Frequently Asked Questions In Quantitative Finance

Chapter 5: Models and Equations 291

Two-factor models

In the two-factor models there are two sources of ran-
domness, allowing a much richer structure of theoretical
yield curves than can be achieved by single-factor mod-
els. Often, but not always, one of the factors is still the
spot rate.

Brennan and Schwartz In the Brennan & Schwartz model
the risk-neutral spot rate process is

dr=(a 1 +b 1 (l−r))dt+σ 1 rdX 1

and the long rate satisfies

dl=l(a 2 −b 2 r+c 2 l)dt+σ 2 ldX 2.

Fong and Vasicek Fong & Vasicek consider the following
model for risk-neutral variables

dr=a(r−r)dt+

√

ξdX 1

and

dξ=b(ξ−ξ)dt+c

√

ξdX 2.

Thus they model the spot rate, andξthe square root of
the volatility of the spot rate.

Longstaff and Schwartz Longstaff & Schwartz consider the
following model for risk-neutral variables

dx=a(x−x)dt+

√

xdX 1

and

dy=b(y−y)dt+

√

ydX 2 ,

where the spot interest rate is given by

r=cx+dy.