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ecg590i_lecture09 - ECG590I Asset Pricing. Lecture 9: The...

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ECG590I Asset Pricing. Lecture 9: The Arbitrage Theorem 1 9 The Arbitrage Theorem 9.1 To arbitrage is to take simultaneous positions in di/erent assets in a way John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 9: The Arbitrage Theorem 2 Types of arbitrage opportunities The opportunity to make investments that have no current For example, short-sell one asset and use the proceeds to buy another asset in such a way as to make the portfolio riskless. The opportunity to make investments that have a neg- ative net commitment today (i.e., money comes to you and you have some left over after you make your asset purchases) and that yield Fair price (correct price). Prices are fair (or correct) if and only if there are no arbitrage opportunities. John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 9: The Arbitrage Theorem 3 9.2 Notation Asset prices. Individual security prices are denoted by S i ( t ) . The set of prices for all (relevant) assets is the vector S t = 2 6 6 6 4 S 1 ( t ) S 2 ( t ) . . . S N ( t ) 3 7 7 7 5 John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 9: The Arbitrage Theorem 4 States of the world W = 2 6 6 6 4 w 1 w 2 . . . w K 3 7 7 7 5 Each w i represents a distinct outcome (or state ) that may occur. The states are mutually exclusive. At least one state is guaranteed to occur. John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 9: The Arbitrage Theorem 5 Returns and payo/s d ij = payment in one period made on security i when state j prevails Components of d ij ± ± Dividends or coupon payments. Payment matrix. For the N assets, we have the payment matrix D = 2 6 4 d 11 ²²² d 1 K . . . . . . . . . d N 1 ²²² d NK 3 7 5 ± Rows are payments from a particular security in di/erent states. ± Columns are payments from di/erent assets in a particular state. John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 9: The Arbitrage Theorem 6 Portfolio A particular combination of the assets in question. Represented by the vector = 2 6 6 6 4 1 2 . . . N 3 7 7 7 5 where i is the amount/number of asset in the portfolio. Negative i represents a short position in the i th asset. John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 9: The Arbitrage Theorem 7 9.3 Example 1. Three assets Asset 1: riskless with initial value B ( t ) and gross return of (1 + r , where r is the riskless rate and is the length of the period. Asset 2: underlying risky asset with value

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ecg590i_lecture09 - ECG590I Asset Pricing. Lecture 9: The...

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