A Practical Guide to Arbitrage-Free Pricing Using Martingales
49 Pages Posted: 22 Jul 2007
Date Written: January 2007
This paper gives a practical, and easy-to-follow introduction to arbitrage-free pricing using martingales with a discrete two-period information structure. Using simple heuristic derivations, we illustrate the concepts of Arrow-Debreu prices, complete and incomplete markets, risk-neutral measure, stochastic discount factor (or pricing kernel), and Radon-Nikodym derivative. We use the discrete-time setup to give a clear and intuitive demonstration of the fundamental theorem of asset pricing, which states that absence of arbitrage is equivalent to the existence of an equivalent martingale measure under which discounted prices are martingales. We also introduce arbitrage-free pricing using martingales in continuous-time, and show the correspondence of the continuous-time results with the discrete-time results. Further, we provide two additional theorems in continuous-time, given as the Girsanov theorem and the Feynman-Kac theorem. The first theorem is used for performing a change of measure, and the second theorem is used for deriving a PDE from an expectation, and vice-versa.
Keywords: arbitrage-free pricing, valuation, martingales, Arrow-Debreu prices, riskneutral measure, forward measure, stochastic discount factor, pricing kernel, Radon-Nikodym derivative, Girsanov theorem, Feynman Kac theorem
JEL Classification: G10, G11, G12, G13, G14
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