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
is a martingale under .
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
Randon-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
and thus
We want to find a , such that
We need the to be independnet of (and to be martingale)
No arbitrage condition:
is independent of . We have , the dervatives are below:
In practice, we start with and choose , this will give us .
Three representation for
Define where , then
Therefore, is a martingale under the measure .
Forward Contracts
A Forward contract on an asset :
- at : agree on a price for at
- at : receive one unit of , pay the agreed price for
We want to derive the forward price of .
No arbitrage argument:
- at : short
- at : buy one unit of
- at :
Then the forward price .
Risk-neutral pricing argument:
Example: An asset pays at time , what is the ?
Solution: first compute current price
Therefore,
Apply change of measure with the R-N process
which can make
a martingale.
Blackβs formula
Suppose is an asset paying no dividends, and there is no cost of carry. There is a call option paying at . Our goal: find .
Note that and are highly correlated, and thus . We have
We already know that is a martingale under and
Assume is constant. Then
and
where
Note that for , we have
where .
Β
Write for the no arbitrage price at , then
The Forward price is
Example: if we want to price
Solution:
then,
where the is the terminal value of quotiant of martingales.
The Quotient of martingales:
Β
Loading Comments...