23_dynamic_replication

23_dynamic_replication - Econ 251a Fall 2005 Dynamic...

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Econ 251a Dynamic Replication Fall 2005 1 Full Spanning with Two States 1.1 Arrow Securities An Arrow security is an asset that pays o f 1 in a single state, and 0 in every other state. Suppose all the Arrow securities are marketed, i.e. are available to be bought or sold at known prices. Then we can replicate every other security with a portfolio of Arrow securities. If the target security pays A 1 A 2 A S then the replicating portfolio of Arrow securities would obviously consist of A 1 of the f rst Arrow security, A 2 of the second Arrow security, and A S o fthelastArrow security. We can also deduce the no-arbitrage price of every targeted asset.by taking the sum of the prices of the Arrow securities in its replicating portfolio.Call the price today of Arrow security s, π s . Then the no-arbitrage price P of the targeted security must be P = X s π s A s If the targeted asset was available to buy or sell at a di f erent price, then we could f nd an arbitrage between it and its replicating portfolio. 1.2 Arrow Securities as Targets Arrow securities are usually not directly traded in the market, although they clearly ought to be. (They are becoming directly traded more often). But just because they are not directly traded does not mean that we cannot deduce what their prices would be if they were traded. Suppose we have a collection of benchmark securities, i.e. securities that are traded at prices we know (and will not change just because we want to trade them). Suppose we can f nd a portfolio of benchmark securities that replicates the f rst Arrow security. Then by no-arbitrage, we can price the f rst Arrow security. Suppose we could f nd another portfolio of benchmark securities that replicated the second Arrow security. Then we could price it by no arbitrage. Suppose in fact we could f nd (di f erent) benchmark portfolios to replicate every Arrow security. Then we could
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price every Arrow security. And then by following the logic in the last section, we could price every targeted security by no-arbitrage. The replicating portfolio of the targeted security with payo f s A described above would consist of a multiple A 1 of the replicating portfolio of the f rst Arrow security, a multiple A 2 of the portfolio replicating the second Arrow security, and so on. One interesting computational trick we shall often employ is to notice that it is easier to f nd all the Arrow prices at the same time then to f nd each of them one at a time. If we know that all the Arrow securities can be replicated, then we can f nd all their prices without having to f nd the replicating portfolios of any one of them. 1.3 Two State Example with Stock and Bond as Benchmarks Imagine a one-period period model with two states. Insert Figure 1
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23_dynamic_replication - Econ 251a Fall 2005 Dynamic...

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