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The approximation is:

$$\sigma \approx \frac{\sum V_j\sigma_j}{\sum V_j}$$

Background information from the first answer to this post:

"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."

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  • $\begingroup$ All options with the same underlying as in your prior post? $\endgroup$ – Charles Fox Apr 12 at 23:49
  • $\begingroup$ Isn't that exactly the vega weighted implied volatility? Are you asking for the volatility such that if it were plugged into the pricing formula of all of the options in the portfolio, it would result in the same total price as when the options are priced individually with their own volatilities? $\endgroup$ – Charles Fox Apr 12 at 23:51
  • $\begingroup$ @CharlesFox, it wasn't my post that I was referencing, but yes, all options should be assumed to have the same underlying asset. $\endgroup$ – Ice101781 Apr 13 at 1:59
  • $\begingroup$ @CharlesFox, I'm looking for a step-by-step explanation as to how the fraction on the right side on the approximation sign was derived. $\endgroup$ – Ice101781 Apr 13 at 2:02
  • $\begingroup$ The issue I have with this is that we are talking of a set of different pricing functions $f_j$. If we want to work out a single volatility $\sigma$, we need to specify into precisely which pricing function it should be plugged to result in the portfolio value. If we do not know which function that is, then the problem is ill-defined $\endgroup$ – ZRH Apr 14 at 17:11
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Intuitively, if an option has 0 Vega (k=0), it has no influence on the single volatility that will correctly price the portfolio. If you have one position with a \$100 vega per vol point, and another with only $50 vega per vol point, then using a volatility that is 1 point above the implied volatility of the first position, but 2 points below the implied volatility of the second results in offsetting mis-pricing for the individual options and an approximately correct priced portfolio.

More generally, portfolio pricing error is the sum of errors for each security (vega times the difference between vol used and true vol), or:

$\sum[V_i*(\sigma-\sigma_i)] = \sum[V_i\sigma]-\sum[V_i\sigma_i] = \sigma\sum V_i-\sum[V_i\sigma_i] $

Setting our error approximation equal to zero:

$0=\sigma\sum V_i -\sum[V_i\sigma_i] $

$\sigma= \frac{\sum[V_i\sigma_i]}{\sum V_i} $

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