Fair Price. Math 5 Crew. Department of Mathematics Dartmouth College. Fair Price p.1/??
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1 Fair Price p.1/?? Fair Price Math 5 Crew Department of Mathematics Dartmouth College
2 Fair Price p.2/?? Historical Perspective We are about ready to explore probability form the point of view of a free and efficient market.
3 Fair Price p.2/?? Historical Perspective We are about ready to explore probability form the point of view of a free and efficient market. As a popular view of probability, this view is rather recent (last 20 years). However, the very first text on probability took this view and is the inspiration for what follows. This was Christian Hygen s amazingly cool Value of all Chances in Games of Fortune.
4 Fair Price p.2/?? Historical Perspective We are about ready to explore probability form the point of view of a free and efficient market. As a popular view of probability, this view is rather recent (last 20 years). However, the very first text on probability took this view and is the inspiration for what follows. This was Christian Hygen s amazingly cool Value of all Chances in Games of Fortune. In summary: Huygens s Rocks!
5 Fair Price p.3/?? The Bush Bet Imagine you are given a credit line to buy and sell shares of various bets. You buy and sell your bets in some market.
6 Fair Price p.3/?? The Bush Bet Imagine you are given a credit line to buy and sell shares of various bets. You buy and sell your bets in some market. Here s one bet. X is 100 dollars if George Bush becomes president and zero otherwise. Let us call a randomly determined number like X a Random Variable.
7 Fair Price p.3/?? The Bush Bet Imagine you are given a credit line to buy and sell shares of various bets. You buy and sell your bets in some market. Here s one bet. X is 100 dollars if George Bush becomes president and zero otherwise. Let us call a randomly determined number like X a Random Variable. Suppose you can buy X for b dollars, and sell X for s dollars. Can you sense any conditions that b and s are guaranteed to satisfy?
8 Fair Price p.4/?? The Efficient Market Hypothesis Well if you buy and sell one share of X, then you will have b s dollars of debt in your credit account until the election.
9 Fair Price p.4/?? The Efficient Market Hypothesis Well if you buy and sell one share of X, then you will have b s dollars of debt in your credit account until the election. After the election you receive and pay 100 dollars, so you still have b s dollars worth or debt.
10 Fair Price p.4/?? The Efficient Market Hypothesis Well if you buy and sell one share of X, then you will have b s dollars of debt in your credit account until the election. After the election you receive and pay 100 dollars, so you still have b s dollars worth or debt. Hence b s or there exist free money! We call this situation an Arbitrage opportunities, and the hypothesis that there are no opportunities for Arbitrage is the No Free Lunch part of the Efficient Market Hypothesis.
11 Fair Price p.5/?? The Fair Price In reality, we find we can buy X for 65 dollars, and sell X for 64.3 dollars.
12 Fair Price p.5/?? The Fair Price In reality, we find we can buy X for 65 dollars, and sell X for 64.3 dollars. As expected, there is no Arbitrage.
13 Fair Price p.5/?? The Fair Price In reality, we find we can buy X for 65 dollars, and sell X for 64.3 dollars. As expected, there is no Arbitrage. A Fair Price for X would be a price that one could buy or sell X at among friends". Let us call this Fair Price E(X). Let s try to make sense out of this rather fishy notion.
14 Fair Price p.6/?? The Transaction Fee If we are not betting among friends", then our transaction will always include a per share Transaction Fee. Call this fee f.
15 Fair Price p.6/?? The Transaction Fee If we are not betting among friends", then our transaction will always include a per share Transaction Fee. Call this fee f. A reasonable notion of Fair Price will satisfy b = E(X) + f and and s = E(X) f. Hence f = (b s)/2.
16 Fair Price p.6/?? The Transaction Fee If we are not betting among friends", then our transaction will always include a per share Transaction Fee. Call this fee f. A reasonable notion of Fair Price will satisfy b = E(X) + f and and s = E(X) f. Hence f = (b s)/2. For our Bush bet, f = =.35 E(X) =
17 Fair Price p.7/?? The Bush Bet: Should You Take It? Suppose you were offered to buy a share of the Bush bet for a mere 55 dollars!
18 Fair Price p.7/?? The Bush Bet: Should You Take It? Suppose you were offered to buy a share of the Bush bet for a mere 55 dollars! How many shares of this Bush Bet bet might you be tempted to buy?
19 Fair Price p.7/?? The Bush Bet: Should You Take It? Suppose you were offered to buy a share of the Bush bet for a mere 55 dollars! How many shares of this Bush Bet bet might you be tempted to buy? There are many possible factors, but in this model it will depend almost entirely on your access to the market.
20 Fair Price p.8/?? Selling and Buying Bets Among Friends. We can view the selling of X as the buying of 1 share of X.
21 Fair Price p.8/?? Selling and Buying Bets Among Friends. We can view the selling of X as the buying of 1 share of X. Among friends there is no problem at all doing this, and
22 Fair Price p.8/?? Selling and Buying Bets Among Friends. We can view the selling of X as the buying of 1 share of X. Among friends there is no problem at all doing this, and if we are buying this bet on the market there will be transaction fees associated both to the bets we buy and those we sell. (Consider the bet X X. )
23 Fair Price p.9/?? In a free market there will be LOTS of bets In our market there are many bets. Since transaction fees exist, pretty much any bet that people are willing to make will exist. We could call this the When There s Cash There s a Way part of our efficient market hypothesis.
24 Fair Price p.9/?? In a free market there will be LOTS of bets In our market there are many bets. Since transaction fees exist, pretty much any bet that people are willing to make will exist. We could call this the When There s Cash There s a Way part of our efficient market hypothesis. For example, let Y be 100 dollars if Lord of the Rings wins the Oscar for best picture and zero other wise. This bet will exist.
25 Fair Price p.9/?? In a free market there will be LOTS of bets In our market there are many bets. Since transaction fees exist, pretty much any bet that people are willing to make will exist. We could call this the When There s Cash There s a Way part of our efficient market hypothesis. For example, let Y be 100 dollars if Lord of the Rings wins the Oscar for best picture and zero other wise. This bet will exist. In reality we find E(Y ) =
26 Fair Price p.10/?? New Bets From Old Let us buy 12 shares of Y and sell 7 shares of X. What is the Fair Price of this bet?
27 Fair Price p.10/?? New Bets From Old Let us buy 12 shares of Y and sell 7 shares of X. What is the Fair Price of this bet? Notice we can think of our bet as buying 12Y 7X, and E(12Y 7X) is the "debt" in our credit account after placing this debt.
28 Fair Price p.10/?? New Bets From Old Let us buy 12 shares of Y and sell 7 shares of X. What is the Fair Price of this bet? Notice we can think of our bet as buying 12Y 7X, and E(12Y 7X) is the "debt" in our credit account after placing this debt. Among friends, we find that after placing this bet we have 12E(Y ) 7E(X) = (12)(83.85) (7)(64.65) = dollars worth of debt in our account hence E(12Y 7X) = 12E(Y ) 7E(X) (by the No Free Lunch and the When There s Cash There s a Way Hypotheses.)
29 Fair Price p.11/?? First Fundamental Mystery of Probability Let X and Y be a pair of Random Variables, let c and d be constants and let E(X) denote the expected value of X. Then we have
30 Fair Price p.11/?? First Fundamental Mystery of Probability Let X and Y be a pair of Random Variables, let c and d be constants and let E(X) denote the expected value of X. Then we have the FFMP E(cX + Y + d) = ce(x) + E(Y ) + d
31 Fair Price p.12/?? The Bush of the Rings Bet Now let us try and find the Fair Price for XY.
32 Fair Price p.12/?? The Bush of the Rings Bet Now let us try and find the Fair Price for XY. This is MUCH trickier. By the When There s Cash There s a Way hypothesis this bet will be offered, but what is its Fair Price?
33 The Bush of the Rings Bet Now let us try and find the Fair Price for XY. This is MUCH trickier. By the When There s Cash There s a Way hypothesis this bet will be offered, but what is its Fair Price? Question: Do we believe that whether or not the Lord of the Rings wins the best picture Oscar will effect Bush s chances of being elected president? If no, we would call X and Y independent. Fair Price p.12/??
34 Fair Price p.13/?? Fair price of XY Suppose we believe X and Y are independent.
35 Fair Price p.13/?? Fair price of XY Suppose we believe X and Y are independent. Now IF E(X) is not changing between the time we make our bet and the time of the Academy Awards, then...
36 Fair Price p.13/?? Fair price of XY Suppose we believe X and Y are independent. Now IF E(X) is not changing between the time we make our bet and the time of the Academy Awards, then... I can purchase E(X) shares of Y now. Once Y is determined (the Academy Awards) I will have Y E(X) dollars. With my Y E(X) dollars (and the above IF) I can purchase Y shares of X, in other words XY. So at the end of the day I will have purchased XY for the same price as Y E(X), which by the first fundamental mystery has fair price E(X)E(Y ).
37 Fair Price p.14/?? Second Fundamental Mystery of Probability Let X and Y be independent random variables and let E(X) denote the expected of value of X. Then we have
38 Fair Price p.14/?? Second Fundamental Mystery of Probability Let X and Y be independent random variables and let E(X) denote the expected of value of X. Then we have the SFMP E(XY ) = E(X)E(Y )
39 Fair Price p.14/?? Second Fundamental Mystery of Probability Let X and Y be independent random variables and let E(X) denote the expected of value of X. Then we have the SFMP E(XY ) = E(X)E(Y ) Notice without the big IF this really is a mystery to us at this point! Getting rid of the big IF would require to figure out how to hedge a bet in an efficient market. Later we will (may?) discuss this concept. For now let us just appreciate the mystery!
40 Fair Price p.15/?? Bush of the Rings Fair Price Since X and Y be independent random variables we have from the SFMT
41 Fair Price p.15/?? Bush of the Rings Fair Price Since X and Y be independent random variables we have from the SFMT E(XY ) = E(X)E(Y ) = (83.85)(64.65) =
42 Fair Price p.15/?? Bush of the Rings Fair Price Since X and Y be independent random variables we have from the SFMT E(XY ) = E(X)E(Y ) = (83.85)(64.65) = Notice if either Bush or Lord of the Rings loses, then I get nothing. If they both win I get That this bet is fair tells me that the current belief is that both Bush AND Lord of the Rings winning is a better than even bet.
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