Ponder This Challenge - August 2006 - Value of future contract

Ponder This Challenge:

Puzzle for August 2006.

Prediction markets trade in contracts that pay off according to whether some future event occurs. If the event occurs the contract pays off 1 unit otherwise the contract becomes worthless. The price of a contract can be interpreted as the market's opinion on the probability that the future event will occur.

Suppose individuals A and B think the probability that the future event will occur is p or q respectively. Suppose A and B have risk capital X and Y respectively. Suppose A and B invest so as to maximize the expected log of their capital. What price r for the contract will clear the market (ie assure that the sum of the demand for the contract from A and B is 0). We are assuming contracts can be sold short which corresponds to risking (1-r) units to gain r units if the event does not occur. We are also ignoring all frictional costs like commissions or interest.

I came up with this problem myself but I doubt it is original. Please don't submit solutions you haven't derived yourself.

NOTE - Some clarifications:

• Log refers to the logarithm function.
  • Don't worry about whether the number of contracts is an integer.

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Solution

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  Answer:

  Let the demand for the contract from A and B be a and b respectively. Negative values of a or b correspond to a desire to sell the contract short. Expected log wealth for A will is

  p*log(X+a*(1-r))+(1-p)*log(X-a*r)

  This has derivative with respect to a of

  p*(1-r)/(X+a*(1-r)-(1-p)*r/(X-a*r)

  Setting the derivative to 0 and solving for a we find

  p*(1-r)*(X-a*r)=(1-p)*r*(X+a*(1-r))

  ( p*(1-r)-(1-p)*r)*X=((1-p)*r*(1-r)+p*r*(1-r))*a

  (p-r)*X=r*(1-r)*a

  a=(p-r)*X/(r*(1-r))

  Similarly

  b=(q-r)*Y/(r*(1-r))

  Setting a+b=0 and solving for r we obtain

  (p-r)*X/(r*(1-r)) + (q-r)*Y/(r*(1-r)) = 0

  (p-r)*X + (q-r)*Y = 0

  p*X+q*Y=(X+Y)*r

  r=(p*X+q*Y)/(X+Y)

  So the market clearing price is just the weighted average of the opinions of A and B as to the correct price.

  ---

  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Ross Millikan (08.02.2006 @04:18:54 PM EDT)
• Joe Fendel (08.02.2006 @06:10:04 PM EDT)
• V. Balakrishnan (08.02.2006 @10:44:56 PM EDT)
• Vincent Vermaut (08.03.2006 @02:44:04 AM EDT)
• Alan Murray (08.03.2006 @10:21:02 AM EDT)
• Dion K. Harmon (08.03.2006 @06:00:45 PM EDT)
• Joseph DiVincentis (08.03.2006 @07:12:35 PM EDT)
• Mark Perkins (08.03.2006 @07:15:48 PM EDT)
• Hagen von Eitzen (08.04.2006 @12:50:20 AM EDT)
• James Dow Allen (08.04.2006 @09:28:38 AM EDT)
• Jan Rubak (08.04.2006 @01:52:06 PM EDT)
• Rob Berger (08.04.2006 @04:51:38 PM EDT)
• Ashish Mishra (08.04.2006 @09:32:10 PM EDT)
• Ed Sheppard (08.05.2006 @06:24:52 PM EDT)
• Jay Sounderpandian (08.05.2006 @06:54:37 PM EDT)
• Se Kwon Kim (08.06.2006 @12:07:46 PM EDT)
• Firat Solgun (08.06.2006 @03:01:41 PM EDT)
• John Fletcher (08.06.2006 @07:23:21 PM EDT)
• Siddharth Joshi (08.07.2006 @04:09:08 AM EDT)
• Frank Mullin (08.07.2006 @08:35:57 PM EDT)
• Paulo Natenzon (08.08.2006 @02:53:39 AM EDT)
• Phil Muhm (08.08.2006 @09:18:10 AM EDT)
• Mark Pilloff (08.09.2006 @02:48:27 AM EDT)
• Arthur Breitman (08.09.2006 @03:22:30 PM EDT)
• Grey Zone (08.10.2006 @04:17:02 PM EDT)
• Didier Bizzarri (08.11.2006 @05:05:49 AM EDT)
• Yoav Raz (08.13.2006 @02:37:53 AM EDT)
• Aleksandar Tamburov (08.13.2006 @04:14:57 PM EDT)
• Dan Dima (08.14.2006 @05:20:07 PM EDT)
• Wolf Mosle (08.15.2006 @12:35:26 PM EDT)
• Dan Stronger (08.16.2006 @03:26:25 AM EDT)
• Dick King (08.16.2006 @02:14:16 PM EDT)
• Du Yang (08.17.2006 @12:53:32 PM EDT)
• Pawan Agarwal (08.18.2006 @09:10:46 AM EDT)
• Oleg Trott (08.18.2006 @06:42:50 PM EDT)
• David McQuillan (08.19.2006 @04:41:00 PM EDT)
• Arjun Acharya (08.28.2006 @12:36:03 PM EDT)
• Ben North (08.29.2006 @05:09:01 AM EDT)
• Michael Quist (08.29.2006 @01:28:35 PM EDT)
• Mahmoud Elhaddad (08.30.2006 @12:34:24 AM EDT)
• Fatih Mehmet Dogu (08.30.2006 @05:00:54 AM EDT)