# Constant elasticity of variance model

In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility model, which 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:

${\displaystyle dS_{t}=\mu S_{t}dt+\sigma S_{t}^{\gamma }dW_{t}}$

The constant parameters ${\displaystyle \sigma ,\;\gamma }$ satisfy the conditions ${\displaystyle \sigma \geq 0,\;\gamma \geq 0}$.

The parameter ${\displaystyle \gamma }$ controls the relationship between volatility and price, and is the central feature of the model. When ${\displaystyle \gamma <1}$ we see the so-called leverage effect, commonly observed in equity markets, where the volatility of a stock increases as its price falls. Conversely, in commodity markets, we often observe ${\displaystyle \gamma >1}$, the so-called inverse leverage effect,[2][3] whereby the volatility of the price of a commodity tends to increase as its price increases.

## References

1. ^ Cox, J. "Notes on Option Pricing I: Constant Elasticity of Diffusions." Unpublished draft, Stanford University, 1975.
2. ^ 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
3. ^ 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