Arbitrage Theory

Abstract

This chapter summarizes arbitrage theory in the framework of mar- tingale theory. First, we introduce an arbitrage-free market and arbitrage price for the general asset market, where the key concepts are the state price de ator and a martingale. Next, a numeraire and a numeraire measure are introduced to generalize arbitrage theory. Accordingly, we will see that the arbitrage price does not vary with the choice of numeraire. Next, we work with a bond market where the bond prices are represented by Ito processes. For this, the market price of risk is introduced to ensure the arbitrage-free condition in the market. The market price of risk widely plays an important role in traditional interest-rate models, as an example, which will appear in the basic theory of the HJM model in Chapter 4. The estimation of the market price of risk is the most important subject of this book and is studied after Chapter 6.

Keywords: Accumulated contribution rate, Arbitrage opportunity, Arbitrage pricing, Bond market, Change of numeraire, Complete market, Contribution rate, Eigenvalue, Equivalent martingale measure, Numeraire, Numeraire mea- sure, Market price of risk, Option pricing, Pricing kernel, Principal compo- nent, Relative price process, Risk-neutral measure, Risk-neutral pricing, Risk- neutral valuation, Self-nancing trading strategy, State price de ator, Time- homogeneous short rate model.