Replicating a Power Option with Asset-or-Nothing Calls

Summary

The note rewrites the payoff of a power option, defined as the underlying raised to a power times a call payoff, as a difference of two asset-or-nothing binary call payoffs. It first expands the payoff into two terms, each gated by the condition that the underlying exceeds the strike. Under a positive underlying and the usual positive-power assumptions, that threshold can be expressed using powers of both the underlying and strike, yielding the two binary payoffs.

This is an algebraic payoff identity rather than a pricing method or trading strategy. The note gives no market data, empirical evidence, or valuation formula. Its threshold manipulation depends on assumptions about the domain and exponent: for arbitrary real powers, especially when the exponent changes the ordering of values, the stated equivalence need not hold. Those conditions should be checked before using the representation.

Key ideas

• A power option payoff can be split into two terms by expanding the call payoff above its strike.
• The terms can be represented with asset-or-nothing binary calls on powered underlying values.
• The transformed strikes are powers of the original strike.
• The threshold equivalence requires suitable assumptions on the underlying and exponent.

Tags

# A Question from "Mathematical Methods for Financial Markets" Chapter 2


# A Question from "Mathematical Methods for Financial Markets" Chapter 2












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 $\beta$.

## Answer by Drmanifold (score 3, accepted)

https://quant.stackexchange.com/a/9742

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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