Decomposing a European Call into Two Digital Option Payoffs

Summary

The document shows that a European call payoff can be represented as a long asset-or-nothing digital call combined with a short cash-or-nothing digital call, with both components expiring at the same time. When the underlying finishes above the strike, the asset-or-nothing payoff contributes the underlying price and the written cash-or-nothing payoff subtracts the strike. When it finishes below the strike, both digital payoffs are zero, matching the ordinary call’s payoff.

This is a payoff identity at maturity, derived by multiplying the difference between the underlying price and strike by an indicator for finishing above the strike. It clarifies the relationship among the three contracts, but does not compare their prices before maturity or discuss replication costs, settlement conventions, or the effects of early exercise. The result applies to European-style exercise at expiry and assumes the digital payoffs are defined consistently with the call’s strike condition.

Key ideas

• A European call pays the positive part of the underlying price minus the strike at expiry.
• A long asset-or-nothing call pays the underlying price when it finishes above the strike.
• A short cash-or-nothing call subtracts the strike amount in that same outcome.
• Both digital payoffs vanish below the strike, matching the European call payoff there.
• The identity compares expiry payoffs and does not by itself establish pre-expiry price equivalence.

Tags

# Answer by ir7 (score 3)


# How is holding an European call option equivalent to holding an asset-or-nothing call option and writing a cash-or-nothing call option?












The cash-or-nothing call option has a payoff that is equal to the strike price. All three options have the same expiry date.

## Answer by ir7 (score 3)

https://quant.stackexchange.com/a/9824

Formally, a long call payoff can be split as follows:

$$(S_T-K)^+ = (S_T-K)\cdot 1_{\{S_T>K\}} $$ $$= S_T\cdot 1_{\{S_T>K\}} - K\cdot 1_{\{S_T>K\}},$$ that is, long an asset-or-nothing digital call payoff and short a cash-or-nothing digital call payoff.

Here, $1_A$ is $1$ if event $A$ takes place, and it is $0$ otherwise.

## Answer by Liwei Zhang (score 1)

https://quant.stackexchange.com/a/9763

The payoff of an European call option is $(S_T-K)^+$. At maturity, if the spot price is greater than (or equal to) the strike price, then holding an asset-or-nothing call option has payoff $S_T$, writing a cash-or-noting call option $K$, which together give the payoff of the European call in this scenario. If the spot price is less than the strike, then all the three assets have 0 payoff. These suffice to show that the two are equivalent.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.