 Solution

ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2015/2/d/solution

a = σ τ

and

m = lo g (S / K) + (r - σ^{2} /2) τ

. Then

lo g (S_{T} / K) = m + a Z

with

Z

d_{-} = m / a

V (S, t) = e^{- r τ} \int_{- d_{-}}^{\infty} (m + a z) ϕ (z) d z = e^{- r τ} [m Φ (d_{-}) + a ϕ (d_{-})],

since

ϕ^{'} (z) = - z ϕ (z)

. Therefore

V (S, t) = e^{- r τ} σ τ [d_{-} N (d_{-}) + \frac{e ^{- d_{-}^{2} /2}}{2 π}] .

Here

ϕ

and

Φ = N

(lo g (S / K))^{+}

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