 Solution

ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2017/2/b/solution

Z \sim N (0, 1)

a = σ τ

, and

m = lo g (S / K) + (r - σ^{2} /2) τ

. Then

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

and

d_{-} = m / a

e^{- r τ} E [(m + a Z)^{+}]

Let

ϕ

and

Φ

E [(m + a Z)^{+}] = \int_{- d_{-}}^{\infty} (m + a z) ϕ (z) d z = m Φ (d_{-}) + a ϕ (d_{-}),

because

ϕ^{'} (z) = - z ϕ (z)

. Therefore

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

For

τ = 0

take the continuous terminal limit

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

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