Chapter 10. Stochastics

Predictability is not how things will go, but how they can go.

— Raheel Farooq

Nowadays, stochastics is one of the most important mathematical and numerical

disciplines in finance. In the beginning of the modern era of finance, mainly in the 1970s

and 1980s, the major goal of financial research was to come up with closed-form solutions

for, e.g., option prices given a specific financial model. The requirements have drastically

changed in recent years in that not only is the correct valuation of single financial

instruments important to participants in the financial markets, but also the consistent

valuation of whole derivatives books, for example. Similary, to come up with consistent

risk measures across a whole financial institution, like value-at-risk and credit value

adjustments, one needs to take into account the whole book of the institution and all its

counterparties. Such daunting tasks can only be tackled by flexible and efficient numerical

methods. Therefore, stochastics in general and Monte Carlo simulation in particular have

risen to prominence.

This chapter introduces the following topics from a Python perspective:

Random number generation

It all starts with (pseudo)random numbers, which build the basis for all simulation

efforts; although quasirandom numbers, e.g., based on Sobol sequences, have gained

some popularity in finance, pseudorandom numbers still seem to be the benchmark.

Simulation

In finance, two simulation tasks are of particular importance: simulation of random

variables and of stochastic processes.

Valuation

The two main disciplines when it comes to valuation are the valuation of derivatives

with European exercise (at a specific date) and American exercise (over a specific

time interval); there are also instruments with Bermudan exercise, or exercise at a

finite set of specific dates.

Risk measures

Simulation lends itself pretty well to the calculation of risk measures like value-at-

risk, credit value-at-risk, and credit value adjustments.