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Binary FX Option Prices Imply Forward-Measure Probabilities

Article Quant Q&A · Author: Zakoff
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Summary

The exchange clarifies how to interpret probabilities inferred from prices of binary FX options. The answer says the probability associated with a binary paying when the terminal forward exceeds its strike is a risk-neutral probability under the forward measure. The option value is the discount factor multiplied by that probability.

This distinction matters because a market-implied probability is not, in general, a forecast of the real-world frequency of an FX event. The answer also gives a special case: when interest rates are deterministic, the forward measure and risk-neutral measure coincide. The explanation is concise and supplies no worked numerical example or discussion of calibration, market frictions, or how to estimate probabilities from a full option surface. Its conclusion concerns the pricing measure implied by the binary payoff.

Key ideas

  • A binary FX option price reflects a probability under the forward measure.
  • The payoff condition compares the terminal forward with the option strike.
  • The binary value equals the discount factor times the forward-measure probability.
  • With deterministic rates, the forward and risk-neutral measures coincide.
  • The implied probability is a pricing quantity rather than automatically a real-world forecast.

Tags

ForexOptionsDerivatives pricing
Full text
# Is probability implied by binary FX options risk neutral or real world?


# Is probability implied by binary FX options risk neutral or real world?












If we consider binary FX options in the market and estimate the market implied probabilities of certain FX rates occurring, would these resulting probabilities be risk neutral or real world?

I hear the term "market implied probability" being used in the work place estimated from binary options, I am not sure if this relates to risk neutral or real world?

## Answer by Mark Joshi (score 4, accepted)

https://quant.stackexchange.com/a/18652

risk-neutral. Really the forward measure. The price of the binary is struck at $K$ is $$ Z P( F_T > K) $$ with $Z$ the discount factor and $F_T$ the forward, and $P$ the probability in the forward measure. If rates are deterministic, the forward measure and the risk-neutral measure will agree.

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.