AUTHOR=Movahed Ali Asghar , Noshad Houshyar 

TITLE=Introducing a new approach for modeling stock market prices using the combination of jump-drift processes

JOURNAL=Frontiers in Physics

VOLUME=12

YEAR=2024

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2024.1402593

DOI=10.3389/fphy.2024.1402593

ISSN=2296-424X

ABSTRACT=<p>The stock price data are sampled at discrete times (e.g., hourly, daily, weekly, etc). When data are sampled at discrete times, they appear as a sequence of discontinuous jump events, even if they have been sampled from a continuous process. On the other hand, distinguishing between discontinuities due to finite sampling of the continuous stochastic process and real jump discontinuities in the sample path is often a challenging task. Such considerations, led us to the question: Can discrete data (e.g., stock price) be modeled using only jump-drift processes, regardless of whether the sampled time series originally belongs to the class of continuous processes or discontinuous processes? To answer this question, we built a stochastic dynamical equation in the general form <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:mi>y</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mo>=</mml:mo><mml:mover accent="true"><mml:mi>μ</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>ξ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>d</mml:mi><mml:msub><mml:mi>J</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, which includes a deterministic drift term (<inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover accent="true"><mml:mi>μ</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:math></inline-formula>) and a combination of stochastic terms with jumpy behaviors (<inline-formula id="inf3"><mml:math id="m3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ξ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>d</mml:mi><mml:msub><mml:mi>J</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula>), and used it to model the log-price time series <inline-formula id="inf4"><mml:math id="m4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>y</mml:mi><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula>. In this article, we first introduce this equation in its simplest form, including a drift term and a stochastic term, and show that such a jump-drift equation is capable of reconstructing stock prices in Black-Scholes diffusion markets. Afterwards, we extend the equation by considering two jump processes, and show that such a drift-jump-jump equation enables us to reconstruct stock prices in jump-diffusion markets more accurately than the old jump-diffusion model. To demonstrate the practical applications of the proposed method, we analyze real-world data, including the daily stock price of two different shares and gold price data with two different time horizons (hourly and weekly). Our analysis supports the practical applicability of the methodology. It should be noted that the presented approach is expandable and can be used even in non-financial research fields.</p>