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

${\displaystyle \mathrm {d} S_{t}=\mu S_{t}\mathrm {d} t+\sigma S_{t}^{\gamma -1}\mathrm {d} W_{t}}$ 

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 ^{[citation needed]}

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

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 an effect, commonly observed in equity markets, where the volatility of a stock increases as its price falls and the leverage ratio increases.^[3] Conversely, in commodity markets, we often observe ${\displaystyle \gamma >1}$ ,^[4][5] whereby the volatility of the price of a commodity tends to increase as its price increases and leverage ratio decreases. If we observe ${\displaystyle \gamma =1}$  this model is considered the model which was proposed by Louis Bachelier in his PhD Thesis "The Theory of Speculation".

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

• 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