Understanding Vega in CEV and Local Volatility Models

Summary

The document asks how to define option vega when volatility varies with the underlying asset price, using the CEV form in which local volatility is proportional to a power of the asset price. It contrasts this setting with Black–Scholes, where volatility is a single model parameter, and questions whether the chain-rule expression that combines delta with the derivative of spot against local volatility is valid.

The post offers no answer or derivation, so it does not establish a method for calculating this sensitivity. Its useful point is the distinction between a scalar volatility input and a state-dependent volatility function: a meaningful Greek requires specifying which model parameter or function is being perturbed and how that perturbation affects the option price. Readers should treat the proposed expression as an unresolved intuition, not a pricing result.

Key ideas

• In Black–Scholes, vega measures sensitivity to a single volatility parameter.
• In CEV and local volatility models, volatility depends on the underlying price.
• A sensitivity to a volatility function needs a clearly specified perturbation.
• The document poses but does not resolve whether its proposed chain-rule expression is valid.

Tags

# Computation of option vega under CEV


# Computation of option vega under CEV












It is easy to define the option vega $\nu=\frac{\partial C}{\partial \sigma}$ under Black Scholes model since volatility is a single quantity.

However, under CEV or local volaility model, it is confusing for me to compute option vega.

For example, the volaility function is defined as $\sigma(S)=\delta S^{\beta}$. Then, how to compute a sensitivity of the option price with respect to $\sigma(S)$??

At first galance, I think $$ \nu = \frac{\partial C}{\partial \sigma(S)}= \frac{\partial C}{\partial S}\frac{\partial S}{\partial \sigma(S)}=\Delta\frac{1}{\delta\beta S^{\beta-1}} $$

Is this right? But it seems wrong..

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