 Solution

ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2015/3/a/ii/solution

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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