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Asset-or-Nothing Option Prices and Put-Call Parity

Article Quant Q&A · Author: Sean Willington
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Summary

The note gives the Black–Scholes put-call parity relation for asset-or-nothing options. It expresses the asset-or-nothing put price as the prepaid underlying value less the corresponding call price, then substitutes the call pricing formula to obtain the put value in terms of the normal cumulative distribution function evaluated at the negative of the usual first Black–Scholes parameter.

The derivation relies on the symmetry of the standard normal distribution, which makes the call and put expressions complementary. It provides a compact pricing identity, but does not develop the underlying payoff definitions, derive the full call and put formulas, or discuss assumptions such as dividends, exercise timing, or market conditions. Readers seeking a complete valuation treatment will need those additional details.

Key ideas

  • Asset-or-nothing options pay the underlying asset or nothing depending on whether the exercise condition is met.
  • The put price equals the prepaid underlying value minus the call price under the stated parity relation.
  • The put formula follows by applying standard normal symmetry to the call probability term.

Tags

OptionsDerivatives pricing
Full text
# Asset-or-nothing Option Valuation in the Black and Scholes model


# Asset-or-nothing Option Valuation in the Black and Scholes model












In standard Black-Scholes Model, compute the price of an asset-or-nothing put and asset-or-nothing call options. Write down the put-call parity relation between the asset-or-nothing call and put option prices.

## Answer by Probilitator (score 2, accepted)

https://quant.stackexchange.com/a/10413

the answer for calculating the prices can be found here - see chapter: Black–Scholes valuation ;)

The put-call parity in that case is pretty straight forward: $P=Se^{-qT}-C$. Using the results presented on the Wikipedia page in the aforementioned section this can be proved as follows

$P=Se^{-qT}-C$

$=Se^{-qT}-Se^{-qT}\Phi(d_1)$

$=Se^{-qT}(1-\Phi(d_1))=Se^{-qT}\Phi(-d_1)$

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.