 Solution

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

Let

a = σ τ

, so

d_{+} = d_{-} + a

. Squaring gives

d_{+}^{2} - d_{-}^{2} = 2 a d_{-} + a^{2} = 2 lo g (S / K) + 2 r τ .

S e^{- d_{+}^{2} /2} = K e^{- r τ} e^{- d_{-}^{2} /2} .

At fixed

S, t, T

, only

lo g (S / K)

depends on

K

\frac{\partial d _{+}}{\partial K} = \frac{\partial d _{-}}{\partial K} = - \frac{1}{K σ τ} .

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