Lecture_04 - 4 The Evolution of the Term Structure of...

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1 4 The Evolution of the Term Structure of Interest Rates A Motivation This chapter introduces the stochastic evolution for the zero-coupon bond price curve. The stochastic structure is introduced sequentially, starting with a one-factor model, then presenting a two-factor model, and so forth. This section motivates the discrete time processes constructed in this chapter. Consider the historical forward rate curve evolution.
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2 Figure 4.1: Forward Rate Curve Evolutions over January 1973 - March 1997
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3 From this evolution, we can generate histograms for changes in a particular maturity forward rate.
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4 Figure 4.2: Histogram of Monthly Changes in Forward Rates from January 1973 - March 1997. -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 0.05 0.1 0.15 0.2 0.25 3 Months Forward Rate Δ f -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 6 Months Forward Rate Δ f -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 Year Forward Rate Δ f -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 3 Years Forward Rate Δ f
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5 Figure 4.2 (continued): Histogram of Monthly Changes in Forward Rates from January 1973 - March 1997. -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 5 Years Forward Rate Δ f -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25 7 Years Forward Rate Δ f -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 10 Years Forward Rate Δ f -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 20 Years Forward Rate Δ f
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6 What we want to do in this chapter is to build a model for the evolution of forward rates such that the model provides a reasonable approximation to the true underlying joint distribution whose marginals are given in Figure 4.2. The binomial (and its generalization – the multinomial) model provides such an approach. This process is illustrated in Figure 4.3.
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7 Figure 4.3: Example of A Binomial Approximation to the True Distribution--A Normal Distribution 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 ... Ó Ó Ó Ó 1/2 - 1/2 - 1/4 - 3/8 - 1/8 - True Distribution f Δ f f f .05 .1 .05 .1 .05 .1 .075 .05 .1 .075 .025 .05 .075 .1 .025 .05 .075 .1 0 . . . .3 Probability Probability Probability
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8 B The One-Factor Economy 1 The State Space Process EXAMPLE: ONE-FACTOR STATE SPACE PROCESS
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9 Figure 4.4: An Example of a One-Factor State Space Tree Diagram 3/4 u 3/4 uu 3/4 uuu uud 1/4 1/4 1/4 d 3/4 1/4 du ud dd 3/4 1/4 3/4 1/4 3/4 1/4 udu udd duu dud ddu ddd tim e 0 1 2 3
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10 At this time, the up state and the down state have no economic interpretation. They are just used as “place-holders” for an economic state, e.g. “good” or “bad”. The state space process in Fig. 4.4 is called a one- factor model because at each node in the tree, only one of two possibilities can happen (up or down). Each branch also occurs with strictly positive probability. One can conceptualize the tree's being constructed by tossing one coin (one-factor). If, instead, at each node of the tree there were three branches, each with strictly positive probability, it would be called a two-factor model.
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This note was uploaded on 12/09/2008 for the course NBA 5550 taught by Professor Jarrow,robert during the Fall '08 term at Cornell University (Engineering School).

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Lecture_04 - 4 The Evolution of the Term Structure of...

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