Spot is 100 and rates are zero. How many times more vega does the one-year at-the-money option carry than the one-month, and roughly what is each vega per volatility point?

Spot is 100 and rates are zero. How many times more vega does the one-year at-the-money option carry than the one-month, and roughly what is each vega per volatility point?

Approach: Use the at-the-money vega approximation proportional to spot times the square root of the time to expiry, then take the ratio of the two maturities.

3.46. At-the-money vega is approximately 0.4*S*sqrt(T) per unit of volatility, so vega grows with the square root of time and the ratio is sqrt(1)/sqrt(1/12), or sqrt(12) = 3.46. In per-point terms the one-year option has vega 0.4*100*1/100 = $0.40 per volatility point and the one-month has 0.4*100*0.2887/100 = $0.115. Vega concentrates in the long end while gamma concentrates in the short end, which is the whole term structure trade: a one-year straddle against 3.46 one-month straddles is vega neutral but very short gamma. That is why a desk hedging a long-dated vega position with front-month options ends up carrying a large and unwanted gamma exposure.

Follow-up: How many one-month straddles against one one-year straddle would instead make the pair gamma neutral, and what vega does that leave?

Key concepts: vega, square root of time, at-the-money approximation, term structure.