Pricing a Binary Contract Under Auctions and Differing Utilities
Summary
The document asks how to choose a quote for a binary contract with a stated probability of paying a fixed amount. Its expected value supplies a simple upper bound for a buyer seeking positive expected profit, but does not by itself determine a precise bid or spread. The responses emphasize that the answer depends on market structure: in a limit-order market, a buyer can set a maximum willingness-to-pay, while an English-style auction can expose bidders to winner's curse because winning may signal that their estimate was overly optimistic.
A separate response frames willingness to pay through utility: traders with different wealth, preferences, or risk tolerance can value the same contract differently. The discussion offers intuition rather than a complete quoting model. It does not derive an optimal bid using beliefs about counterparties, inventory, competition, or risk aversion, and the auction argument is presented without supporting data. The practical lesson is that expected value is a reference point, while market mechanism and trader utility shape an actual price decision.
Key ideas
- Expected value provides a reference ceiling for a buyer seeking positive expected profit, not a unique quote.
- The appropriate price depends partly on the market's trading mechanism.
- Auction bidders may face winner's curse when winning indicates that their estimate was unusually optimistic.
- Different utility functions and risk preferences can lead traders to assign different values to the same contract.
- The discussion is conceptual and does not provide a calibrated optimal-pricing method.
Tags
Full text
# How should the spread be determined after calculation of expected value? # How should the spread be determined after calculation of expected value? Suppose I am willing to buy a contract which I believe has a 15% chance to settle to $100 and 0 otherwise. The EV of this contract is therefore 15. How much should I buy this for? I would answer at most 15 (in order to ensure I make money on this contract). But going beyond this, how can the price on (0, 15) be determined? Is it arbitrary? Should the price be altered until other people are willing to make a trade with me? How can we make a better answer to this than "at most 15"? Suppose that we are trading against other people who do not necessarily agree on how much this contract may be worth. To make this more concrete, we were playing a trading game which was a toy model of a real trading scenario. I believed the scenario above, and I did not know which price on (0, 15) to quote, so I just quoted a pretty much arbitrary price until someone was willing to trade. How can I do better? ## Answer by Dave Harris (score 0, accepted) https://quant.stackexchange.com/a/70550 It depends upon the market structure. For example, if it were traded in a double auction with limit and market orders, you could simply issue an offer of buy with a limit of 15. With that said, if the contract is trading at 7, that would be no different than a market order. If it was trading at 20, then you don't want it anyway. In other types of markets, however, you would just want to be certain that the price did not exceed 15. In English-style auctions, you would want to bid less than 15 because the winning bid will follow a Gumbel distribution as its sampling distribution. Because of that, you will end up with a winner's curse and your estimated expectation will be too high. For a large enough number of bidders, and with infinite repetition, you will almost only win when you are certain to take a loss. ## Answer by TickaJules (score 0) https://quant.stackexchange.com/a/73319 Say I'm a person with current utility $u(x,y)$ (where $x$ is money and $y$ is everything else). Paying $p$ for this contract leaves me in only two possible states. I lose so end up with utility $u(x-p,y)$ or I win and end up with utility $u(x-p+100,y)$. So the shape of $u$ is going to determine my answer. Since each of us has different utility functions, we will each give our own answer. So sampling seems like all you can do.
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