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Put-Call Parity for Asset-or-Nothing Binary Options

Article Quant Q&A · Author: Ryan J. Shrott
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Summary

The document explains a parity relationship for asset-or-nothing binary options. The call pays the underlying asset at maturity when its price is above the strike, while the put contributes a negative asset payoff when the price is below the strike. Subtracting the put payoff from the call payoff therefore reproduces the underlying asset payoff, except at the strike, an event treated as having zero probability in the answer.

By no-arbitrage pricing, the difference between the option prices must then equal the current underlying asset price. This gives a concise way to connect the prices of these binary contracts to the asset itself. The explanation assumes matching contract terms and ignores the exact-strike case; it does not discuss discounting, dividends, funding, or market frictions. The result is therefore a basic payoff identity under the stated assumptions, not a full treatment of every market convention or possible implementation.

Key ideas

  • An asset-or-nothing call pays the underlying asset when the terminal price exceeds the strike.
  • The corresponding put payoff is negative the underlying asset when the price is below the strike.
  • Subtracting these payoff definitions reproduces the underlying asset payoff away from the strike.
  • The no-arbitrage price difference between the options equals the current asset price under the stated assumptions.

Tags

OptionsDerivatives pricingArbitrage
Full text
# Put-Call Parity Arbitrage Exploitation for Binary-Asset-or-Nothing Options


# Put-Call Parity Arbitrage Exploitation for Binary-Asset-or-Nothing Options












Is the Put-Call-Parity valid for binary (asset-or-nothing) options? If not, is there another formula for such exotic options?

I know that for regular options, there are arbitrage opportunities when the put-call-parity does not hold.

Please note that I am very new to learning finance and I am not looking for overly complex answers.

## Answer by Mark Joshi (score 2, accepted)

https://quant.stackexchange.com/a/19566

the call version pays $$ I_{S_T > K } S_T $$ the put version pays $$ -I_{S_T < K } S_T $$

Subtract to get a pay-off $$ S_T. $$ (ignoring the probability zero event of $S_T=K.$)

So the prices subtract to give $S_0.$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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