Inmathematical finance, theCEV orconstant elasticity ofvariance model is astochastic volatility model, although technically it would be classed more precisely as alocal volatility model, that attempts to capture stochastic volatility and theleverage effect. The model is widely used by practitioners in the financial industry, especially for modellingequities andcommodities. It was developed byJohn Cox in 1975.^[1]

The CEV model is a stochastic process which evolves according to the followingstochastic differential equation:

${\displaystyle \mathrm{d}{S}_t=\mu S_t\mathrm{d}t+\sigma S_t^{\gamma }\mathrm{d}{W}_t}$

in whichS is the spot price,t is time, andμ is a parameter characterising the drift,σ andγ are volatility parameters, andW is a Brownian motion.^[2]It is a special case of a generallocal volatility model, written as

${\displaystyle \mathrm{d}{S}_t=\mu S_t\mathrm{d}t+v(t,S_t)S_t\mathrm{d}{W}_t}$

where the price return volatility is

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

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

The parameter${\displaystyle \gamma }$ controls the relationship between volatility and price, and is the central feature of the model. When 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 ageometric Brownian motion as in theBlack-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 anarithmetic Brownian motion, the model which was proposed byLouis Bachelier in his PhD Thesis "The Theory of Speculation", known asBachelier 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

• Options (finance)
• Derivatives (finance)
• Financial models