290 Frequently Asked Questions In Quantitative Finance
witha,bandcbeing constant. As long asais suffi-
ciently large this process cannot go negative.
There is a solution for bonds of the form exp(A(t;T)−
B(t;T)r).
Ho and Lee In this model the risk-neutral process is
dr=a(t)dt+cdX,
withcbeing constant. It is possible forrto go negative
in this model.
There is a solution for bonds of the form exp(A(t;T)−
B(t;T)r).
The time-dependent parametera(t) is chosen so that
the theoretical yield curve matches the market yield
curve initially. This is calibration.
Hull and White There are Hull and White versions of the
above models. They take the forms
dr=(a(t)−b(t)r)dt+c(t)dX,
or
dr=(a(t)−b(t)r)dt+c(t)r^1 /^2 dX.
The functions of time allow various market data to be
matched or calibrated.
There are solutions for bonds of the form exp(A(t;T)−
B(t;T)r).
Black–Karasinski In this model the risk-neutral spot-rate
process is
d(lnr)=(a(t)−b(t)lnr)dt+c(t)dX.
There are no closed-form solutions for simple bonds.
