Characterize the theoretical environment

The first step in the model-building process is characterization of the theoretical
environment in which the model is going to be used. The model can be a normative
one (which describes how the things should be) or a positive one (which describes
how they are). In both cases, the goal is to build an abstraction close enough to
reality. What does the world look like? Does trading take place continuously or at
selected time intervals? Are all prices and information observable, or is there some
form of asymmetric information? Is the market free of frictions, or do we face
transaction costs or differential taxes? Is the market for the instruments considered
always liquid enough, or should we also consider the possibility of excess demand
or supply? Are borrowing and short selling allowed without restriction? Do investors’
actions have an effect on prices? Are all required hedging instruments available to
ensure market completeness? The answers to all these questions will result in a set
of hypotheses that are fundamental for the model to be developed.
In any case, understanding and being comfortable with the model’s assumptions
is essential. Challenging them is recommended. In fact, most of the main strands of
theoretical work in finance are concerned with the relaxation of assumptions in
existing models. However, this should not be considered as a sport, as it results in
an abundant profusion of similar models. We must always remember that a theory
should not be judged by the restrictiveness of its assumptions, but rather by the
contribution of its conclusions to improve our understanding of the real world and
the robustness of these conclusions. Important objections were already raised with
the original Markowitz (1952) model; despite this, even the attacks on Markowitz
have triggered new concepts and new applications that might have never come about
without his innovative contributions.

As an illustration, Table 14.2 lists the set of idealizing assumptions that characterize
the original Black and Scholes (1973) world. All of them are necessary to the
further development of the model. Of course, one may criticize or reject some of
them, and argue that they are highly unrealistic with respect to the real world. This
is often a subjective point of view. For instance:

Ω Assumptions A1 (no market frictions) and A3 (short selling) are highly unrealistic
for the individual investor; they may hold as a first approximation for large market
participants, at least in normal market conditions. What are considered as
‘normal’ conditions remain to be defined. In the recent collapse of the Long Term
Capital Management hedge fund, the partners (including Robert Merton and
Myron Scholes) looked for fancy mathematical models, but they failed to appreciate
the liquidity of investments and the ability to execute their trades. After the
Russian debacle, markets lost confidence and liquidity dried up. There were no
more buyers, no more sellers, and therefore no trades. This is a typical violation
of the Black and Scholes necessary hypotheses.
Ω Assumption A8 (no default risk) is also acceptable for futures and options that
trade in organized exchanges, where a clearinghouse acts as a counterpart for
both sides of the contracts; it is subject to caution when pricing over-the-counter
derivatives.
Assumptions can also be visionary. Consider, for instance, assumption A2 (continuous
time). In the 1970s, this was a fiction, justified by saying that if the length
of time between successive market quotes is relatively small, the continuous-time
solution given by the model will be a reasonable approximation to the discrete-time
solution. But with electronic stock exchanges, continuous time trading has now
become a reality. As Merton initially noticed, ‘reality will eventually imitate theory’. 419