# 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