The Mathematics of Arbitrage

(Tina Meador) #1

150 9 Fundamental Theorem of Asset Pricing


As an example that this technique has in fact a long history, we quote the
use of mortality tables in insurance. The actual mortality table is replaced
by a table reflecting more mortality if a life insurance premium is calculated
but is replaced by a table reflecting a lower mortality rate if e.g. a lump sum
buying a pension is calculated. Changing probabilities is common practice in
actuarial sciences. It is therefore amazing to notice that today’s actuaries are
introducing these modern financial methods at such a slow pace.
The present paper focuses on the question: “What is the precise meaning
of the wordessentiallyin the first paragraph of the paper?” The question has
a twofold interest. From an economic point of view one wants to understand
the precise relation between concepts of no-arbitrage type and the existence
of an equivalent martingale measure in order to understand the exact lim-
itations up to which the above sketched approach may be extended. From
a purely mathematical point of view it is also of natural interest to get a bet-
ter understanding of the question which stochastic processes are martingales
after an appropriate change to an equivalent probability measure. We refer to
the well-known fact that a semi-martingale becomes a quasi-martingale under
a well-chosen equivalent law (see [P 90]); from here to the question whether
we can obtain a martingale, or more generally a local martingale, is natural.
We believe that the main theorem (Theorem 9.1.1 below) of this paper
contributes to both theories, mathematics as well as economics. In economic
terms the theorem contains essentially two messages. First that it is possible
to characterise the existence of an equivalent martingale measure for a general
class of processes in terms of the concept of no free lunch with vanishing risk,
a concept to be defined below. In this notion the aspect of vanishing risk
bears economic relevance. The second message is that — in a general setting
— there is no way to avoid general stochastic integration theory. If the model
builder accepts the possibility that the price process has jumps at all possible
times, he needs a sophisticated integration theory, going beyond the theory
for “simple integrands”. In particular the integral of unbounded predictable
processes of general nature has to be used. From a purely mathematical point
of view we remark that the proof of the Main Theorem 9.1.1 below, turns
out to be surprisingly hard and requires heavy machinery from the theory
of stochastic processes, from functional analysis and also requires some very
technical estimates.
The processS, sometimes denoted (St)t∈R+is supposed to beR-valued,
although all proofs work with ad-dimensional process as well. However, we
prefer to avoid vector notation inddimensions. If the reader is willing to
accept the 1-dimensional notation for thed-dimensional case as well, nothing
has to be changed. The theory ofd-dimensional stochastic integration is a little
more subtle than the one-dimensional theory but no difficulties arise.
The general idea underlying the concept of no-arbitrage and its weaken-
ings, stated in several variants of “no free lunch” conditions, is that there
should be no trading strategyHfor the processS, such that the final pay-
off described by the stochastic integral (H·S)∞, is a non-negative function,

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