TABLE OF CONTENTS A as minimal as possible introduction to the background and methods in infinitesimal analysis is given here. This will be needed for construction of hypermodels for options. Only results are mentioned, the interested reader can find proofs and other mathematical details in Chapter 7.
The original basis of infinitesimal analysis is to make sense of the concept of a positive number which is small in comparison with respect to any positive real numbers.
Such a notion is not as far-fetched as it first sounds.
For example, everybody knows that trading takes place at finite and discrete time moments; stocks are traded only finitely many times say between 10:30 a.m. and 3:30 p.m., trades never take place at every moment between 10:30 a.m. and 3:30 p.m., i.e. activities do not take place continuously. So, in order to model trading mathematically, one would like to divide the time period from 10:30 a.m. to 3:30 p.m. into finitely many intervals so that trades take place only at the endpoints of each interval.
Now the question is:
How do we set the size of such an interval?
Suppose we divide the above period into intervals given by finitely many moments
then we can let 
denote the smallest (positive) size among such intervals [ t i , t i +1]. No matter how small we set the positive size , as long as it is a real number, there will be a chance that in the future,...
