When Option Volatility Scales with Stock Volatility and Elasticity

Summary

The document examines the relation between an option’s volatility, the underlying stock’s volatility, and the option’s elasticity. It asks whether this relation depends on the Black–Scholes model, referencing a textbook proof but supplying no derivation of its own.

The answer says the result follows from definitions and hedging under a risk-neutral measure, with absence of arbitrage and market completeness as stated conditions. It also assumes constant volatility, while suggesting that an extension to time-varying volatility may be possible. A further caveat is that the framework presumes volatility itself is well-defined. The discussion therefore distinguishes the conditions claimed to support the relation from the specific Black–Scholes assumptions, but it offers no proof or details on how the result changes in incomplete markets or under alternative volatility dynamics.

Key ideas

• The stated relationship scales stock volatility by the absolute elasticity of the option.
• The answer identifies a risk-neutral measure and the ability to hedge as requirements in the proof’s setup.
• Absence of arbitrage and market completeness are given as assumptions.
• Constant volatility is assumed, though a time-varying extension is suggested as possible.
• The argument still presumes that a meaningful volatility measure exists.

Tags

# Volatility of Option


# 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?

## Answer by Lucas Morin (score 1, accepted)

https://quant.stackexchange.com/a/11456

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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