Constant elasticity of variance model

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

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

in which S is the spot price, t is time, and μ is a parameter characterising the drift, σ and γ are other parameters, and W is a Brownian motion.^[2] And so we have

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

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 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,^[3][4] whereby the volatility of the price of a commodity tends to increase as its price increases.

참고 항목

• 변동성(금융)
• 확률적 변동성
• SABR 변동성 모델

참조

외부 링크

• CEV 및 SABR 모델에 대한 점근법
• Monte-Carlo 및 유한차이법을 적용한 CEV 모델에서의 가격 및 묵시적 변동성
• CEV Model delamotte-b.fr에서 유럽 옵션의 가격 및 암시적 변동성