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Unformatted text preview: Economics 395 Risk and Uncertainty Problem Set 5 Answer Key Question Nineteen A market maker (MM) trades in an asset with uncertain value, V. The MM’s prior over the asset’s value is uniformly distributed between $0 and $100. The MM will quote a bid price, P B (the price at which the MM will buy one unit of the asset from a trader), and an ask price, P A (the price at which the MM will sell one unit of the asset to a trader). There are two types of trader in the market: Informed traders know with certainty the value of the asset, and will trade on the basis of this information. An informed trader will buy if V > P A and will sell if V < P B . Liquidity traders employ no information in their trades. Liquidity traders will buy with probability 1/3, will sell with probability 1/3 and will choose not to trade with probability 1/3. Liquidity traders account for a proportion of the market denoted by α. The proportion of the market that is informed is (1 – α). Suppose a market maker selects P A = $74 and P B = $26. (a) Calculate the expected profits from trading with an uninformed (liquidity) trader. The liquidity trader buys, sells or doesn’t trade each with probability 1/3. The liquidity trader provides no information, so the expected value of the asset is simply 50. If the liquidity trader buys or sells, then, MM makes expected profits: E[profit| trade with liquidity trader] = (1/3)[74 – 50] + (1/3) + (1/3)[50 – 26] = 17.33 (b) If MM meets an informed trader, what is the probability that the informed trader will wish to buy the asset at price P A ? Conditional on an informed trader buying the asset, what is MM’s expected value of the asset? Conditional on an informed agent buying the asset, what is MM’s expected profit from the trade? If an informed trader can buy at a price of 74, the probability of buying is equal to the probability that the value of the asset is greater than 74. As the value is uniformly distributed over the range 0 to 100, this probability is (100-74)/100 = 0.26. If the informed trader buys, this implies that the value of the asset must be between 74 and 100. As the prior distribution is uniform over the range 0 to 100, the posterior distribution is uniform over the range 74 to 100. The expected value, conditional on this information, is E[v| infomed trader buys] = (100+74)/2 = 87 Conditional on the informed trader buying, the expected profit is E[profit| informed trader buys] = 74 – E[v| infomed trader buys] = -13. (c) If MM meets an informed trader, what is the probability that the informed trader will wish to sell the asset to MM at price P B ? Conditional on an informed trader selling the asset, what is MM’s expected value of the asset? Conditional on an informed agent selling the asset to her, what is MM’s expected profit from the trade?...
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This note was uploaded on 04/13/2010 for the course ECON 330 taught by Professor Minetti during the Fall '08 term at Michigan State University.
- Fall '08