1.5. MATHEMATICAL MODELING 47
The additional constantsaandbrepresent the new elements of intermolecular
attraction and volume effects respectively. Ifaandbare small because we are
considering a monatomic gas under ordinary conditions, the Van der Waals
equation of state can be well approximated by the ideal gas law. Otherwise
we must use this more complicated relation for engineering our needs with
gases.
It is now realized that because of the complex nature of the intermolecu-
lar forces, a real gas cannot be rigorously described by any simple equation
of state. It can be honestly said that the assumptions of the ideal gas are
not correct, yet are sometimes useful. Likewise, the predictions of the van
der Waals equation of state describe quite accurately the behavior of carbon
dioxide gas in appropriate conditions. Yet for very low temperatures, carbon
dioxide deviates from even these modified predictions because we know that
the van der Waals model of the gas is wrong. Even this improved mathe-
matical model is wrong, but it still is useful.
Later we will make a limited number of idealized assumptions about se-
curities markets. We start from empirical observations of economists about
supply and demand and the role of prices as a quantifiable element relating
them. We will ideally assume that
- a very large number of identical, rational traders,
- all traders always have complete information about all assets they are
trading, - prices may be random, but are continuous with some probability dis-
tribution, - trading transactions take negligible time,
- trading transactions can be made in any amounts.
These assumptions are very similar to the assumptions about an ideal gas.
From the assumptions we will be able to make some standard economic
arguments to derive some interesting relationships about option prices. These
relationships can help us manage risk, and speculate intelligently in typical
markets. However, caution is necessary. In discussing the economic collapse
of 2008-2009, blamed in part on the overuse or even abuse of mathematical
models of risk, Valencia [51] says “Trying ever harder to capture risk in
mathematical formulae can be counterproductive if such a degree of accuracy