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Unformatted text preview: DiscreteTime Term Structure Models References: Chapter 34 (Fabozzi) & Chapter 7 (FE) Lecture Notes for Actsc 445/845 Department of Statistics and Actuarial Science University of Waterloo Notes by K.S. Tan/Actsc 445/845 DiscreteTime Term Structure Models p. 1/69 Topics Introduction to term structure models Pricing interest rate derivatives/contingent claims backward recursion/dynamic programming approach Calibration forward induction approach Notes by K.S. Tan/Actsc 445/845 DiscreteTime Term Structure Models p. 2/69 Introduction Term structure models Modelling the evolution of term structure of interest rates Singlefactor all relevant information in the term structure of interest rate is captured by a single factor e.g. short rate , forward rate, bond price vs. multifactor Discretetime e.g. binomial lattice (or model), trinomial model vs continuoustime e.g. Vasicek (1977) Model: dr = a ( b r ) dt + dZ Arbitragefree Notes by K.S. Tan/Actsc 445/845 DiscreteTime Term Structure Models p. 3/69 Notation T { , 1 , 2 , ... ,T } : time space partition i t : short rate (r.v.) at time period t T for discretetime model, short rate is the spot rate of interest for the relevant time unit N { , 1 , 2 , ... ,N } : state space partition i ( t,n ) : realized short rate at time period t T and in state n N Notes by K.S. Tan/Actsc 445/845 DiscreteTime Term Structure Models p. 4/69 A Generic OnePeriod Binomial Branching Model If i t = i ( t,n ) , then in the next period the short rate takes one of only two possible values: either i t +1 = i ( t + 1 ,n ) with probability 1 q ( t,n ) or i t +1 = i ( t + 1 ,n + 1) with probability q ( t,n ) i ( t,n ) i ( t + 1 ,n ) i ( t + 1 ,n + 1) 1 q ( t,n ) q ( t , n ) Generalization to multiperiod? Notes by K.S. Tan/Actsc 445/845 DiscreteTime Term Structure Models p. 5/69 A General Binomial Interest Rate Lattice i (0 , 0) i (1 , 0) i (1 , 1) 1 q (0 , 0) q ( , ) i (2 , 0) 1 q (1 , 0) q ( 1 , ) i (2 , 1) i (2 , 2) 1 q (1 , 1) q ( 1 , 1 ) i (3 , 0) i (3 , 1) 1 q (2 , 0) q ( 2 , ) i (3 , 2) i (3 , 3) 1 q (2 , 2) q ( 2 , 2 ) 1 q (2 , 1) q ( 2 , 1 ) b b b b b b time 0 time 1 time 2 time 3 Notes by K.S. Tan/Actsc 445/845 DiscreteTime Term Structure Models p. 6/69 Remarks on the Interest Rate Lattice Recombining: At time t , there are t + 1 possible states Markovian or pathindependent process: the conditional probability of an upjump:" q ( t,n ) Pr { i t +1 = i ( t + 1 ,n + 1)  i t = i ( t,n ) } , n t < T. the conditional probability of a down jump" is 1 q ( t,n ) Pr { i t +1 = i ( t + 1 ,n )  i t = i ( t,n ) } . A sample path for the short rate process corresponds to a path through the binomial lattice, starting from node (0 , 0) , and proceeding to one of the time T nodes....
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This note was uploaded on 01/13/2011 for the course ACTSC 445 taught by Professor Christianelemieux during the Spring '09 term at Waterloo.
 Spring '09
 ChristianeLemieux

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