# Higher Vega with ATM options when Spot is higher

Which would have larger vega, an ATM call option at spot 100 or an ATM call option at spot 200.

Apparently the answer is the one with ATM at spot 200. I am not sure how you get this answer. Why should vega be multiplicative? I suppose the further question would be how do we know whether the greeks would be additive or multiplication or neither?

• It depends on your units of vega, if you're talking $/change in vol, then all else the same, it will increase. If you're talking % value/change in vol, then it will be the same. – will Apr 29 '18 at 15:06 • The$ change one – Permian Apr 29 '18 at 15:09
• In models with constant returns to scale (the distribution of log-returns is not level dependent), such as under geometric Brownian motion, the European vanilla price is a homogeneous function of degree one in spot and strike. Thus, the derivatives such as vega scale as well for at-the-money options at different spots. See also Theorem 6 in Merton (1973). – LocalVolatility Apr 29 '18 at 18:10