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Pricing a Pay-Later Option with a Zero Upfront Premium

Article Quant Q&A · Author: glork
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Summary

The document asks how to set the contingent payment for a pay-later option whose holder receives the call payoff when the underlying finishes above the strike and then pays a fixed amount. Because the holder pays nothing at inception, the answer treats the option’s initial value as zero and imposes a risk-neutral valuation condition: the discounted expected terminal payoff must equal zero.

This gives the basic pricing equation for the contingent fee, with discounting canceling from both sides when the risk-neutral expectation is set to zero. The explanation is brief and does not derive a closed-form fee, specify a model for the underlying asset, or discuss assumptions such as interest rates, dividends, or settlement details. It therefore illustrates the no-arbitrage valuation principle for this payoff structure rather than providing a full calibration or implementation method.

Key ideas

  • A pay-later option has no upfront premium, so its initial value is set to zero.
  • The contingent fee is incorporated into the terminal payoff only when the underlying finishes above the strike.
  • Under risk-neutral valuation, the discounted expected payoff must equal the option’s zero initial value.
  • The explanation states the valuation condition but does not provide a model-specific fee formula.

Tags

OptionsDerivatives pricing
Full text
# no arbitrage condition for paylater option


# no arbitrage condition for paylater option












a paylater option has the folowing payoff: $(S_{T}-K)_{+}-P1_{S_{T}>K}$. To determine the fee P that the option holder must pay, we must write the non arbitrage condition. Why is it this: $E_{Q}[(S_{T}-K)_{+}-P1_{S_{T}>K}]=0$ ? I mean it would say the no arbitrage condition consists to write that the price of a paylater option is 0 ??

Thank you.

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/22038

The value of an option is the premium that is paid to own this option. For this paylater option, since nothing is paid upfront, the value of the option is zero. That is, \begin{align*} e^{-rT}E\big((S_{T}-K)^{+}-P1_{S_{T}>K}\big)=0, \end{align*} or \begin{align*} E\big((S_{T}-K)^{+}-P1_{S_{T}>K}\big)=0. \end{align*}

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.