Constant elasticity of variance model

In mathematical finance, the CEV model is a stochastic volatility model, which attempts to capture stochastic volatility and the leverage effect. The name stands for "Constant Elasticity of Variance. 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

Dynamics

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

$dS_{t}=\mu S_{t}d_{t}+\sigma S_{t}^{\gamma },dW_{t}$

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 the so-called leverage effect, commonly observed in equity markets, where the volatilty of a stock increases as its price falls. Conversely, in commodities, we often observe $\gamma >1$, the so-called inverse leverage effect, whereby the volatilty of the price of a commodity tends to increase as its price increases.

See also

• Volatility (finance)
• Stochastic Volatility
• SABR Volatility Model