Vega at spot 100 versus spot 200

Question

Two at-the-money European calls have the same maturity and volatility, one on a stock priced at $100$ and one on a stock priced at $200$ . Without writing down a formula, which has the larger vega?

Short answer

The one at spot 200, by a factor of two. Option prices are homogeneous: a stock at $200$ with strike $200$ is exactly two copies of the $100$ -strike- $100$ situation scaled up, so its price, and therefore its sensitivity to volatility, is twice as large.

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q2.46, p.20; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Greeks and Numerical Sensitivities
Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit
The Black-Scholes Formula Derived via Change of Measure

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