# General Implied Processes and Direct Implementation of the Fundamental Theorem of Asset Pricing

28 Pages Posted: 2 Jul 2020 Last revised: 7 Jul 2020

Date Written: June 9, 2020

### Abstract

This paper studies the following conceptual question: in what sense is the Fundamental Theorem of Asset Pricing similar to the two-period no-arbitrage theorem (a.k.a., Farkas lemma)? The purpose of studying this question is (1) to study the information that can be extracted from prices of derivatives in a multi-period context, generalizing the result in a two-period case in Breeden and Litzenberger (1978); (2) to find a way to write down explicitly a multi-period arbitrage process, just as a two-period arbitrage can be written down as a vector.

To answer the above conceptual question, I break it down into three more specific questions: (1) How to generalize the concept of states to a multi-period model? (2) How to generalize the concept of state price to a multi-period model? (3) In what sense is a multi-period arbitrage process similar to a two-period arbitrage strategy which is just a vector?

The key to answering those questions is to explicitly describe the probability space on which price processes are defined, especially what “information flow” means. I adopt the canonical probability space (i.e., the space of all possible paths of some price process) and propose to consider the whole path of as the state variable and the “path prices”(i.e., the equivalent martingale measure) as the analogue of state prices. This paper discusses how we can recover prices of paths using prices of associated derivative securities and then use them to price other derivatives, which contributes to the literature of implied processes. In addition, it also shows that a multi-period arbitrage process can be reduced to a random vector. The theoretical contribution of this paper is that it sheds new light on the nature of arbitrage processes and the Fundamental Theorem of Asset Pricing. Practically it provides a general framework to precisely extract the information contained in prices of frequently-traded derivatives and then price other derivatives.

**Keywords:** Fundamental Theorem of Asset Pricing, No Arbitrage, Martingale Condition, Equivalent Martingale Measure, Path Prices, Information Flow, Relative Pricin

**JEL Classification:** G00, G13, G12

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