Constant elasticity of variance model: Difference between revisions

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In [[mathematical finance]], the '''CEV''' or '''constant elasticity of [[variance]] model''' is a [[stochastic volatility]] model, although technically it would be classed more precisely as a [[local 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 Carrington Cox|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 Unpublished draft, Stanford University, 1975.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 describesis a stochastic process which evolves according to the following [[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 othervolatility 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>

It is a special case of a general [[local volatility]] model, written as

And so we have

:<math>\sigma(mathrm{d}S_t, t)= \sigmamu S_t^ \mathrm{d}t + v(t,S_t) S_t \gamma-1mathrm{d}W_t</math>

where the price return volatility is

:<math>v(t, S_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 = 01</math> this model becomes a [[geometric Brownian motion]] as in the [[Black-Scholes model]], whereas if <math>\gamma = 0</math> and either <math>\mu = 0</math> or the drift <math>\mu S</math> is consideredreplaced by <math>\mu</math>, this model becomes an [[Geometric_Brownian_ motion#Arithmetic_Brownian_Motion|arithmetic Brownian motion]], the model which was proposed by [[Louis Bachelier]] in his PhD Thesis "The Theory of Speculation", known as [[Bachelier model]].

==See also==

*[[Volatility (finance)]]

*[[Stochastic volatility]]

*[[Local volatility]]

*[[SABR volatility model]]

*[[CKLS process]]