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Brownian Motion & the Stochastic Behavior of Stocks
Department of Mathematics, Imperial College London, London, UK
Department of Mathematics, University College London, London, UK
- 1 Department of Mathematics, Imperial College London, London, UK
- 2 Department of Mathematics, University College London, London, UK
Journal of Mathematical Finance·Volume 12 (2021)·Pages 138–149·Published 28 December 2021·DOI10.4236/jmf.2022.121009
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Abstract
In this paper, we test the effectiveness of predicting the behavior of stocks utilizing stochastic calculus. We begin by exploring the intuition of Brownian motion by explaining its birth through the observations of Robert Brown and later through Bachelier’s work on its applications to the financial market and finally its rigorous and concretized form proposed by Norbert Wiener. The aforementioned motivates a stochastic differential equation to model the future price fluctuations of a stock wherein It? integration is prominent and consequently expanded upon. The final part of this paper focuses on the accuracy of the model by back testing it with Apple stock and deriving a correlation coefficient.
KeywordsBrownian MotionStochastic CalculusStochastic Differential EquationStocksItôIntegral
- Brown, R., et al. (1828) XXVII. A Brief Account of Microscopical Observations Made in the Months of June, July and August 1827, on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies. The Philosophical Magazine, 4, 161-173. https://doi.org/10.1080/14786442808674769
- Bachelier, L. (1900) Théorie de la Spéculation. Annales Scientifiques de l'Ecole Normale Supérieure, 17, 21-86. https://doi.org/10.24033/asens.476
- Wiener, N. (1923) Differential-Space. Journal of Mathematics and Physics, 2, 131-174. https://doi.org/10.1002/sapm192321131
- Kolmogorov, A.K. and Bharucha-Reid, A.T. (2018) Foundations of the Theory of Probability. 2nd Edition, Courier Dover Publications, New York.
- Huo, H. (2006) G Introduction to Stochastic Integration. Springer-Verlag, Berlin.
- Øksendal, B. (2003) Stochastic Differential Equations. Springer, New York. https://doi.org/10.1007/978-3-642-14394-6_5
- Fernholz, E.R. (2002) Stochastic Portfolio Theory. Springer, Princeton. https://doi.org/10.1007/978-1-4757-3699-1_1
- Bremaud, P. (2020) Probability Theory and Stochastic Processes. Springer Nature, Berlin.
- Blitzstein, J.K. and Hwang, J. (2019) Introduction to Probability. Chapman and Hall/ CRC, Boca Raton. https://doi.org/10.1201/9780429428357
- Heston, S.L. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
- Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654. https://doi.org/10.1086/260062
- Moalosi-Court, K.I. (2019) Pricing European Option When the Stock Price Process Is Being Driven by Geometric Brownian Motion. Open Access Library Journal, 6, e5568. https://doi.org/10.4236/oalib.1105568