Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Exercise 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 $\beta$.

share|improve this question
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:
$$h(S_t) =\Phi_\text{Asset-or-nothing}^\text{dig}(S_{t}^{\beta + 1},K^{\beta + 1}) - K \,\Phi_\text{Asset-or-nothing}^\text{dig}(S_{t}^{\beta}, K^{\beta}) $$

share|improve this answer
Thank you Drmanifold for providing a great answer and thank you William for editing the answer. – Liwei Zhang Dec 20 '13 at 0:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.