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

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Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 3 / b / iv / Solution by

Codex  0 2026-10-25

0 < B < K

, every surviving terminal price is already below the strike, so this is a barrier digital put paying one at expiry exactly when the upper barrier has been avoided. To use the Black-Scholes reflection symmetry, cap the free digital-put strike at

B

, rather than continuing to use

K

H (S, t) = D_{B} (S, t) - (S / B)^{γ} D_{B} (B^{2} / S, t), 0 < S < B < K .

(1)

Indeed,

D_{B}

has terminal payoff one throughout the alive region, while its reflected argument exceeds

B

and hence has zero terminal payoff. The barrier values again cancel. Using

D_{K}

in both terms would generally leave an unwanted reflected terminal payoff near the barrier.

Writing

b = r - σ^{2} /2

makes the discounted survival probability explicit:

H (S, t) = e^{- r τ} [Φ (\frac{lo g ( B / S ) - b τ}{σ τ}) - (S / B)^{γ} Φ (\frac{lo g ( S / B ) - b τ}{σ τ})] .

(2)

The price is independent of

K

in this case. Both cases combine by replacing the free strike with

min (K, B)

in the up-and-out digital put formula.

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