# 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 $$\frac{\delta}{\delta v}(O)$$.

• There are various small variations in definition of vega, for example $\partial / \partial \sigma$, or $\partial / \partial \sigma^2$, or even $\partial / \partial \sigma \sqrt{\tau}$. In Derman's paper he defines vega as $\partial / \partial \sigma^2 = t \partial / \partial (\sigma^2 t)$ $= t\partial / \partial v$ Oct 30 '20 at 6:52
• I agree with you. Then, it means that here, it is not the sensitivity to the total variance but just to the variance. I am quite surprise that it is possible to find a typo in such a famous paper, Oct 30 '20 at 20:52