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Valuation Commentary - Nov. '07
Using Tuning Dials for Prepay Risk-Neutrality
by Alex Levin
This issue of The Pipeline is dedicated to how various tuning parameters embedded in AD&Co.’s prepayment, LoanDynamics™, and valuation models can be used. The articles written by Dan Szakallas and Anne Ching describe how using this important tool can improve the performance of empirical models. This was the initial purpose of exposing the tuning knobs, or dials, to users.
In this article, I will discuss a different use of the tunings that relates to MBS pricing. At first, this may seem to be a purely ad-hoc exercise. In reality, this simple step is marked by full quantitative rigor; it reflects how the MBS risk is priced by the market and how this way of pricing, often termed “risk-neutral,” should be reflected by tuning a prepayment model.
What is Risk-Neutrality?
Let us assume that a single risk factor x exists and that this factor is a continuous random process (diffusion). Let us denote volatility of x as s . The price of an asset (P) should depend on x; let Px denote the derivative of price with respect to x. Volatility of the asset’s price is going to be equal to the volatility of the factor (s ) times the exposure (Px); dividing this by price we get the volatility of return: s Px/P.
The arbitrage pricing theory is based on two key statements:
1. Risky return. Each asset exposed to the risk factor x is expected to return the risk-free rate plus a risky return proportional to the systematic volatility. The coefficient of proportionality is common for all instruments and called price of risk (p). The risky return is therefore ps Px/P for all instruments.
Suppose we have 2 stocks, A and B, and the systematic volatility of A is twice that of B. Considering no other risk factors, we can invest $100 in A and sell $200 in B. It is easy to see that the total returns’ volatilities are offset. The total position becomes free of risk and, as such, should earn only the risk-free rate. This is possible if and only if stock A generates a double risky return to that of stock B. This way of reasoning is often referred to as” arbitrage argument.”
