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discrete_time_interest_models

# discrete_time_interest_models - Discrete-Time Term...

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Discrete-Time 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 Discrete-Time Term Structure Models – p. 1/69

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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 Discrete-Time Term Structure Models – p. 2/69
Introduction Term structure models Modelling the evolution of term structure of interest rates Single-factor 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. multi-factor Discrete-time e.g. binomial lattice (or model), trinomial model vs continuous-time e.g. Vasicek (1977) Model: dr = a ( b r ) dt + σdZ Arbitrage-free Notes by K.S. Tan/Actsc 445/845 Discrete-Time Term Structure Models – p. 3/69

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Notation T ∈ { 0 , 1 , 2 , . . . , T } : time space partition i t : short rate (r.v.) at time period t ∈ T for discrete-time model, short rate is the spot rate of interest for the relevant time unit N ∈ { 0 , 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 Discrete-Time Term Structure Models – p. 4/69
A Generic One-Period 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 multi-period? Notes by K.S. Tan/Actsc 445/845 Discrete-Time Term Structure Models – p. 5/69

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A General Binomial Interest Rate Lattice i (0 , 0) i (1 , 0) i (1 , 1) 1 q (0 , 0) q (0 , 0) i (2 , 0) 1 q (1 , 0) q (1 , 0) 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 , 0) i (3 , 2) i (3 , 3) 1 q (2 , 2) q (2 , 2) 1 q (2 , 1) q (2 , 1) time 0 time 1 time 2 time 3 Notes by K.S. Tan/Actsc 445/845 Discrete-Time Term Structure Models – p. 6/69
Remarks on the Interest Rate Lattice Recombining: At time t , there are t + 1 possible states Markovian or path-independent process: the conditional probability of an “up-jump:" q ( t, n ) Pr { i t +1 = i ( t + 1 , n + 1) | i t = i ( t, n ) } , 0 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. How many distinct interest rate paths for a T -period binomial lattice? Notes by K.S. Tan/Actsc 445/845 Discrete-Time Term Structure Models – p. 7/69

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An Example of Interest Rate Lattice 6.00 6.52 9.54 6.95 9.81 13.85 7.39 10.09 13.79 18.85 7.90 10.49 13.87 18.37 24.33 year 0 year 1 year 2 year 3 year 4 Assume q ( t, n ) = 0 . 5 Notes by K.S. Tan/Actsc 445/845 Discrete-Time Term Structure Models – p. 8/69
Bond Pricing Based on Interest Rate Lattice Let P (0 , t ) be the time-0 price of a zero-coupon bond with \$1 face value with t time periods maturity

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