Deriving Vega-Weighted Implied Volatility for an Options Portfolio
Summary
The document derives an approximate single implied volatility for a portfolio of options whose components have different implied volatilities. It begins with the condition that the portfolio’s modeled value at the common volatility should equal the sum of the individual option prices. Each option’s pricing function is then approximated locally with a first-order expansion around its own implied volatility, using vega as the sensitivity to volatility.
Summing these approximated pricing errors and setting the total to zero yields a vega-weighted average: options with greater vega have more influence on the portfolio volatility, while a zero-vega position contributes none under this approximation. A simple comparison of two positions illustrates how differently sized vegas balance volatility deviations. The method is only a local linear approximation; it does not account for higher-order effects such as volga or for large changes in volatility, so the resulting volatility may not exactly reprice the portfolio.
Key ideas
- A portfolio’s common implied volatility can be approximated by equating its modeled value with the sum of component prices.
- A first-order expansion approximates each option’s pricing error as vega times the volatility difference.
- The resulting portfolio volatility is the component implied volatilities weighted by their vegas.
- An option with zero vega has no influence on this estimate under the linear approximation.
- Higher-order price effects can make the approximation less accurate when volatility differences are large.
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# Answer by Charles Fox (score 1)
# Is there a simple, intuitive derivation (using Taylor series) of the following approximation to Vega-weighted Implied Volatility?
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."
## Answer by Charles Fox (score 1)
https://quant.stackexchange.com/a/45100
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} $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.