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ecg590i_lecture09

# 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 De°nition ° To arbitrage is to take simultaneous positions in di/erent assets in a way that guarantees a riskless pro°t higher than the return on the riskless asset. 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 ± First kind. The opportunity to make investments that have no current net commitment but that have a positive pro°t. For example, short-sell one asset and use the proceeds to buy another asset in such a way as to make the portfolio riskless. ± Second kind. 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 non-negative pro°t. ° 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 ± Capital gain ±change in the value of the asset. Can be negative. ± Dividends or coupon payments. ± Payment matrix. For the N assets, we have the payment matrix D = 2 6 4 d 11 ² ² ² d 1 K . . . . . . . . . d N 1 ² ² ² 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.

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