CHAPTER 7

Stochastic Models

Inveniemus viam aut faciemus. (We will find a way, or we will make one.)

—Hannibal

7.1 STATIC VERSUS STOCHASTIC

Previous chapters describe a static asset-backed security (ABS) model for deriving the price of a portfolio of loans given interest rates, prepayments, losses, and recoveries. These are in reality stochastic variables; that is, their future values are functions of probabilistic events. The simplest static model takes one interest rate curve, one prepayment curve, one default curve, and one recovery curve. Each curve represents one view of its variable over time—it may be the expectation, a stressful scenario, or something else.

The static model evolved to use alternative curves in certain cases. For example, each bond of a different rating uses a different interest rate curve to derive its collateral cash flows. Hybrid adjustable-rate mortgage (ARM) loans having different initial resets use different prepay curves. First liens and second liens use different loss curves, and so on. These evolved static models are intended to give more accurate results. For example, an ARM that first resets in two years will have a flurry of prepayments around two years. An ARM that first resets in five years will effectively delay that peak of prepayments to around five years. Fundamentally, however, use of a finite number of values to represent a stochastic variable, rather than its entire distribution, is an approximation.

To help understand the approximation, ...