Nonparametric Finance

Chapter 14Pricing by Arbitrage

Pricing by arbitrage means that an asset is priced with the unique arbitrage-free price. Pricing by arbitrage can be applied in two different settings: (1) We can price by arbitrage linear securities, like forwards and futures, in any markets. (2) We can price by arbitrage nonlinear securities, like options, in complete markets. We discuss both of these cases in this chapter.

If two assets have the same terminal value with probability one, then the assets should have the same price. Otherwise, we could obtain a risk-free profit by selling the more expensive asset and by buying the cheaper asset. Linear assets, like futures, can be defined as a linear function of the underlying assets. Thus, they can be replicated, and their price is the initial value of the replicating portfolio. Nonlinear assets, like options, can be replicated only under restrictive assumptions on the markets (under the assumption of complete markets). However, the restrictive assumptions are often not too far away from the real properties of the markets.

The concepts of an arbitrage-free market and a complete market were studied in Chapter 13. The results of Chapter 13 imply that if the market is arbitrage-free and complete, then there is only one arbitrage-free price. In fact, Theorem 13.2 states that the arbitrage-free prices are obtained as expected values with respect to the equivalent martingale measures, and Theorem 13.3 (the second fundamental theorem of asset pricing) ...