the model.

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URL: https://economyandsociety.in.ua/index.php/journal/article/view/3038

DOI: 10.32782/2524-0072/2023-56-98

Fischer Black and Myron Scholes. The pricing of options and corporate li-abilities. Journal of Political Economy, 81(3):637–654, 1973. ISSN 0022-3808. DOI: 10.1086/260062.