 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2009/b/5/a/solution

Write

τ = T - t

d_{1}

and

d_{2}

depend on

x, K

only through

lo g (x / K)

. Replacing them by

λ x, λ K

leaves that ratio unchanged. Each prefactor in the put formula acquires the same factor

λ

p (t, λ x; λ K, T) = λ K e^{- r τ} Φ (- d_{2}) - λ x Φ (- d_{1}) = λ p (t, x; K, T) .

This is degree-one homogeneity of the European put option price in its spot and strike together.

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