 Put-call symmetry in the Black-Scholes model

ID: put-call-symmetry-in-the-black-scholes-model

F (v, m) = Φ (d_{1}) - m Φ (d_{2})

, with

d_{1} = (- lo g m + v /2) / v

and

d_{2} = d_{1} - v

. The identities

d_{1} (v, 1/ m) = - d_{2} (v, m)

and

Φ (- x) = 1 - Φ (x)

imply

F (v, m) = 1 - m + m F (v, 1/ m) .

For a stock with risk-neutral drift

r

τ

, and positive strike

K

P_{t} (T, K) = K e^{- r τ} F (σ^{2} τ, \frac{S _{t} e ^{r τ}}{K}) .

The positive sign of

r τ

in the reciprocal argument is forced by taking the reciprocal of

K e^{- r τ} / S_{t}

. The identity includes

v = 0

by continuity and requires

m > 0

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