# Which greeks do you need to hedge if you want to implement an implied-volatility security?

Assume you want to create a security which replicates the implied volatility of the market, that is when $\sigma$ goes up, the value of the security $X$.

The method you could use is to buy call options on that market for an amount $C$.

We know that call options have a positive vega $\nu = \frac{\partial C}{\partial \sigma}= S \Phi(d_1)\sqrt{\tau} > 0$, so if the portfolio was made of the call $X=C$, then the effect of $\sigma$ on the security is as we desired.

However, there is of course a major issue: the security $X$ would also have embedded security risk, time risk and interest rate risk. You can use the greeks to hedge against $\Delta$, $\Theta$ and $\rho$ (which are the derivative of the call option respective to each source of risk).

In practice, I think you definitely need $X$ to be $\Theta$-neutral and $\Delta$-neutral, but would you also hedge against $\rho$ or other greeks? Have the effect of these variable been really important on option prices to make a significant impact, or would the cost of hedging be too high for the potential benefit?

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For non-interest rate derivatives with not-so-long maturities worrying about rho is uncommon. Think about it: interest-rates do not change that often relative to options expiring next week, next month or at most next year. LEAPS are obviously another turf. You could think about gamma, but the intimate relation of gamma and vega (at least in BS model) makes hedging difficult from a standard model point of view.

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disagree quite a bit with this notion of rho. Sadly the OP chose this as the correct answer. Please see my suggested answer below because I think rho can expose you to significant risks even if the underlying asset is not interest bearing. –  Matt Wolf May 29 '12 at 2:10

You cannot be theta neutral: if your security has vega, then it has gamma and theta.

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I agree, it was part of the answer I expected. –  SRKX May 29 '12 at 9:12

I somewhat disagree (partially) with the other answers so I offer my own. First, most importantly is that you specify exactly what underlying asset you really talk about. Even for non-interest bearing assets an unhedged rho can sometimes have devastating results on your profitability. Imagine a stock denominated in a highly inflated currency. If you buy options you need to finance such investment through borrowing cash or shorts in fixed income securities which directly expose you to interest rate risk. Some economies struggle with so high inflation rates that the impact of just a few days of lending/borrowing in such money markets expose you to significant inflation/interest rate risk.

You also want to pay close attention to fx risk. I know of some Kospi index options traders who lost a huge bunch of their positive pnl because they badly hedged fx risk through NDFs (= non-deliverable forwards). Well to be fair, I should not say they badly hedged it but hedging fx risk in certain markets can be extremely tricky especially when the currency is not freely convertible.

You named the basic greeks so I wont get into this but may I point you to papers that introduce you to the mechanics of variance and volatility swaps? The replication of those may be exactly what you are looking for and some of those (especially the Deutsche Bank paper and JPM) did a pretty good job at highlighting the basic greek hedges and residual higher order risk. Hope this helped a little.

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Yes please! Put the links at the bottom of the post if you have them. –  SRKX May 29 '12 at 9:14
Excellent points Freddy! –  onlyvix.blogspot.com May 29 '12 at 15:05
The replicating var swap can be achieved by buying options within a fixed expiry according to weights 1/strike^2. Typically you buy the OTM call/put, i.e. calls for strikes > ATM and puts for strikes < ATM, since these are more liquid. You can check out the JPM paper at scribd.com/doc/19606780/JP-Morgan-Variance-Swaps –  muratoa Jun 7 '12 at 17:56
@muratoa, thanks for the link, I was about to post it when I saw the request by SRKX. If you google var swaps and deutsche bank you should also be able to find a paper covering the same topic –  Matt Wolf Jun 12 '12 at 5:48