View source for Finance:Constant elasticity of variance model

In [[Finance:Mathematical finance|mathematical finance]], the '''CEV''' or '''constant elasticity of [[Variance|variance]] model''' is a [[Finance:Stochastic volatility|stochastic volatility]] model that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling equities and commodities. It was developed by John Cox in 1975.<ref>Cox, J. "Notes on Option Pricing I: Constant Elasticity of Diffusions." Unpublished draft, Stanford University, 1975.</ref>
== Dynamic ==

The '''CEV''' model describes a process which evolves according to the following [[Stochastic differential equation|stochastic differential equation]]:

:<math>\mathrm{d}S_t = \mu S_t \mathrm{d}t + \sigma S_t ^ \gamma \mathrm{d}W_t</math>

in which ''S'' is the spot price, ''t'' is time, and ''μ'' is a parameter characterising the drift, ''σ'' and ''γ'' are other parameters, and ''W'' is a Brownian motion.<ref>[https://web.archive.org/web/20180220152049/http://rafaelmendoza.org/wp-content/uploads/2014/01/CEVchapter.pdf <nowiki>Vadim Linetsky & Rafael Mendozaz, 'The Constant Elasticity of Variance Model', 13
July 2009</nowiki>]. (Accessed 2018-02-20.)</ref>
:<math>\sigma(S_t, t)=\sigma S_t^{\gamma-1}</math>
The constant parameters <math>\sigma,\;\gamma</math> satisfy the conditions <math>\sigma\geq 0,\;\gamma\geq 0</math>. 

The parameter <math>\gamma</math> controls the relationship between volatility and price, and is the central feature of the model. When <math>\gamma < 1</math> we see an effect, commonly observed in equity markets, where the volatility of a stock increases as its price falls and the leverage ratio increases.<ref> Yu, J., 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127, 165–178.</ref> Conversely, in commodity markets, we often observe <math>\gamma > 1</math>,<ref>Emanuel, D.C., and J.D. MacBeth, 1982. "Further Results of the Constant Elasticity of Variance Call Option Pricing Model." Journal of Financial and Quantitative Analysis, 4 : 533–553</ref><ref>Geman, H, and Shih, YF. 2009. "Modeling Commodity Prices under the CEV Model." The Journal of Alternative Investments 11 (3): 65–84. {{doi|10.3905/JAI.2009.11.3.065}}</ref> whereby the volatility of the price of a commodity tends to increase as its price increases and leverage ratio decreases. If we observe <math>\gamma = 0</math> this model is considered the model which was proposed by [[Biography:Louis Bachelier|Louis Bachelier]] in his PhD Thesis "The Theory of Speculation".

==See also==
*[[Finance:Volatility|Volatility]]
*[[Finance:Stochastic volatility|Stochastic volatility]]
*[[Finance:SABR volatility model|SABR volatility model]]
*CKLS process

==References==
{{Reflist}}

==External links==
*[https://ssrn.com/abstract=1850709 Asymptotic Approximations to CEV and SABR Models]
*[https://web.archive.org/web/20150605050301/http://www.serdarsen.somee.com/CEV.aspx Price and implied volatility under CEV model with closed formulas, Monte-Carlo and Finite Difference Method]
*[http://www.delamotte-b.fr/CEV.aspx Price and implied volatility of European options in CEV Model] delamotte-b.fr
 

{{Derivatives market}}
{{Volatility}}
{{Stochastic processes}}

[[Category:Options (finance)]]
[[Category:Derivatives (finance)]]
[[Category:Financial models]]

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