Constant elasticity of variance model explained

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 Cox in 1975.^[1]

Dynamic

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

d S _{t} = \mu S _{t} d t + \sigma S ^{\gamma} _{t} 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.^[2]

Notes and References

1. Cox, J. "Notes on Option Pricing I: Constant Elasticity of Diffusions." Unpublished draft, Stanford University, 1975.
2. Yu, J., 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127, 165–178.
3. 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
4. Geman, H, and Shih, YF. 2009. "Modeling Commodity Prices under the CEV Model." The Journal of Alternative Investments 11 (3): 65–84.
5. 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.)In terms of general notation for a local volatility model, written asd S _{t} = \mu S _{t} d t + v(t,S _{t)} S _{t} d W _{t} we can write the price return volatility as v(t, S _{t)=\sigma} S ^{\gamma -1} _{t} The constant parameters \sigma, \gamma satisfy the conditions \sigma\geq 0 , \gamma\geq 0 .

   The parameter

   \gamma controls the relationship between volatility and price, and is the central feature of the model. When \gamma < 1 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.^[2] Conversely, in commodity markets, we often observe \gamma > 1 ,^{[3] [4]} whereby the volatility of the price of a commodity tends to increase as its price increases and leverage ratio decreases. If we observe \gamma = 1 this model becomes a geometric Brownian motion as in the Black-Scholes model, whereas if \gamma = 0 and either \mu = 0 or the drift \mu S is replaced by \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.

   See also

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

   External links

   • Asymptotic Approximations to CEV and SABR Models
   • Price and implied volatility under CEV model with closed formulas, Monte-Carlo and Finite Difference Method
   • Price and implied volatility of European options in CEV Model delamotte-b.fr