ATM Vega-Vol Graph

Quantitative Finance Level 1

For an at-the-money option, which of the following best describes the graph of vega against volatility?

Straight line through the origin A smoothed out graph of

\max( 0, \frac{1}{x-1})

Horizontal line The right half of the bell-curve

2 solutions

Caleb Townsend
Mar 6, 2015

For an at-the-money option, $V \approx \frac{1}{\sqrt{2\pi}}S\sigma\sqrt{T}$ Differentiating with respect to $\sigma,$ $\nu = \frac{1}{\sqrt{2\pi}}S\sqrt{T}$ which is a constant (as long as the graph is of a single point in time). Thus, vega vs. volatility is a horizontal line for an at-the-money option.

Chew-Seong Cheong
Mar 19, 2015

Vega of an option at the money is maximum, therefore, the graph of vega $\nu$ against volatility $\sigma$ has a gradient $\dfrac {\partial \nu}{\partial \sigma} = 0$ , implying that it is a $\boxed{horizontal \space line}$ .

You have to be careful, your statements do not imply each other. We are taking many different partial derivatives here, so you have to be careful what you are taking it with respect to.

When you say "Vega of an option at the money is maximum", you mean "with respect to underlying price / strike price". What this means is that $\frac { \partial \nu } { \partial S } = 0$ .
It does not imply that "As we change the volatility, the vega of the ATM option is at a maximum / stays constant / minimum", which is what we need to conclude that $\frac{\partial \nu } { \partial \sigma} = 0$ .

Calvin Lin Staff - 6 years, 2 months ago

ATM Vega-Vol Graph

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