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 Incoming links: Put-price strike derivative

Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 3 / a / ii / Solution 2026-10-25

Differentiate the Black-Scholes formula for the European put option with respect to its strike. The chain rule gives

\frac{\partial P}{\partial K} = e^{- r τ} Φ (- d_{-}) - K e^{- r τ} ϕ (d_{-}) \frac{\partial d _{-}}{\partial K} + Sϕ (d_{+}) \frac{\partial d _{+}}{\partial K} .

The final two terms cancel by both identities in part i. Hence the put-price strike derivative is

\frac{\partial P}{\partial K} = e^{- r τ} N (- d_{-}) .

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Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 3 / b / i / Solution 2026-10-25

For a terminal price distinct from

K

, differentiating the put's financial payoff gives

\partial_{K} (K - S_{T})^{+} = 1_{{S_{T} < K}}

. The difference quotients are bounded by one, and the continuous lognormal distribution has zero probability at the strike. The dominated convergence theorem therefore permits differentiation inside its discounted expectation. The put-price strike derivative is exactly the price of the digital put option:

P_{d} (S, t) = \frac{\partial P}{\partial K} = e^{- r τ} N (- d_{-}) .

Equivalently, risk-neutral pricing discounts the probability

Q (S_{T} < K ∣ S_{t} = S) = Φ (- d_{-})

.

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