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Lecture

ACTSC445 Lecture Notes - Vasicek Model, Callable Bond, Arbitrage


Department
Actuarial Science
Course Code
ACTSC445
Professor
Jiahua Chen

Page:
of 19
ACTSC 445: Asset-Liability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 8 (Part I) – Discrete-Time 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 fixed income securities (such as callable bonds),
and interest-rate derivatives (such as interest caps and floors).
Most of the models we will look at in Part I are discrete-time,single-factor,no-arbitrage models...
What does it mean?
discrete-time: rates change at each period (e.g., 6 months, one year), rather than continuously
(e.g., Vasicek model dr =a(br)dt +σdZt).
single-factor: model only has one source of randomness (e.g., short rates), by contrast with
multi-factor models, where e.g., we would model short rate + another asset
no-arbitrage: model prevents arbitrage opportunities. Alternative is equilibrium model, in which
economic agents determine, through their behavior/preferences, equilibrium prices (e.g., Cox-
Ingersoll-Ross model)
When dealing with discrete-time 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 find 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 interest-rate derivatives, and explain how to price them.
An important tool both for pricing and calibrating is the use of Arrow-Debreu 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 effective duration/convexity.
1
A Generic Binomial Model
Let us first introduce some notation:
T= number of time periods.
it= short rate at time t(random variable), t= 0, . . . , T 1.
i(t, n) = nth possible value that itcan take, n= 0, . . . , Nt.
In other words, we will be modeling the process i0, . . . , iT1by assuming that the state space for each
short rate itis of the form {i(t, 0), . . . , i(t, Nt)}.
Abinomial model for the short-rate process {i0, . . . , iT1}means that we are making the following
assumptions:
i0is fixed to some value i(0,0).
For each time t0, it+1 can only take two possible values, which depend on the value i(t, n)
taken by the previous short rate: with probability q(t, n), it will take the value i(t+ 1, n + 1),
and with probability 1 q(t, n), it will take the value i(t+ 1, n).
Figure 1 illustrates how the process evolves from time tto time t+ 1.
i(t,n)
time time t t+1
i (t+1,n+1)
i (t+1,n)
1−q(t,n)
q(t,n)
Figure 1: Binomial model: one step
Hence if we start from time 0, the short rate process will proceed along one path of a binomial interest
rate lattice as shown on Figure 2:
A few remarks are in order:
The probabilities q(t, n) in our model are conditional probabilities, i.e.,
q(t, n) = P(it+1 =i(t+ 1, n + 1)|it=i(t, n))
is associated with an up-move from state i(t, n), and
1q(t, n) = P(it+1 =i(t+ 1, n)|it=i(t, n))
is associated with a down-move from state i(t, n).
The binomial interest rate lattice (or tree) is recombining. That is, at any given node in the
lattice, an up move followed by a down move (the latter being represented by a straight line on
the graph) will reach the same node as a down move followed by an up move. Hence the number
of possible states at each time tis equal to t+ 1. A non-recombining tree would double the
number of nodes at each step, ending with 2Tnodes at time T.
2
0 1 2 3
i(0,0) i(1,0) i(2,0)
i(1,1) i(2,1)
i(3,3)
i(3,0)
i(3,1)
i(3,2)i(2,2)
1−q(0,0)
q(1,0)
q(0,0)
1−q(1,0)
q(1,1)
1−q(1,1)
q(2,2)
q(2,1)
q(2,0)
1−q(2,0)
1−q(2,2)
1−q(2,1)
i(T,0)
i(T,1)
i(T,T−1)
i(T,T)
i(T,T−2)
q(T,T−2)
q(T,T−1)
1−q(T,T−1)
1−q(T,T−2)
1−q(T,0)
q(T,0)
Figure 2: Binomial interest rate lattice
The behavior of itonly depends on the previous value taken by it1. In other words, the short-rate
process {it, t = 0, . . . , T }is Markovian, i.e.,
P(it=i(t, n)|it1, it2, . . . , i0) = P(it=i(t, n)|it1) = 1q(t1, n) if it1=i(t1, n)
q(t1, n 1) if it1=i(t1, n 1).
Figure 3 illustrates the idea.
i (t,n)
i(t−1,n−1)
i(t−1,n) 1−q(t−1,n)
q(t−1,n−1)
Figure 3: How to get to a state i(t, n)
How many possible paths from time 0 to time Tare there?
Example: Consider the binomial interest-rate lattice shown on Figure 4, where we assume for sim-
plicity that q(t, n)=1/2 for n= 0, . . . , t, t = 0, . . . , 4. In what follows we’ll illustrate how to perform
various tasks with interest-rate models using this particular example.
3