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Risk-Neutral Prices of Cash-or-Nothing and Asset-or-Nothing Calls

Article Quant Q&A · Author: John
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Summary

The document raises a derivation question about two binary call options under a risk-neutral geometric Brownian motion model. It states the familiar form for an asset-or-nothing call, whose payoff is the underlying asset when its terminal price exceeds the strike, and for a cash-or-nothing call, which pays a fixed amount when that condition is met. The formulas use the standard normal cumulative distribution function and distinguish the terms associated with the two payoff types.

No derivation or answer is included in the supplied text. As a result, it identifies the pricing problem and the target formulas but does not show how the risk-neutral expectation is evaluated, explain the roles of the normal variables, or discuss assumptions such as dividends. Its value is mainly as a concise statement of the pricing topic and formulas to be derived.

Key ideas

  • The document asks how to derive binary option prices from risk-neutral expectations under geometric Brownian motion.
  • An asset-or-nothing call pays the underlying asset when its terminal price exceeds the strike.
  • A cash-or-nothing call pays a fixed amount when the terminal price exceeds the strike.
  • The supplied text gives target pricing formulas but contains no derivation or response.

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OptionsDerivatives pricing
Full text
# Cash-or-nothing and Asset-or-nothing price derivation


# Cash-or-nothing and Asset-or-nothing price derivation












I was wondering how to derive the price of a cash-or-nothing and asset-or-nothing option by trying to work out the expectation under the risk-neutral measure, while assuming that the underlying follows a Geometric Brownian motion.

I know that the value of the Asset-or-nothing call is supposed to be $Value = S_0\Phi(d_1)$

Furthermore the value of the Cash-or-nothing call should be $Value = e^{-rT}A\Phi(d2)$, if we assume that it pays out A if $S_T > K$.

Yet I don't know how to derive these results myself, and I haven't been able to find a book that does it

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