8 Discrete Time Interest Rate Models

8 Discrete Time Interest Rate Models - ACTSC 445:...

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Unformatted text preview: 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 ( 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 find 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 effective duration/convexity. 1 A Generic Binomial Model Let us first 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 = 0 ,...,N t . In other words, we will be modeling the process i ,...,i T- 1 by assuming that the state space for each short rate i t is of the form { i ( t, 0) ,...,i ( t,N t ) } . A binomial model for the short-rate process { i ,...,i T- 1 } means that we are making the following assumptions: i is fixed to some value i (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 )....
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This note was uploaded on 10/24/2010 for the course ACTSC 445 taught by Professor Christianelemieux during the Fall '09 term at Waterloo.

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8 Discrete Time Interest Rate Models - ACTSC 445:...

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