Ways to obtain an option's delta

Question

How do you compute an option's delta when a closed-form price is available, and how when it is not?

Short answer

Differentiate the price with respect to spot if you can; otherwise approximate the derivative numerically. In Black-Scholes, $\Delta = N(d_1)$ for a call. Without a formula, bump the spot up and down and reprice ( $\frac{V(S+h)-V(S-h)}{2h}$ ), using the same random numbers in Monte Carlo; read it off a tree as $\frac{V_u - V_d}{S_u - S_d}$ ; or use pathwise or likelihood-ratio estimators in simulation.

Adapted from Crack, Heard on the Street, Q2.8, p.24; 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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