8models - ACTSC 445: Asset-Liability Management Department...

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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 ²oors). 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 + σdZ 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 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 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 e³ective duration/convexity. 1
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A 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 short-rate 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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8models - ACTSC 445: Asset-Liability Management Department...

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