FEATURES OF PRICE DYNAMICS MODELING BASED ON CEV MODEL
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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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.
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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.
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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).
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Dan Pirjol and Iingjiong Zhu. Short maturity Asian options for the CEV model. Preprint. Arxiv:1702.03382 v1 [q-fin.pr] 11 feb. 2017.
