152 9 Fundamental Theorem of Asset Pricing
(see [S 92, KK 94, R 94] for elementary proofs). For the case of discrete but
infinite time sets, the problem is solved in [S 94]. The case of continuous and
bounded processes in continuous time, is solved in [D 92]. In these two cases
the theorems are stated in terms of simple integrands and limits of sequences
and by using the concept of no free lunch with bounded risk. We shall review
these issues in Sect. 9.6.
In the general case, i.e. a time set of the form [0,∞[or[0,1] and with
a possibility of random jumps, the situation is much more delicate. The ex-
istence of an equivalent martingale measure can be characterised in terms of
“no free lunch” involving the convergence of nets or generalised sequences,
see e.g. [K 81, L 92]. S. Kusuoka [K 93] used convergence in Orlicz spaces and
Fuffie, Huang and Stricker [DH 86, Str 90] usedLpconvergence for 1≤p<∞.
In the latter case the restrictions posed onSwere such that the new mea-
sure has a density inLqwhereq=p−p 1. Contrary to the case of continuous
processes or to the case of discrete time sets, no general solution was known
in terms of “no free lunch” involving convergent sequences. Hence there re-
mained the natural question whether for a general adapted processS,the
existence of an equivalent martingale measure could be characterised in such
terms.
The answer turns out to be no if one only uses simple integrands. In
Sect. 9.7, we give an example of a processS=M+AwhereMis a uniformly
bounded martingale,Ais a predictable process of finite variation,Sadmits
no equivalent martingale measure but there is “no free lunch with bounded
risk” if one only uses simple integrands. A closer look at the example shows
that if one allows strategies of the form: “sell before each rational number
and buy back after it”, then there is even a “free lunch with vanishing risk”.
Of course such a trading strategy is difficult to realise in practice but if we
allow discontinuities for the price process at arbitrary times, then we should
also allow strategies involving the same kind of pathology. The example shows
that we should go beyond the simple integration theory to cover these cases as
well. To back this assertion let us recall that the basis of the whole theory of
asset pricing by arbitrage is, of course, the celebrated Black-Scholes formula
(see [BS 73, M 73]), widely used today by practitioners in option trading. Also
in this case the trading strategyH, which perfectly replicates the payoff of the
given option, is not a simple integrand. It is described as a smooth function
of time and the underlying stock price. Being a smooth function of the stock
price, its trajectories are in fact of unbounded variation. One can argue that
in practice already this strategy is difficult to realise. In this case, however,
one shows that the integrand can be approximated by simple integrands in
a reasonable way; for details we refer the reader to books an stochastic inte-
gration theory with special emphasis an Brownian motion, e.g. [KS 88]. In the
case of the example of Sect. 9.7, this reduction is not possible and as already
advocated, general integrands are really needed.
Summing up we are forced to leave the framework of simple integrands.
However, we immediately face new problems. First the processSshould be re-