Mathematical Techniques in Finance

CHAPTER 3Investment Theory

The future cash flows and, hence, returns of many investments are uncertain. Even for fixed income instruments like bonds where the cash flows are known, the future yields are not known, hence, the future price of a bond until its maturity is random: we know the current price, and we also know its final price at maturity (100%), but for any date between now and maturity, the bond price is random. Faced with this uncertainty, much of finance uses techniques and insights from probability theory and shares concepts and language of games of chance and gambling: a sound investment is a gamble with good odds and risk/reward profile.

One of the oldest studies of games of chance that combines elements of probability theory, decision‐making under uncertainty, and behavioral finance is the following: consider a game based on successive coin flips with the payoff where is the number of tosses to get to the first tail (see Table 3.1).

Let be the random payoff of this game. Its expected value is infinite

What is the fair price of this random payoff, that is, how ...