Skip to content
All library documents

Expected Probability as the Fair Price of a Binary Claim

Article Quant Q&A · Author: wainwain
Read the original

Summary

The answer explains an intuitive link between a probability and the price of a binary claim. If an event pays one unit when it occurs and nothing otherwise, its expected payoff under the relevant pricing assumptions is the event probability. On that basis, the response interprets a probability-valued quantity as a fair price for a claim paying according to the event outcome, and says that an estimate based on finite samples may differ from the underlying expectation.

The explanation argues that an Arrow-Debreu framing may add formality without changing this basic expected-payoff intuition. Its limits are material: the answer assumes a standard fair-value treatment and does not specify a risk-neutral measure, discounting, market frictions, or whether the probability is subjective. It also notes that preferences with nonlinear utility across outcomes can produce a valuation that depends on more than the mean. The paper motivating the question is not examined directly.

Key ideas

  • A unit-payoff binary claim has expected payoff equal to the probability of its event under the stated pricing assumptions.
  • Finite-sample estimates of an event probability can differ from its underlying expectation.
  • The answer treats expected value as the key input to fair pricing in its simplified framework.
  • Nonlinear utility or differing outcome preferences can make valuation depend on more than the mean.

Tags

Derivatives pricingStatisticsOptions
Full text
# Why a probability distribution can be viewed as a price?


# Why a probability distribution can be viewed as a price?












in this paper : at page 111, left part, we pass from the distribution p to price p. But I don't see why it's the case. I've searched, and it seems like we can see the LMSR as a n Arrow-Debreu security or a binary option. But I don't really see why it's the case. The transition in this paper is quite fast an unexplained. Thank you for any kind of help!

## Answer by demully (score 4)

https://quant.stackexchange.com/a/59782

I didn't want to pay to download the paper, but the intuitive answer is that if the probability of event X is P, then a binary option on X (with a 0-vs-1 payoff) will have a fair price of FV = E(P).

The distribution around P will obviously differ in finite samples; but as sample size -> infinity, this will/should converge towards E(P). Anything else would represent a FV estimate that is biased with respect to P. In truth, FV does not care about the distribution around P. It cares only for the expected value of P = E(P).

This one is easy to overthink ;-) Formally tying everything back to Arrow-Debreu very often just serves to complicate the simple, giving the same answer. My answer is "wrong", if different segments of the distribution of P have different utility values. Then FV is an integral function of F(X,P,utility). Which differs from case to case with respect to one's subjective utility function in any case.

Assuming that the fair price of any distribution P is E(P) is a quite standard financial assumption, which is probably why your article didn't justify its assumptions in any detail.

hope this helps, DEM

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.