# Volatility of Option

I hope I'm asking this at the right place.

This pertains to actuarial exam MFE/3F on Financial Economics. If $\sigma$ is "volatility" and $\Omega$ the elasticity of the stock, one formula that is taught in this course is

$$\sigma_{\text{option}} = \sigma_{\text{stock}} \cdot |\Omega|\text{,}$$

where "option" means a call or a put.

Finan (Proposition 31.1, pp. 234-235) proves this statement.

My question is, does this formula make an implicit assumption that the Black-Scholes assumptions have to hold?

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Can you provide more info about this formula? Is Ω the option gearing (or leverage)? –  Quantopic May 27 at 11:38
I haven't heard of those terms before, but I can tell you that $\Omega = \dfrac{\Delta S_0}{C}$, where $C$ is the call price, $S_0$ is the initial stock price, and $\Delta = \dfrac{\partial C}{\partial S_0}$, the option Greek. –  Clarinetist May 27 at 15:13

## 1 Answer

From the definitions and the proof given in the paper you only need a risk neutral measure and the possibility to hedge.

The assumptions you need to make are the absence of arbitrage opportunities (AOA) and the market completeness.

You also work with a constant volatility. I think the result can be generalized to non-constant volatility. There is still an embeded assumption we often forget: the existence of a volatility. The use of a given model will guarantee the existence of the vol, But no need of a bs model.

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...which then basically sums up most B-S assumptions ;-) –  Matt Wolf May 28 at 14:56
The calculation are made with bs, but it shoul work with time varying r and sigma, whitout brownian motion. –  lmorin May 28 at 15:37
yes as long as you have deterministic r and sigma, as far as I understood the paper. –  Matt Wolf May 28 at 16:27