132 8 Arbitrage Theory in Continuous Time: an Overview
(i) (EMM), i.e., there is a probability measureQ, equivalent toP, such that
Sis a local martingale underQ.
(ii)(NFLVR), i.e.,Ssatisfies the condition of no free lunch with vanishing
risk.
The present theorem is a sharpening of the Kreps-Yan Theorem 5.2.2, as
it replaces the weak-star convergence in the definition of “no free lunch” by
the economically more convincing notion of uniform convergence. The price
to be paid for this improvement is, that now we have to place ourselves into
the context ofgeneral admissible, instead ofsimple admissibleintegrands.
The proof of Theorem 8.2.1 as given in Chap. 9 and its extension in
Chap. 14 is surprisingly long and technical; despite of several attempts, no
essential simplification of this proof has been achieved so far. We are not able
to go in detail through this proof in this “guided tour”, but we shall try to
motivate and help the interested reader to find her way through the arguments
in Chap. 9 below.
We start by observing that the implication (i)⇒(ii) is still the easy one:
supposing thatSis a local martingale underQandHis an admissible trading
strategy, we may deduce from the Ansel-Stricker Theorem 7.3.7 and the fact
thatH·Sis bounded from below, thatH·Sis a local martingale underQ,
too. Using again the boundedness from below ofH·S, we also conclude that
H·Sis a super-martingale underQ,sothat
EQ[(H·S)∞]≤ 0. (8.3)HenceEQ[g]≤0, for allg∈C, and this equality extends to the norm
closureC ofC (in fact, it also extends to the weak-star-closure ofC, but
we don’t need this stronger result here). Summing up, we have proved that
(EMM)implies(NFLVR).
Before passing to the reverse implication let us still have a closer look at
the crucial inequality (8.3): its message is that the notion of equivalent local
martingale measuresQand admissible integrandsHhas been designed in
such a way, that the basic intuition behind the notion of a martingale holds
true:you cannot win in average by betting on a martingale.Note,however,
that the notion of admissible integrands does not rule out the possibilityto
lose in average by betting onS. An example, already noted in [HP 81], is the
so-called “suicide strategyH” which is just the doubling strategy considered
at the beginning of this chapter with opposite signs. Consider, similarly as
above, a simplified roulette, where red and black both have probability^12 and,
as usual, when winning, your bet is doubled. The strategy consists in placing
oneeon red and then walking to the bar of the casino and regarding the
roulette from a distance: if it happens that consecutively only red turns up
in the next couple of games, you may watch a huddle of chips piling up with
exponential growth (assuming, of course, that there is no limit to the size of
the bets). But inevitably, i.e., with probability one, black will eventually turn