The delta of a call minus the delta of a put

Question

A non-dividend stock trades at 100. The 105-strike call (three months out) has delta 0.62. What is the delta of the 105-strike put with the same expiry, and why can you answer without any pricing model?

Accepted Answer

Put–call parity ties the two options together: C - P = S - K e^{-rT}. Differentiate both sides with respect to the stock price S. The right side has \partial S/\partial S = 1 and the discounted strike is constant, so \DeltaC - \DeltaP = 1 \quad\Longrightarrow\quad \DeltaP = \DeltaC - 1. Here \DeltaP = 0.62 - 1 = -0.38. The put's delta is negative (puts gain when the stock falls) with magnitude 0.38. Why this must hold: buying the call and selling the put (same strike, same expiry) is a synthetic long stock position, its payoff is ST - K no matter what, so it moves one-for-one with the stock a…

The delta of a call minus the delta of a put

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