Comparing one-day risk of a call and its stock

Question

Measure risk over one day by the percentage change in value for a given move in the stock. Is a call option riskier than the underlying stock?

Short answer

Yes, the call is riskier. The call's percentage sensitivity is its elasticity $\tfrac{S\Delta}{C}$ . Because a call's value is convex and below $S\Delta$ (indeed $C=S N(d_1)-Ke^{-rT}N(d_2)<S\Delta$ ), the elasticity exceeds $1$ , so the call's percentage moves are larger than the stock's: it behaves like a levered position in the stock.

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q2.19, p.18; question and answer rewritten by SelfMadeQuant.com.

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