 Solution

ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2021/2/d/solution

p > 0

and

K > 0

, the payoff is positive exactly when

S_{T} > k := K^{1/ p}

. Splitting it into two truncated moments gives

e^{- r τ} [s^{p} V_{p} (t, s, K^{1/ p}) - K V_{0} (t, s, K^{1/ p})] .

For

p = 0

, the payoff is the constant

(1 - K)^{+}

, so its price is

e^{- r τ} (1 - K)^{+}

. For

K = 0

and

p > 0

, its price is

e^{- r τ} s^{p} M_{p} (τ)

. These boundary cases avoid taking

K^{1/ p}

when

p = 0

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