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Pricing in Linear Algebra: Understanding Complete and Incomplete Markets

Explore the concept of complete and incomplete markets in pricing through a simple example. Learn how assets are priced in relation to existing assets, and how to identify arbitrage opportunities. Understand the importance of vector spaces in determining unique asset prices.

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Pricing in Linear Algebra: Understanding Complete and Incomplete Markets

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  1. Linear Algebra and Pricing Sy-Ming Guu College of Management, Yuan Ze University 2008/3/4

  2. 1 3 1 2 1 1 A Simple Example • There are two assets in the market.

  3. 5 ? 3 A Simple Example • There is a new asset….

  4. A Simple Example The price of this new asset is

  5. Complete Market • What is a complete market? • The previous example is a complete market. Why?

  6. x ? y Complete Market • Given any new asset, we can find its price.

  7. 1 6 1 4 1 5 1 3 Incomplete Market • There are two assets in the market.

  8. 5 3 ? 2 Incomplete Market • Case 1: There is a new asset….

  9. Case 1 The price of this new asset is

  10. 0 1 ? 2 • Case 2: There is a new asset….

  11. Case 2 We cannot find a unique price of this new asset. What price is reasonable? $1 ??

  12. 1 3 0 6 1 4 3 1 5 1 1 0 3 1 0 2 Short 3 units buy 21 units Short 6 units

  13. Incomplete Market • If the price of this new asset is $1 in Case 2, we can find arbitrage opportunities. • There are no arbitrage opportunities if $1/2<price<$2/3 in Case 2. (This proof can be verified with risk neutral probability measures)

  14. What we have observed? • The Market is very much like our vector (sub)space. • New asset (new vector) is decomposed into pieces of the existing assets (basis) in the market (subspace). If the new asset can be represented, then we can determine its price uniquely. Otherwise, we can at best provide an interval for its price. • Thus, if you can do equation-solving or determine a column space of a matrix, you should be able to have a good start in Pricing.

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