Constant elasticity of variance model

In mathematical finance, the CEV or constant elasticity of variance model is a 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.^[1]

Dynamic

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

\mathrm {d} S_{t}=\mu S_{t}\mathrm {d} t+\sigma S_{t}^{\gamma }\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 other parameters, and W is a Brownian motion.^[2] And so we have

\sigma (S_{t},t)=\sigma S_{t}^{\gamma -1}

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 $\gamma <0$ we see an effect, commonly observed in equity markets, where the volatility of a stock increases as its price falls. Conversely, in commodity markets, we often observe $\gamma >0$ ,^[3]^[4] whereby the volatility of the price of a commodity tends to increase as its price increases. However those effects cannot be considered leverage-effects as in any case for $\gamma$ the effect is symmetric for positive and negative price shocks ^[5]. If we observe $\gamma =0$ this model is considered the model which was proposed by Louis Bachelier in his PhD Thesis "The Theory of Speculation".

References

^ Cox, J. "Notes on Option Pricing I: Constant Elasticity of Diffusions." Unpublished draft, Stanford University, 1975.
^ Vadim Linetsky & Rafael Mendozaz, 'The Constant Elasticity of Variance Model', 13 July 2009. (Accessed 2018-02-20.)
^ 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
^ 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
^ Yu, J., 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127, 165–178.

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