Implied volatility of a complex options position

Assume I have a "complex" options position like a straddle, strangle, or iron condor. In other words, several options traded together as a single position against one underlying asset (not a basket option).

I know the implied volatility of each of the options within the position. Is there an accepted method of generating an implied volatility for the overall position?

• Why the downvote? Is this as easy as it sounds or are you just having a bad day? Oct 16 '15 at 9:04

You can guesstimate by vega weighted implied vol. This is why:

Say that you have a portfolio of options with prices $P_j$. Each one of them has a different pricing function $f_j$ (as function of vol) and a different implied vol $\sigma_j$. For each option $f_j(\sigma_j)=P_j$.

Now you put them together in a single product. If the implied vol of the product is $\sigma$ then $\sum f_j(\sigma)=\sum P_j$. Now, approximately each pricing function will satisfy $f_j(\sigma)\approx P_j+V_j (\sigma-\sigma_j)$ as a linear expansion around its price, with $V_j$ the Vega.

If you substitute and solve you end up with the vega weigted vol $$\sigma \approx \frac{\sum V_j\sigma_j}{\sum V_j}$$

First note that implied volatility only makes sense with respect to an etablished pricing model, like the Black-Scholes or Bachelier model, and it is the quantity which has to be put into a closed form pricing formula obtained in one of those models to get the market dollar price.

For a straddle with strike $K$ holds $$\text{Price_Straddle}(K) = \text{Price_Call}(K) + \text{Price_Put}(K) \\ = 2\text{Price_Call}(K) - \text{UnderlyingPrice} + K,$$ due to the Put-Call parity. So the implied vola of a straddle equals the implied vola of a call.

To my knowledge, for strangles such a formula does not exists the same as for iron condors.

• "So the implied vola of a straddle equals the implied vola of a call": shouldn't it be twice the implied vola of a call? Oct 21 '20 at 9:35

Yes, and in fact this is a quoting convention in FX derivatives; for flys straddles and reversals. It is applied to Legs by often by symmetric delta, and strike-by-delta formulas is used to convert out. Keep in mind that put call parity as it relates to option type is relevant here.

Reference a see: "Foreign Exchange Option Pricing: A Practitioner's Guide by Iain J. Clark". The method is simple but long for a thread here.

The weighted approach previously answered, is a clever approach, and will give better results in relation to sensitivity as things change which is ultimately what I think you are interested in.

In general, the implied volatility is based on vanilla European or American options. In your case, since the positions depend on only a single underlier, if you can have an analytical formula, or approximation, for each individual position, then, in principle, you can compute an implied volatility based on the market price of your portfolio. However, note that, in general, there is no liquid market price quote for such "complex" options.

• Sometimes people take the implied vol of a complex position like this to see what "vol" a market-maker is charging them for a package. Nov 16 '15 at 15:02
• @BrianB: That makes perfect sense. As long as there is an analytical formula, or approximation, it is possible. Nov 16 '15 at 15:07
• You should also note that for something like a risk reversal, or another structure that can be largely vega neutral, it can produce a misleading spread.
– will
Oct 21 '20 at 21:55