Constant elasticity of variance model

The CEV model is a stochastic process which evolves according to the following stochastic differential equation:

:$\backslash mathrm\{d\}S\_t\; =\; \backslash mu\; S\_t\; \backslash mathrm\{d\}t\; +\; \backslash sigma\; S\_t\; ^\{\backslash gamma\}\; \backslash mathrm\{d\}W\_t$

in which S is the spot price, t is time, and μ is a parameter characterising the drift, σ and γ are volatility parameters, and W is a Brownian motion.[https://web.archive.org/web/20180220152049/http://rafaelmendoza.org/wp-content/uploads/2014/01/CEVchapter.pdf Vadim Linetsky & Rafael Mendozaz, 'The Constant Elasticity of Variance Model', 13

July 2009]. (Accessed 2018-02-20.)

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

:$\backslash mathrm\{d\}S\_t\; =\; \backslash mu\; S\_t\; \backslash mathrm\{d\}t\; +\; v(t,S\_t)\; S\_t\; \backslash mathrm\{d\}W\_t$

where the price return volatility is

:$v(t,\; S\_t)=\backslash sigma\; S\_t^\{\backslash gamma-1\}$

The constant parameters $\backslash sigma,\backslash ;\backslash gamma$ satisfy the conditions $\backslash sigma\backslash geq\; 0,\backslash ;\backslash gamma\backslash geq\; 0$.

The parameter $\backslash gamma$ controls the relationship between volatility and price, and is the central feature of the model. When 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. Yu, J., 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127, 165–178. Conversely, in commodity markets, we often observe $\backslash gamma\; >\; 1$,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 : 533–553Geman, 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}} whereby the volatility of the price of a commodity tends to increase as its price increases and leverage ratio decreases. If we observe $\backslash gamma\; =\; 1$ this model becomes a geometric Brownian motion as in the Black-Scholes model, whereas if $\backslash gamma\; =\; 0$ and either $\backslash mu\; =\; 0$ or the drift $\backslash mu\; S$ is replaced by $\backslash mu$, this model becomes an arithmetic Brownian motion, the model which was proposed by Louis Bachelier in his PhD Thesis "The Theory of Speculation", known as Bachelier model.

• Volatility (finance)
• Stochastic volatility
• Local volatility
• SABR volatility model
• CKLS process

{{Reflist}}

• [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