Why Theta Cannot Be Isolated by Hedging Only Gamma and Vega

Summary

The document considers whether an options seller can collect time decay while keeping a position neutral to gamma and vega, and whether contracts can be selected to do so at the lowest cost. Its answer is that neutralizing those two exposures alone does not make theta a free return. Theta is connected to a broader set of sensitivities, including vanna and volga, so a hedge that leaves them exposed retains risks tied to changes in volatility and the relationship between price and volatility.

The practical implication is that earning theta requires accepting some unhedged risk or having a market view; the document does not provide a contract-selection algorithm or cost-minimization method. It points to a working paper on a gamma-vanna-volga framework for constructing implied volatility curves, while noting that the paper may omit vega's contribution to theta. The discussion is conceptual and offers no empirical tests, hedge examples, or quantitative comparison of hedge costs, so it does not establish a universally efficient hedge.

Key ideas

• Neutralizing gamma and vega does not by itself eliminate the risks that offset option theta.
• Vanna and volga are additional sensitivities relevant to the relationship between theta and volatility exposure.
• Leaving vanna and volga unhedged amounts to taking views on correlation and volatility of volatility.
• The discussion gives no method for choosing the cheapest contracts or preserving a specified amount of theta.

Tags

# Effective gamma/vega hedging


# Effective gamma/vega hedging












I want an options position where I can short some options to pocket the premiums and benefit from the time decay. I also want to be vega and gamma neutral.

Is there an established way to find which are the most efficient contracts to hedge your gamma and vega for lowest cost, whilst maintaining as much theta as possible?

## Answer by user34971 (score 3)

https://quant.stackexchange.com/a/49794

I think you may be missing two other important greeks here: vanna and volga

Theta is not balanced by gamma only, it is balanced by vega, gamma, vanna, and volga.

So, when you ask is there an established way, by which I think you mean is there a way to more or less have a free lunch, the answer is no, not really.

You will need to take risks, i.e. leave some things unhedged because you have a view on the market, to earn (or lose) money. Going back to your question: no you can't earn theta and hedge gamma and vega, unless you were really meaning (but I don't think you were) to leave your vanna and volga unhedged - which boils down to having a view on correlation and the vol of vol.

EDIT:

I would recommend everyone interested in this topic to try to get a hold of the not publicly available paper by:

M. Arslan, G. Eid, J. El Khoury and J. Roth, "The Gamma-Vanna-Volga Cost Framework for Constructing Implied Volatility Curves", Deutsche Bank Working Paper

This is a very illuminating paper. The only thing they missed is the Vega contribution to theta (which arguably could be smaller than the other components).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.