固定收益证-券Interest-Rate-Mod课件.ppt

1、Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-1Chapter 17 Interest-Rate Models Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-2Learning ObjectivesAfter reading this chapter, you will understandqwhat an interest-rate model isqhow an interest-rate model is

2、represented mathematicallyqthe characteristics of an interest-rate model: drift, volatility, and mean reversionqwhat a one-factor interest-rate model isqthe difference between an arbitrage-free model and an equilibrium modelqthe different types of arbitrage-free models and why they are used in pract

3、iceqthe difference between a normal model and a lognormal modelqthe empirical evidence on interest rate changesqconsiderations in selecting an interest rate modelqhow to calculate historical volatilityCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-3Mathematical Description of O

4、ne-Factor Interest-Rate Modelsq Interest-rate models must incorporate statistical properties of interest-rate movements.q These properties are i. driftii. volatilityiii.mean reversionq The commonly used mathematical tool for describing the movement of interest rates that can incorporate these proper

5、ties is stochastic differential equations (SDEs).q A rigorous treatment of interest-rate modeling requires an understanding of this specialized topic in mathematics.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-4Mathematical Description of One-Factor Interest-Rate Models (cont

6、inued)q The most common interest-rate model used to describe the behavior of interest rates assumes that short-term interest rates follow some statistical process and that other interest rates in the term structure are related to short-term rates.q The short-term interest rate (i.e., short rate) is

7、the only one that is assumed to drive the rates of all other maturities.q Hence, these models are referred to as one-factor models where the “one factor” is the short rate.q The other rates are not randomly determined once the short rate is specified.q Using arbitrage arguments, the rate for all oth

8、er maturities is determined.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-5Mathematical Description of One-Factor Interest-Rate Models (continued)q There are also multi-factor models that have been proposed in the literature.q The most common multi-factor model is a two-factor

9、 model where a long-term rate is the second factor.q In practice, however, one-factor models are used because of the difficulty of applying even a two-factor model as well as empirical evidence that supports one-factor models.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-6Math

10、ematical Description of One-Factor Interest-Rate Models (continued)q While the value of the short rate at some future time is uncertain, the pattern by which it changes over time can be assumed.q In statistical terminology, this pattern or behavior is called a stochastic process.q Thus, describing t

11、he dynamics of the short rate means specifying the stochastic process that describes the movement of the short rate.q It is assumed that the short rate is a continuous random variable and therefore the stochastic process used is a continuous-time stochastic process.Copyright 2010 Pearson Education,

12、Inc. Publishing as Prentice Hall17-7Mathematical Description of One-Factor Interest-Rate Models (continued)q There are different types of continuous-time stochastic processes used in interest-rate modeling.q In all of these models because time is a continuous variable, the letter d is used to denote

13、 the “change in” some variable.q Specifically, in the models we let r = the short rate and therefore dr denotes the change in the short ratet = time and thus dt denotes the change in time or equivalently the length of the time interval (for a very small interval of time)z = a random term and dz deno

14、tes a random processCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-8Mathematical Description of One-Factor Interest-Rate Models (continued)q A Basic Continuous-Time Stochastic ProcessA basic continuous-time stochastic process for describing the dynamics of the short rate (r) is

15、 given by:dr = bdt + dz dr = the change in the short rateb = expected direction of rate changedt = the change in time or equivalently the length of the time interval (for a very small interval of time) = standard deviation of the changes in the short ratez = a random term and dz denotes a random pro

16、cessThe expected direction of the change in the short rate (b) is called the drift term and is called the volatility term.The change in the short rate (dr) over the time interval (dt) depends oni.the expected direction of the change in the short rate (b)ii. a random process (dz) that is affected by

17、volatilityCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-9Mathematical Description of One-Factor Interest-Rate Models (continued)q A Basic Continuous-Time Stochastic ProcessThe random nature of the change in the short rate comes from the random process dz. The assumptions are t

18、hati.the random term z follows a normal distribution with a mean of zero and a standard deviation of one (i.e., is a standardized normal distribution)ii. the change in the short rate is proportional to the value of the random term, which depends on the standard deviation of the change in the short r

19、ateiii. the change in the short rate for any two different short intervals of time is independent The expected value of the change in the short rate is equal to b, the drift term.In the special case where the drift term is zero and the variance is one, it can be shown that the variance of the change

20、 in the short rate over some interval of length T is equal to T and therefore the standard deviation is the square root of T.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-10Mathematical Description of One-Factor Interest-Rate Models (continued)q It ProcessNotice that in the eq

21、uation, dr = bdt + dz, that neither the drift term (b) nor the standard deviation of the change in the short rate () depends on either the level of the short rate (r) and time (t).There are economic reasons that might suggest that the expected direction of the rate change will depend on the level of

22、 the current short rate; the same is true for .We can change the dynamics of the drift term and the dynamics of the volatility term by allowing these two parameters to depend on the level of the short rate and/or time.We can denote that the drift term depends on both the level of the short rate and

23、time by b(r,t); the same is true for , e.g., (r,t). Thus, we can writedr = b(r,t) dt + (r,t)dz The continuous-time stochastic model given by the above equation is called an Ito process.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-11Mathematical Description of One-Factor Inter

24、est-Rate Models (continued)q Specifying the Dynamics of the Drift TermIn specifying the dynamics of the drift term, one can specify that the drift term depends on the level of the short rate by assuming it follows a mean reversion process.By mean reversion it is meant that some long-run stable mean

25、value for the short rate is assumed.We denote this value by .So, if r is greater than , the direction of change in the short rate will move down in the direction of the long-run stable value and vice versa.The mean reversion process that specifies the dynamics of the drift term is: b(r,t) = (r ) whe

26、re is called the speed of adjustment because it indicates the speed at which the short rate will move or converge to the long-run stable mean value.rrCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-12Mathematical Description of One-Factor Interest-Rate Models (continued)q Specif

27、ying the Dynamics of the Volatility TermThere have been several formulations of the dynamics of the volatility term.If volatility is not assumed to depend on time, then (r,t) = (r).In general, the dynamics of the volatility term can be specified as follows:rdzwhere is equal to the constant elasticit

28、y of variance.The above equation is called the constant elasticity of variance model (CEV model).The CEV model allows us to distinguish between the different specifications of the dynamics of the volatility term for the various interest-rate models suggested by researchers.Copyright 2010 Pearson Edu

29、cation, Inc. Publishing as Prentice Hall17-13Mathematical Description of One-Factor Interest-Rate Models (continued)q Specifying the Dynamics of the Volatility TermFor the Vasicek interest rate model, we look at the case for = 0.Substituting zero for into the equation rdz, we get the following the C

30、EV model identified by Vasicek who first proposed it: = 0: (r,t) = In the Vasicek specification of the CEV model, volatility is independent of the level of the short rate as in the equation of dr = bdt + dz (where b is the drift term) and is referred to as the normal model.In the normal model, it is

31、 possible for negative interest rates to be generated.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-14Mathematical Description of One-Factor Interest-Rate Models (continued)q Specifying the Dynamics of the Volatility TermFor the Dothan interest rate model, we look at the case

32、for = 1.Substituting one for into the equation rdz, we get the following the CEV model specified by Dothan who first proposed it: = 1: (r,t) = rIn the Dothan specification of the CEV model, volatility is proportional to the short rate.This model is referred to as the proportional volatility model.Co

33、pyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-15Mathematical Description of One-Factor Interest-Rate Models (continued)q Specifying the Dynamics of the Volatility TermFor the Cox-Ingersoll-Ross (CIR) interest rate model, we look at the case for = .Substituting one for into the e

34、quation rdz, we get the following the CEV model proposed by CIR: = : (r,t) = The CIR specification, referred to as the square-root model, makes the volatility proportional to the square rate of the short rate.Negative interest rates are not possible in this square-root model.rCopyright 2010 Pearson

35、Education, Inc. Publishing as Prentice Hall17-16Arbitrage-Free Versus Equilibrium Modelsq Arbitrage-Free ModelsIn arbitrage-free models, also referred to as no-arbitrage models, the analysis begins with the observed market price of a set of financial instruments.The financial instruments can include

36、 cash market instruments and interest-rate derivatives, and they are referred to as the benchmark instruments or reference set.The underlying assumption is that the benchmark instruments are fairly priced.A random process for the generation of the term structure is assumed.Based on the random proces

37、s and the assumed value for the parameter that represents the drift term, a computational procedure is used to calculate the term structure of interest rates.The model is referred to as arbitrage-free because it matches the observed prices of the benchmark instruments.Copyright 2010 Pearson Educatio

38、n, Inc. Publishing as Prentice Hall17-17Arbitrage-Free Versus Equilibrium Models (continued)q Arbitrage-Free ModelsThe most popular arbitrage-free interest-rate models used for valuation are:the Ho-Lee modelthe Hull-White modelthe Kalotay-Williams-Fabozzi modelthe Black-Karasinki modelthe Black-Derm

39、an-Toy modelthe Heath-Jarrow-Morton modelCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-18Arbitrage-Free Versus Equilibrium Models (continued)q Arbitrage-Free ModelsThe first arbitrage-free interest-rate model was introduced by Ho and Lee in 1986.In the Ho-Lee model, there is n

40、o mean reversion and volatility is independent of the level of the short rate.That is, it is a normal model where = 0.In the Kalotay-Williams-Fabozzi model, changes in the short-rate are modeled by modeling the natural logarithm of r; no allowance for mean reversion is considered in the model.The He

41、ath-Jarrow-Morton (HJM) model is a general continuous time, multi-factor model; it has received considerable attention in the industry as well as in the finance literature.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-19Arbitrage-Free Versus Equilibrium Models(continued)q Equi

42、librium ModelsA fair characterization of arbitrage-free models is that they allow one to interpolate the term structure of interest rates from a set of observed market prices at one point in time assuming that one can rely on the market prices used.Equilibrium models, however, are models that seek t

43、o describe the dynamics of the term structure using fundamental economic variables that are assumed to affect the interest-rate process.In the modeling process, restrictions are imposed allowing for the derivation of closed-form solutions for equilibrium prices of bonds and interest rate derivatives

44、.In these modelsi.a functional form of the interest-rate volatility is assumedii.how the drift moves up and down over time is assumedCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-20Arbitrage-Free Versus Equilibrium Models (continued)q Equilibrium ModelsIn characterizing the di

45、fference between arbitrage-free and equilibrium models, one can think of the distinction being whether the model is designed to be consistent with any initial term structure, or whether the parameterization implies a particular family of term structure of interest rates.Arbitrage-free models have th

46、e deficiency that the initial term structure is an input rather than being explained by the model.Basically, equilibrium models and arbitrage models are seeking to do different things.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-21Arbitrage-Free Versus Equilibrium Models(cont

47、inued)q Equilibrium ModelsIn practice, there are two concerns with implementing and using equilibrium models.i.Many economic theories start with an assumption about the class of utility functions to describe how investors make choices.ii. These models are not calibrated to the market so that the pri

48、ces obtained from the model can lead to arbitrage opportunities in the current term structure.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall17-22Empirical Evidence on Interest-Rate Changesq In a review of interest-rate models, one can encounter the following issues:i. the choice

49、between normal models (i.e., volatility is independent of the level of interest rates) and logarithm modelsii. if interest rates are highly unlikely to be negative, then interest-rate models that allow for negative rates may be less suitable as a description of the interest-rate processCopyright 201

50、0 Pearson Education, Inc. Publishing as Prentice Hall17-23Empirical Evidence on Interest-Rate Changes (continued)q Volatility of Rates and the Level of Interest RatesThe dependence of volatility on the level of interest rates has been examined by several researchers.The earlier research focused on s