Constant elasticity of variance model

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

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

in which S is the spot price, t is time, and μ is a parameter characterising the drift, σ and γ are volatility parameters, and W is a Brownian motion.^[2] In terms of general notation for a local volatility model, written as

${\displaystyle \mathrm {d} S_{t}=\mu S_{t}\mathrm {d} t+v(t,S_{t})S_{t}\mathrm {d} W_{t}}$ 

we can write the price return volatility as

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

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 becomes a geometric Brownian motion as in the Black-Scholes model, whereas if ${\displaystyle \gamma =0}$  and either ${\displaystyle \mu =0}$  or the drift ${\displaystyle \mu S}$  is replaced by ${\displaystyle \mu }$ , 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.

• Volatility (finance)
• Stochastic volatility
• Local 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