# How to price this contract in this scenario?

Here is the question:

You want to sell a painting you inherited from a grandparent. There is a 10% chance it is painted by a famous artist, in which case it's worth 100k. There is a 30% chance it is painted by an art student, in which case it's worth 2k. Otherwise, your grandparent must have painted it at a wine and paint night, in which case it's worth \$0.

What is the fair price to charge, right now, for a contract that entitles someone to the right, but not the obligation to pay 10k for the painting after we discover who painted it?

So the expected value of the painting is easy to calculate which is 10,600. But how do I use this information to price the contract? Should I price the contract slightly below the expected value to account for the risk involved? If so how much do I discount?

• Think of the situation as a wager. Then the wager amount plus the wagering gain must equal to \$90000 (so as to buy the painting at \$100000 overall). A 10% chance is given so that's \$90000 * 0.10 or \$9000 for the wager amount. The odds are 9 to 1 since ((\$9000 * 9) + \$9000) = \$90000. Well, since the chance was given at 10%, the wager breaks even if correct one time in ten. kbhscape.com/turf.htm . Or consider a prediction pool for the situation. Jul 15, 2023 at 22:36
• Or call it a a pari-mutuel pool. Numbers of speculators are encouraged to buy-in at a low price if the likelihood seems small. The pool can have something like a 20% take-out such that the winners are proportionally paid with the remaining funds. The odds change as the wagers build-up and re-weight. Or with a prediction pool then each speculator has locked in the price that they paid. Jul 15, 2023 at 23:21

## 1 Answer

Ignoring issues such as risk discounting, interest costs, etc., the value of an underlying is the integral from zero to infinity of p(x) x dx, where p(x) is the probability of the underlying being worth x at some future time. An option is worth max(p-s, 0) at expiry, where p is the spot price and s is the strike price. Since this is zero for x<s, the value of an option is the integral from s to infinity of p(x)(x-s) dx. In your case, you have a discrete distribution, so you can just take the sum over all x>s of p(x)(x-s).