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ACTSC 445: AssetLiability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 8 (Part I) – DiscreteTime Interest Rate Models
References
(recommended readings): Chap. 7 of Financial Economics, chapter 37 of Fabozzi.
Introduction
In this unit, we will discuss
term structure models
, i.e., models for the evolution of the term structure
of interest rates. Such models can be used to price ±xed income securities (such as callable bonds),
and interestrate derivatives (such as interest caps and ²oors).
Most of the models we will look at in Part I are
discretetime
,
singlefactor
,
noarbitrage
models.
..
What does it mean?
•
discretetime: rates change at each period (e.g., 6 months, one year), rather than continuously
(e.g., Vasicek model
dr
=
a
(
b

r
)
dt
+
σdZ
t
).
•
singlefactor: model only has one source of randomness (e.g., short rates), by contrast with
multifactor models, where e.g., we would model short rate + another asset
•
noarbitrage: model prevents arbitrage opportunities. Alternative is
equilibrium model
, in which
economic agents determine, through their behavior/preferences, equilibrium prices (e.g., Cox
IngersollRoss model)
When dealing with discretetime interest rate models, sometimes we move forward in time, sometimes
we move backward:
•
To use these models for pricing, one approach that we’ll see is based on
backward induction
.
•
To calibrate these models (which in our case means ±nd parameters from data so that there is
no arbitrage), we’ll use
forward induction
.
The plan for Part I of this unit is as follows:
•
First, we’ll look at a simple generic model and see how we can use it to price bonds.
•
Second, we’ll go over interestrate derivatives, and explain how to price them.
•
An important tool both for pricing and calibrating is the use of
ArrowDebreu securities
, which
we’ll discuss next.
•
We’ll then look more closely at models and how we can calibrate them.
•
Finally, we’ll discuss how to price embedded options in bonds, and use this to revisit the notion
of e³ective duration/convexity.
1
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View Full DocumentA Generic Binomial Model
Let us frst introduce some notation:
T
=
number oF time periods.
i
t
=
short rate at time
t
(random variable),
t
=0
,...,T

1.
i
(
t,n
)=
n
th
possible value that
i
t
can take,
n
,...,N
t
.
In other words, we will be modeling the process
i
0
,...,i
T

1
by assuming that the state space For each
short rate
i
t
is oF the Form
{
i
(
t,
0)
(
t,N
t
)
}
.
A
binomial model
For the shortrate process
{
i
0
T

1
}
means that we are making the Following
assumptions:
•
i
0
is fxed to some value
i
(0
,
0).
•
±or each time
t
≥
0,
i
t
+1
can only take two possible values, which depend on the value
i
(
)
taken by the previous short rate: with probability
q
(
), it will take the value
i
(
t
+1
,n
+ 1),
and with probability 1

q
(
), it will take the value
i
(
t
).
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 Spring '09
 ChristianeLemieux

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