How the CEV Exponent Shapes Price-Dependent Volatility

Summary

The document clarifies how the constant elasticity of variance model links stock price to volatility. Its price process has a diffusion term proportional to the stock price raised to an exponent, but volatility in percentage terms is found by factoring the stock price out of the dynamics. The resulting volatility function is proportional to price raised to the exponent minus one.

This resolves the apparent sign confusion: when the exponent is below one, that power is negative, so percentage volatility rises as the stock price falls. The answer also describes the qualitative behavior across exponent values: lower values resemble constant volatility at higher prices, while larger values associate higher prices with higher volatility. The discussion is conceptual and does not provide parameter-estimation procedures, calibration evidence, or guidance on choosing an exponent for a particular asset.

Key ideas

• In the CEV model, the diffusion coefficient in price units is proportional to price raised to the exponent.
• Percentage volatility is obtained by dividing the diffusion coefficient by the stock price.
• When the exponent is below one, percentage volatility increases as price decreases.
• Larger exponent values associate higher stock prices with higher volatility in the described comparison.

Tags

# Confusion about CEV model


# Confusion about CEV model












Under the CEV model the stock price has the following dynamics:

$dS_t=\mu S_tdt+\sigma S_t^\gamma dW_t$, where $\sigma\geq0, $ $\gamma\geq0$.

According to Wikipedia, if $\gamma <1$ the volatility of the stock increases as the price falls.

But why is this true? Shouldn't be the exponent negative in order to have an inverse relationship between stock price and the volatility term?

## Answer by Gordon (score 4, accepted)

https://quant.stackexchange.com/a/32058

Note that \begin{align*} dS_t = S_t\left(\mu dt+\sigma S_t^{\gamma-1} dW_t \right). \end{align*} That is, the volatility function is defined by $\sigma S_t^{\gamma-1}$. Then, if $\gamma <1$, the volatility increases as the price falls.

## Answer by RUser4512 (score 0)

https://quant.stackexchange.com/a/45571

On top of @Gordon answers which gives the mathematical reason of why this happens, have a look at the graph below which illustrates the behavior of the CEV process for various values of $\gamma$.

As you can see, a low $\gamma$ looks like a constant volatility for high prices (similar to a brownian motion with drift) while a high value of $\gamma$ shows high volatilities for high prices.

Source : estimating a CEV model.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.