# FEATURES OF PRICE DYNAMICS MODELING BASED ON CEV MODEL

• Vasyl Yanishevskyi Lviv Polytechnic National University
• Yuri Fulmes Lviv Polytechnic National University
Keywords: stochastic equations, Brownian motion, model of geometric Brownian motion, CEV model, Fokker – Planck equation, Feller model, option price

### Abstract

A known constant elasticity of variance (CEV) option pricing model is investigated for the purpose of determining stochastic price dynamics of assets and option price. The CEV model is an attempt to generalize the geometric Brownian motion of Black-Scholes model. As it is known the stochastic dynamics of stock price (asset) that is defined by geometric Brownian motion is quite logical, however it doesn't take into account dispersion change, and assumes it to be constant. The peculiarity of the CEV model is that according to it the volatility changes according to the base price which aligns with theoretical and a lot of empirical data. Because of that the CEV pricing model is considered an important step in the Black-Scholes model evolution which allows to cover in one way all other known stochastic models depending on values of parameter β. However it was found out that for the density of transition probability the dynamics of which is given by CEV model, one should use different solutions based on positive and negative values of parameter β. The solution which is normalized by a unit for positive values of parameter β, is different for negative values of parameterβ. Despite this in works of many authors a single solution is used for the entire domain of parameter β. In the work a transition probability density of stochastic variable (asset price) was found for model with arbitrary value of elasticity density parameter β. Depending on value of parameter β two solutions for transitional probability density were defined. One solution describe a positive domain of parameter β, the other - negative domain of β. In addition it was shown that the given solution for β<0 describes only interval -1<β<-1/2 . The parameter domain -1/2<β<0 needs separate research. A detailed research of pricing in CEV model for β=-1/2 was carried out. Pricing of European call option for a given parameter βwas defined for a model based on two given solutions for transition probability density. The solution for transition probability density of Feller process was used, which in limit matches the CEV model for β=-1/2. A comparative analysis of numeric calculations for option price was carried out.

### References

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P. Carr, A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the time dependent CEV and CIR models, Journal of Derivatives, 28 (2020), 26–50.

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Tim Bollerslev, Julia Litvinova, and George Tauchen. Leverage and volatility feedback effects in highfrequency data. Journal of Financial Econometrics, 4(3):353–384, 2006.

Jens Carsten Jackwerth and Mark Rubinstein. Recovering probability distributions from option prices. The Journal of Finance, 51(5):1611–1631, 1996.

Yuh-Dauh Lyuu. Financial Engineering and Computation: Principles, Mathematics, and Algorithms. Cambridge University Press, 648 p. (2004).

Neil Shephard and Torben G Andersen. Stochastic volatility: origins and overview. In Thomas Mikosch, Jens-Peter Kreiß, Richard A. Davis, and Torben Gustav Andersen, editors, Handbook of Financial Time Series, pages 233–254. Springer, Berlin, Heidelberg, 2009.

Fabrice D. Rouah, Steven L. Heston. The Heston Model and its Extensions in Matlab and C#, + Website. Published by John Wiley & Sons, Inc., Hoboken, New Jersey, Year: 2013. 411 p.

John C Cox. Notes on option pricing i: Constant elasticity of variance diffusions. Working paper, Stanford University, 1975.

John C Cox. The constant elasticity of variance option pricing model. The Journal of Portfolio Management, 23(5):15–17, 1996.

VK Singh and N Ahmad. Forecasting performance of constant elasticity of variance model: Empirical evidence from India. International Journal of Applied Economics and Finance, 5(1):87–96, 2011.

KC Yuen, H Yang and KL Chu. Estimation in the constant elasticity of variance model. British Actuarial Journal, 7(2):275–292, 2001.

P. Carr, A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the time dependent CEV and CIR models, Journal of Derivatives, 28 (2020), 26–50.

John C Cox and Stephen A Ross. The valuation of options for alternative stochastic processes. Journal of financial economics, 3(1-2):145–166, 1976.

C. F. Lo and p. H. Yuen. Constant elasticity of variance option pricing model with time-dependent parameters. International Journal of Theoretical and Applied Finance Vol. 3, No. 4 (2000) 661–674.

Y. L. Hsua, T. I. Lina, C. F. Leebc. Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation. Mathematics and Computers in Simulation. Volume 79, Issue 1, October 2008, Pages 60–71.

Stan Beckers. The constant elasticity of variance model and its implications for option pricing. The Journal of Finance, Vol. 35, N 3, p. 661–673, 1980.

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (2 nd Ed.), Springer-Verlag, New York (1985).

Yanishevsky V. S. Application of the path integral method to some stochastic models of financial engineering. Journal of Physical Studies, 25 (2), 2801: 1–10 (2021).

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Axel A. Araneda and Marcelo J. Villena. Computing the CEV option pricing formula using the semiclassical approximation of path integral. arXiv:1803.10376 v1 [q-fin. CP] 28 Mar 2018

Linetsky, V., and R. Mendoza. “Encyclopedia of Quantitative Finance.” In Constant Elasticity of Variance (CEV) Diffusion Model. Hoboken, NJ: Wiley & Sons. 2010.

Dan Pirjol and Iingjiong Zhu. Short maturity Asian options for the CEV model. Preprint. Arxiv:1702.03382 v1 [q-fin.pr] 11 feb. 2017.

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