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Exercise 2.3.1.5: The payoff of a power option is $h(S_T)$, where the function h is given by $h(x) = x^\beta(x-K)^+$. Prove that the payoff can be written as the difference of European payoffs on the underlying assets $S^{\beta+1}$ and $S^\beta$ with strikes depending on K and $\beta$.

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What have you tried already? –  olaker Dec 16 '13 at 7:00
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up vote 3 down vote accepted

We have,
$$ h(x) = x^\beta(x-K)^+ = x^\beta (x - K) \, \mathbf{1}_{[x>K]}$$ Thus we get, $$ h(x) = x^{\beta+1}\mathbf{1}_{[x>K]} - K\,x^{\beta}\mathbf{1}_{[x>K]}$$ now $x \in [x>K]$ if and only if $ x \in [x^{\beta}>K^{\beta}]$ Therefore,
$$ h(x) = x^{\beta+1}\mathbf{1}_{[x^{\beta + 1}>K^{\beta + 1}]} - K\,x^{\beta}\mathbf{1}_{[x^{\beta}>K^{\beta}]}$$

Thus if we let $\Phi_\text{Asset-or-nothing}^\text{dig}$ the contract function for an asset or nothing binary european call. we can Write:
$$h(S_t) =\Phi_\text{Asset-or-nothing}^\text{dig}(S_{t}^{\beta + 1},K^{\beta + 1}) - K \,\Phi_\text{Asset-or-nothing}^\text{dig}(S_{t}^{\beta}, K^{\beta}) $$

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Thank you Drmanifold for providing a great answer and thank you William for editing the answer. –  Zhangtingale Dec 20 '13 at 0:39
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