Generating Random Numbers with Specified Distribution

Simulation and valuation of finance instruments require numbers with specified distributions. For example, in Section 1.6 we have used numbers Z drawn from a standard normal distribution, Z ~ N(0, 1). If possible the numbers should be random. But the generation of “random numbers” by digital computers, after all, is done in a deterministic and entirely predictable way. If this point is to be stressed, one uses the term pseudo-random ¹.

Computer-generated random numbers mimic the properties of true random numbers as much as possible. This is discussed for uniformly distributed numbers in Section 2.1. Suitable transformations generate normally distributed numbers (Sections 2.2, 2.3). Another approach is to dispense with randomness and to generate quasi-random numbers, which aim at avoiding one disadvantage of random numbers, namely, the potential lack of equidistributedness. The resulting low-discrepancy numbers will be discussed in Section 2.5. These numbers are used for the deterministic Monte Carlo integration (Section 2.4).