Positivity 6: 359–368, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
No Arbitrage: On the Work of David Kreps
Department of Financial and Actuarial Mathematics, Vienna University of Technology, Wiedner
Hauptstrasse 8–10/105, A-1040 Vienna, Austria. E-mail: email@example.com
Received 18 May 2001; accepted 10 September 2001
Abstract. Since the seminal papers by Black, Scholes and Merton on the pricing of options (Nobel
Prize for Economics, 1997), the theory of No Arbitrage plays a central role in Mathematical Finance.
Pioneering work on the relation between no arbitrage arguments and martingale theory has been done
in the late seventies by M. Harrison, D. Kreps and S. Pliska. In the present note we give a brief survey
on the relation of the theory of No-Arbitrage to coherent pricing of derivative securities. We focus
on a seminal paper published by D. Kreps in 1981, and give a solution to an open problem posed in
AMS Classiﬁcation: 91B28, 46A40
Key words: No Arbitrage, No Free Lunch, Coherent Pricing of Derivative Securities
1. Introduction: “No arbitrage” and “No Free Lunch”
The principle of no arbitrage formalizes a very convincing economic argument: in
a ﬁnancial market it should not be possible to make a proﬁt with zero net investment
and without bearing any risk. It is surprising, how much can be deduced from this
primitive principle. For example, in the celebrated model used by Black, Scholes
, and Merton , which is based on geometric Brownian motion, the price of
any derivative security (e.g., a European call option) is already determined by this
In addition, the fundamental theorem of asset pricing, as isolated in the work of
Harrison and Kreps , Harrison and Pliska , and Kreps ) allows to relate
the no arbitrage arguments with martingale theory.
Here is the mathematical formulation of the concept of no arbitrage as formal-
ized in : we start with an ordered topological vector space X, equipped with
a locally convex topology τ , and its positive cone K (with the origin deleted).
A typical situation is X = L
(, F , P), equipped with its natural order structure
and topology, where, as usual 1 p ∞,and(, F , P) is a probability space
modeling the “possible states of the world” ω at some ﬁxed time horizon T .In
this setting the elements x = x(ω) ∈ X are random variables modeling contingent
claims at time T (denoted in terms of the single consumption good under consid-
eration, which we choose as numéraire). The arch-example of a contigent claim x
is a European call option on a stock modeled by a stochastic process (S