8
Arbitrage Theory in Continuous Time:
an Overview
8.1 Notationand Preliminaries ...............................
After all this preliminary work we are finally in a position to tackle the theme
of no-arbitrage in full generality, i.e., for general modelsSof financial mar-
kets in continuous time, and for general (i.e., not necessarily simple) trading
strategiesH. The choice of the proper class of trading strategies will turn
out to be rather subtle. In fact, for different applications (e.g., portfolio op-
timisation with respect to exponential utility to give a concrete example; see
[DGRSSS 02] and [S 03a]) it will sometimes be necessary to consider differ-
ent classes of appropriate trading strategies. But for the present purpose the
concept ofadmissiblestrategies developed below will serve very well.
When defining an appropriate class of trading strategies, then, first of
all, one has to restrict the choice of the integrandsHto make sure that the
processH·Sexists. Besides the qualitative restrictions coming from the theory
of stochastic integration considered in the previous chapter, one has to avoid
problems coming from so-called doubling strategies. This was already noted
in the paper by Harrison and Pliska [HP 81]. To explain this remark, let us
consider the classical doubling strategy. We take the framework of a fair coin
tossing game. We toss a coin, and when heads comes up, the player is paid 2
times his bet. If tails comes up, the player loses his bet.
The so-called “doubling strategy” is known for centuries and in French it
is still referred to as “la martingale” (compare, e.g., [B 14, p. 77]^1 ). The player
(^1) We cannot resist citing from Bachelier’s book where he discusses the “suicide
strategy” (see after Theorem 8.2.1 below), which is a close relative to the doubling
strategy.
“La martingale est la cause unique des grosses fortunes, .... Pour devenir tres riche, il faut ˆetre favoris ́e par des concours de circonstances extraordinaires et par des hasards constamment heureux. Jamais un homme n’ est devenu tres riche par
sa valeur.”
The martingale is the unique cause for big fortunes, .... To become very rich,
you have to be favoured by extraordinary circumstances and by constantly lucky
bets. Never a man became very rich by his value.