Ho-Lee model
, and
The bond price is then
therefore, the is a martingale.
Besides, we have
We can use the 4-step procedure to obtain the PDE for :
step 1
step 2
step 3
step 4
Claim: can be computed explicitly
We firstly guess the is in the form of
Plug in this we have
cancel out the , we have
The conditions for are
the solution is
The conditions for are
which is equal to
and thus
The Bond price is thus
Denote , to obtain the , we need to find the
using the ItΗ formula, we have
Besides,
the solution is
Forward rate
and thus
Random-Nikodym:
note that is a martingale under .
from to (a risk-neutral measure).
Example: interest rate caplet pays
where
Calibration:
to let
comparing the term, we have
Health-Jarrow-Morton Framework
The procedure to go from to
where and are both random and adapted in . is the starting point.
Calibration: match to the market. Our primary assets are
we have
thus
where
Β
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