Market Risk Analysis Volume I: Quantitative Methods in Finance

I.3

Probability and Statistics

I.3.1 INTRODUCTION

This chapter describes the probabilistic and statistical models that we use to analyse the evolution of financial asset prices or interest rates. Prices or returns on financial assets, interest rates or their changes, and the value or P&L of a portfolio are some examples of the random variables used in finance. A random variable is a variable whose value could be observed today and in the past, but whose future values are unknown. We may have some idea about the future values, but we do not know exactly which value will be realized in the future. This is what we mean by uncertainty. For instance, we cannot say that the return on an investment over the next year will be 25%, we can only say that there is some probability that the return will be 25%. When we write ‘P(R = 25%) = 1/2’ we mean that there is a 50:50 chance that the investment will make a return of 25%.

Most of the random variables we consider are assumed to be real continuous random variables. Loosely speaking, this means that we assume their value can be anything within a defined range of real numbers. By contrast, discrete random variables can only take certain discrete values such as 0, 1, 2,…. Any real numbers are allowed, not just non-negative integers, provided that the possibilities are discrete.

Since P&L, returns, prices and interest rates are usually regarded as continuous random variables the statement P(R = 25%) = 1/2 is nonsense, because for any continuous ...