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Unit 8 (Part I) – Discrete-Time Interest Rate Models

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University of Waterloo
Actuarial Science
Jiahua Chen

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 xed income securities (such as callable bonds), and interest-rate derivatives (such as interest caps and oors). 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(b r)dt + dZ ). t 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 well 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), well use forward induction. The plan for Part I of this unit is as follows: First, well look at a simple generic model and see how we can use it to price bonds. Second, well 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 well discuss next. Well then look more closely at models and how we can calibrate them. Finally, well discuss how to price embedded options in bonds, and use this to revisit the notion of eective duration/convexity. 1 A Generic Binomial Model Let us rst introduce some notation: T = number of time periods. i = short rate at time t (random variable), t = 0,...,T 1. t th i(t,n) = n possible value that i ctn take, n = 0,...,N . t In other words, we will be modeling the process i ,...,i 0 T1 by assuming that the state space for each short rate i ts of the form {i(t,0),...,i(t,N )}. t A binomial model for the short-rate process {i ,...,i 0 T1 } means that we are making the following assumptions: i is xed to some value i(0,0). 0 For each time t 0, i t+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 t to time t + 1. i t 1 + q 1 n ) ) 1 ( q ( n t 1 ) i t ) i n + m m ) 1 e e 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(i t+1 = i(t + 1,n + 1)|i = t(t,n)) is associated with an up-move from state i(t,n), and 1 q(t,n) = P(i = i(t + 1,n)|i = i(t,n)) t+1 t 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 t is equal to t + 1. A non-recombining tree would double the number of nodes at each step, ending with 2 T nodes at time T. 2 T ( T T ) ) q i , T T ( ) 1 3 ( ) ( ) , 2 1 1 ) i ( , i 2 i T 2 2 3 2 ) ) ) ) ) 1 ( 1 , ) ) i ( 2 ( 3 , 1 1 1 1 T ) q ) ) q ) ) q 1 ( 1 2 T ) 0 ) ) 0 ) ) 0 1 2 ( T 0 1 0 0 1 , 1 0 ) q ) ( ) q ) q ) 0 0 1 0 2 2 3 T ) ) ) ) Figure 2: Binomial interest rate lattice The behavior of i only depends on the previous value taken by i . In other words, the short-rate t t1 process {i ,t =t0,...,T} is Markovian, i.e., 1 q(t 1,n) if i = i(t 1,n) P(i = i(t,n)|i ,i ,...,i ) = P(i = i(t,n)|i ) = t1 t t1 t2 0 t t1 q(t 1,n 1) if i = i(t 1,n 1).
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