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ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2021/1/c/solution

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Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 1 / c / Solution by

Codex  0 2026-10-25

The discounted stock over the remaining horizon satisfies

e^{- r τ} S_{T} = s + σ \int_{t}^{T} e^{- r (u - t)} d W_{u}^{Q} .

The Itô integral on the right is a centered normal random variable with variance

a_{τ} = (1 - e^{- 2 r τ}) / (2 r)

. Therefore

Q (S_{T} \geq 0 ∣ S_{t} = s) = Φ (d)

, where

d = s / (σ a_{τ})

. The digital option price and its option delta are

V (t, s) = e^{- r τ} Φ (d), Δ (t, s) = \frac{e ^{- r τ} ϕ ( d )}{σ a _{τ}} .

The formula also shows

V (t, 0) = e^{- r τ} /2

for every

t < T

. At maturity use the specified step payoff, including its chosen value one at zero; the terminal discontinuity is not removed by the earlier-time smoothing.

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