Share count for delta-hedging an ATM option

Question

About how many shares of stock do you need to hedge one at-the-money option on that stock?

Short answer

Roughly half a share per option (about 50 shares per 100-share contract). An at-the-money option has delta close to $0.5$ in absolute value: slightly above for a call because of rates and lognormal skew, slightly below for a put. Buy shares to hedge a short call or long put, and short shares to hedge a long call or short put.

Adapted from Crack, Heard on the Street, Q2.40, p.30; 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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