(BQ) Part 2 book "The mathematics of financial modeling and investment management" has contents: Fat tails, scaling, and stable laws; arbitrage pricing - finite state models, capital asset pricing model, equity portfolio management; multifactor models and common trends for common stocks,.and other contents. | CHAPTER 13 Fat Tails, Scaling, and Stable Laws ost models of stochastic processes and time series examined thus far assume that distributions have finite mean and finite variance. In this chapter we describe fat tailed distributions with infinite variance. Fat-tailed distributions have been found in many financial economic variables ranging from forecasting returns on financial assets to modeling recovery distributions in bankruptcies. They have also been found in numerous insurance applications such as catastrophic insurance claims and in value-at-risk measures employed by risk managers. In this chapter, we review the related concepts of fat-tailed, powerlaw and Levy-stable distributions, scaling and self-similarity, as well as explore the mechanisms that generate these distributions. We discuss the key intuition relative to the applicability of fat-tailed or scaling processes to finance: In a fat-tailed or scaling world (as opposed to an ergodic world), the past does not offer an exhaustive set of possible configurations. Adopting, as an approximation, a scaling description of financial phenomena implies the belief that only a small space of possible configurations has been explored; vast regions remain unexplored. We begin with the mathematics of fat-tailed processes, followed by a discussion of classical Extreme Value Theory for independent and identically distributed sequences. We then explore the consequences of eliminating the assumption of independence and discuss different concepts of scaling and self similarity. Finally, we present evidence of fat tails in financial phenomena and discuss applications of Extreme Value Theory. M 351 352 The Mathematics of Financial Modeling and Investment Management SCALING, STABLE LAWS, AND FAT TAILS Let’s begin with a review of the different but related concepts and properties of fat tails, power laws, and stable laws. These concepts appear frequently in the financial and economic literature, applied to both random .
