Deriving an Option’s Vega from Total Variance

Summary

The document asks why an option’s sensitivity to total variance includes a factor of time. It defines total variance as volatility squared times time and refers to a Black–Scholes option value expressed as a function of spot price, strike, and total variance. The question compares a derivative with respect to total variance to the conventional sensitivity called vega.

The post does not provide an answer or derivation, so it leaves the key relationship unresolved. It points to the need to distinguish differentiation with respect to volatility from differentiation with respect to total variance, which are related through the chain rule. Readers should treat this as a question prompt rather than a complete explanation; it gives no numerical example, portfolio construction details, or evidence beyond its reference to an earlier options analysis.

Key ideas

• The post asks how option sensitivity to total variance relates to conventional vega.
• It defines total variance as volatility squared multiplied by time.
• It cites a result that constant-vega option portfolios weight options inversely to the square of strike.
• The document poses the derivative question but does not answer it.

Tags

# Sensitivity to total variance for an option


# Sensitivity to total variance for an option












In the famous article of Demertifi, Derman et al (1999), the authors, in the appendix, show that it it necessary to have options weighted inversely proportional to the Square of the Strike in order to have a constant vega in a portfolio of options. $O(S,K,v)$ represents a standard Black-Scholes option of strike $K$ and of total variance $v = \sigma^2t$ when the stock price is $S$.

To show that, the authors start by deriving the sensitivity $V_O$ to the total variance of an individual option.

And then, they write the following equation:

$$ V_O = t \frac{\delta}{\delta v}(O) $$

I do not understand why there is a $t$ in front of the derivative $\frac{\delta}{\delta v}(O)$.

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.