The Mathematics of Arbitrage

(Tina Meador) #1

78 5 The Kreps-Yan Theorem


Hence,g 0 +g 1 wouldbeanelementofGwhose support hasP-measure strictly
bigger thanP[{g 0 > 0 }], a contradiction.
Normaliseg 0 so that‖g 0 ‖ 1 =1andletQbe the measure onFwith
Radon-Nikod ́ym derivativeddPQ=g 0. We conclude from Lemma 5.1.3 thatQ
is a local martingale measure forS,sothatMe(S)=∅. 


Some comments on the Kreps-Yan theorem seem in order: this theorem was
obtained by D. Kreps [K 81] in a more abstract setting and under a — rather
mild — additional separability assumption; the reason for the need of this
assumption was that D. Kreps did not use the above exhaustion argument, but
rather some sequential procedure relying on the separability ofL^1 (Ω,F,P).
Independently and at about the same time, Ji-An Yan [Y 80] proved in a
different context, namely the characterisation of semi-martingales as good
integrators (which is the theme of the Bichteler-Dellacherie theorem), and
without a direct relation to finance, a general theorem which is similar in
spirit to Theorem 5.2.2. Ch. Stricker [Str 90] observed that Yan’s theorem
may be applied to quickly prove the theorem of Kreps without any separability
assumption. We therefore took the liberty to give Theorem 5.2.2 the name of
these two authors.
The message of the theorem is, that the assertion of the “fundamental
theorem of asset pricing” 2.2.7 is valid for general processes, if one is willing to
interpret the notion of “no-arbitrage” in a somewhat liberal way, crystallised
in the notion of “no free lunch” above.
What is the economic interpretation of a “free lunch”? By definitionS
violates the assumption(NFL)if there is a functiong 0 ∈L∞+(Ω,F,P),g 0 =0,
and nets (gα)α∈I,(fα)α∈IinL∞(Ω,F,P), such thatfα=(Hα·S)∞for some
admissible, simple integrandHα,gα≤fα, and limα∈Igα =g 0 , the limit
converging with respect to the weak-star topology ofL∞(Ω,F,P). Speaking
economically: an arbitrage opportunity would be the existence of a trading
strategyHsuch that (H·S)∞≥0, almost surely, andP[(H·S)∞>0]>0. Of
course, this is the dream of each arbitrageur, but we have seen, that — for the
purpose of the fundamental theorem to hold true — this is asking for too much
(at least, if we only allow for simple admissible trading strategies). Instead, a
free lunch is the existence of a contingent claimg 0 ≥0,g 0 =0,whichmay,in
general, not be written as (or dominated by) a stochastic integral (H·S)∞
with respect to a simple admissible integrandH; but there are contingent
claimsgα“close tog 0 ”, which can be obtained via the trading strategyHα,
and subsequently “throwing away” the amount of moneyfα−gα.
This triggers the question whether we can do somewhat better than the
above — admittedly complicated — procedure. Can we find a requirement
sharpening the notion of “no free lunch”, i.e., being closer to the original
notion of “no-arbitrage” and such that a — properly formulated — version of
the “fundamental theorem” still holds true?


Here are some questions related to our attempt to make the process of
taking the weak-star-closure more understandable:

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