Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 5

Probability

Basic Concepts

Standard finance theory generally rules out the conditions that investors have about uncertainty vis-à-vis the probability distribution of asset returns. However, Frank Knight in his book Risk, Uncertainty and Profit, published in 1921, draws a distinction between risk and true uncertainty and argues that uncertainty is more common in the decision-making process. He points out that risk occurs where the future is unknown, however, the probability of all possible outcomes is known. Uncertainty occurs where the probability distribution is itself unknown. Knight’s distinction between risk and uncertainty implies that risk is related to the objective distribution of return or the subjective distribution of return commonly agreed on by all investors, whereas uncertainty is related to the probability distribution unique to an individual investor. Probability theory is useful to understand and investigate the changes in prices, riskiness, and uncertainty about financial instruments. Probability theory is also useful to understand and investigate the changes in state variables, such as financial factors and macroeconomic fundamentals, that affect consumption and investment opportunities for investors. Using probability theory:

• One can examine the empirical return distribution of assets such as stocks, bonds, currencies, and commodities.
• One can estimate expected return and risk of assets.
• One can determine how expected prices of assets change from ...