1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Can you provide more info about this formula? Is Ω the option gearing (or leverage)? $\endgroup$ – Quantopik May 27 '14 at 11:38
  • $\begingroup$ 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. $\endgroup$ – Clarinetist May 27 '14 at 15:13
1
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.