The implied volatility surface and the option Greeks - to what extent is the information contained in their daily movements the same?

Question

What is the link between option Greeks (i.e. vega, delta, gamma, theta) and implied volatility surface (IVS) movements? Could you say that their 'information content' is the same. i.e. that out of movements of the one you could derive the movements of the other at the same point in time?

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Some Background to why I am asking this question:

I have two sets of data from Optionmetrics:

• An interpolated IVS (of the SPY) as constant point of time to maturity (30,60,..180 days) and constant points of Delta
• Option greeks data (vega, delta, gamma, theta) for an option that is modeled to be perpetually at the money and at 30 days of expiration.

Regression these sets on each other, and also using principal components techniques reveals that both sets of data are essentially the same 'information' (or you could say they have the same variation).

My question is whether is due to the manner in which Optiometrics handles/interpolates the data or that this is something structural and founded in theory. That why I ask the above - I would like to know whether to what extent theoretically this similarity is also the case.

Answer 1

If you want to calculate the change of a greek, lets say Delta, from a change in the volatility, you would need Vanna:

$$Vanna=\partial_\sigma\Delta=\partial_\sigma\partial_S C$$, which under Black-Scholes becomes:

$$ Vanna=\left(\sqrt{T-t+\frac{1}{\sigma}}\right)\phi\left(d_1\right) $$

where $\phi\left(\cdot\right)$ is the standardnormal density and

$$ d_1=\frac{\ln\left(\frac{S}{K}\right)+\left(r+\frac{\sigma^{2}}{2}\right)(T-t)}{\sigma\sqrt{T-t}} $$ (from: Franke, J. Haerdle, W.K., Hafner, C.M., 'Statistics of Financial markets - An Introduction', 2nd Edition, Springer, 2008, pg. 110)

Consequently the new greek is approximately: $\Delta_{t+1}\approx\Delta_t+Vanna(\sigma_{t+1})$

As you can see, it is not straightforward, but in general you just derive the corresponding greek with respect to $\sigma$ to find the relationship.

Answer 2

Gamma and delta are related, insofar as gamma is just the second derivative (or delta) of the delta (first derivative) of the value of the option with respect to price.

Theta and vega are related insofar as they are the derivative of price with respect to time (theta) and volatility (vega) respectively. They can be linked by a "cross derivative" of volatility with respect to time.