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14_fullspan - Econ 251a Uncertainty and Arbitrage Full...

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Econ 251a Fall 2008 Uncertainty and Arbitrage: Full Spanning Arbitrage under uncertainty is exactly the same as arbitrage over time without uncertainty. We look for ways of combing some securities into a portfolio which reproduces exactly the payo ff s of another security. 1 Spanning We consider a world with only two time periods, today (time 0), and next year or tomorrow (time 1). Tomorrow one of S states may occur. The states of the world, or possible worlds for short, is one of the most brilliant conceptualizations in history. It is usually attributed to the great philosopher and mathematician Leibnitz, who among other things invented calculus at the same time as Isaac Newton. Leibnitz coined the immortal (and very optimistic) expression “we live in the best of all possible worlds.” Let us begin with a collection of J securities that can easily be bought or sold. We shall call these the benchmark securities. We have used the coupon Treasury bonds as an example already. If the benchmark securities can be combined into a portfolio that perfectly reproduces the payo ff s of some new security, then we say that the new security is spanned by the benchmarks. A portfolio of securities is de fi ned by the holdings (positive or negative) of each security. We denote the holding of security j by θ j . We choose a Greek letter to distinguish security holdings from commodity consumption, and the letter θ is the easiest greek letter to type on my word processor after π , which is reserved for the prices of special benchmark securities (like zeroes). Denote the payo ff s of each security j J by A j 1 A j 2 . . . A j S Denote the payo ff s of the new security by b 1 b 2 . . . b S The new security is spanned by the benchmark securities if there exist ( θ 1 , ..., θ J ) 1
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solving the system of equations A 1 1 θ 1 + A 2 1 θ 2 + ... + A J 1 θ J = b 1 A 1 2 θ 1 + A 2 2 θ 2 + ... + A J 2 θ J = b 2 . . A 1 S θ 1 + A 2 S θ 2 + ... + A J S θ J = b S When a θ j is positive, then asset j is held in the portfolio, and when θ j is negative, asset j is sold short in the portfolio. 2 Full Spanning If every imaginable security is spanned, then we say that the benchmark securities are fully spanning. Theorem: Suppose there are as many benchmark securities as there are states of the world. As long as the benchmark security payo ff s are independent, they will be fully spanning. Proof: Given an arbitrary security with payo ff s ( b 1 , ..., b S ) in the S states, and given the benchmark securities A 1 , ..., A J , the security is spanned by the assets if and only if we can solve the above S equations in J unknowns ( θ 1 , ..., θ J ) . When J = S, we have the same number of equations as unknowns and this is always possible, provided the asset payo ff s are linearly independent.
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