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# Chp.19 Term Structure of Interest Rates (II) PowerPoint PPT Presentation

Chp.19 Term Structure of Interest Rates (II). Continuous time models. Term structure models are usually more convenient in continuous time. Specifying a discount factor process and then find bond prices. A wide and popular class models for the discount factor:. Implications.

Chp.19 Term Structure of Interest Rates (II)

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## Chp.19 Term Structure of Interest Rates (II)

### Continuous time models

• Term structure models are usually more convenient in continuous time.

• Specifying a discount factor process and then find bond prices.

• A wide and popular class models for the discount factor:

### Implications

• Different term structure models give different specification of the function for

• r starts as a state variable for the drift of discount factor process, but it is also the short rate process since

• Dots(.) means that the terms can be function of state variables.(And so are time-varying)

• Some orthogonal components can be added to the discount factor with on effect on bond price.

### Some famous term structure models

• 1.Vasicek Model:

Vasicekmodel is similar to AR(1) model.

• 2.CIRModel

• The square root terms captures the fact that higher interest rate seem to be more volatile, and keeps the interest rate from zero.

### Continuous time models

Having specified a discount factor process, it is simple matter to find bond prices

Two way to solve

• 1. Solve the discount factor model forward and take the expectation

• 2. Construct a PDE for prices, and solve that backward

### Implication

• Both methods naturally adapt to pricing term structure derivatives : call options on bonds, interest rate floors or caps, swaptions and so forth, whose payoff is

• We can take expectation directly or use PDE with option payoff as boundary conditions.

### expectation approach

• Example: in a riskless economy

• With constant interest rate,

### Remark

• In more situations, the expectation approach is analytically not easy.

• But in numerical way, it is a good way. We can just stimulate the interest rate process thousands of times and take the average.

### Differential Equation Approach

• Similar to the basic pricing equation for a security price S with no dividend

• For a bond with fixed maturity, the return is

• Then we can get the basic pricing equation for the bonds with given maturity:

### Differential Equation Solution

• Suppose there is only one state variable, r. Apply Ito’s Lemma

• Then we can get:

### Market Price of Risk and Risk-neutral Dynamic Approach

• The above mentioned PDE is derived with discount factors.

• Conventionally the PDE is derived without discount factors.

• One approach is write short-rate process and set market price of risk to

### Implication

• If the discount factor and shocks are imperfectly correlated,

• Different authors use market price of risk in different ways.

• CIR(1985) warned against modeling the right hand side as , it will lead to positive expected return when the shock is zero, thus make the Sharpe ration infinite.

• The covariance method can avoid this.

### Risk-Neutral Approach

A second approach is risk-neutral approach

• Define:

• We can then get

• price bonds with risk neutral probability:

### Remark

• The discount factor model carries two pieces of information.

• The drift or conditional mean gives the short rate

• The covariance generates market price of risk.

• It is useful to keep the term structure model with asset pricing, to remind where the market price of risk comes from.

• This beauty is in the eye of the beholder, as the result is the same.

### Solving the bond price PDE numerically

• Now we solve the PDE with boundary condition

• numerically.

• Express the PDE as

• The first step is

### Solving the bond price PDE

• At the second step

### 5. Three Linear Term Structure Models

• Vasicek Model, CIR Model, and Affine Model gives a linear function for log bond prices and yields:

• Term structure models are easy in principle and numerically. Just specify a discount factor process and find its conditional expectation or solve the differential equation.

### Overview

• Analytical solution is important since the term structure model can not be reverse-engineered. We can only start from discount factor process to bond price, but don’t know how to start with the bond price to discount factor. Thus, we must try a lot of calculation to evaluate the models.

• The ad-hoc time series models of discount factor should be connected with macroeconomics, for example, consumption, inflation, etc.

### Vasicek Model

• The discount factor process is:

• The basic bond differential equation is:

• Method: Guess and substitute

### PDE solution:(1)

• Guess

• Boundary condition: for any r,

so

• The result is

### PDE solution:(1)

• To substitute back to PDE ,we first calculate the partial derivatives given

### PDE solution:(1)

• Substituting these derivatives into PDE

• This equation has to hold for every r, so we get ODEs

### PDE solution(2)

• Solve the second ODE with

### PDE solution(3)

• Solve the first ODE with

### PDE solution(4)

• Remark: the log prices and log yields are linear function of interest rates

• means the term structure is always upward sloping.

### Vasicek Model by Expectation

• The Vasicek model is simple enough to use expectation approach. For other models the algebra may get steadily worse.

• Bond price

### Vasicek Model by Expectation

• First we solve r from

• The main idea is to find a function of r, and by applying Ito’s Lemma we get a SDE whose drift is only a function of t. Thus we can just take intergral directly.

• Define

### Vasicek Model by Expectation

• Take intergral

• So

• We have

### Vasicek Model by Expectation

• Next we solve the discount factor process

• Plugging r

### Vasicek Model by Expectation

• The first integral includes a deterministic function, so gives rise to a normally distributed r.v. for

• Thus is normally distributed withmean

• And variance

### Vasicek Model by Expectation

• So

• Plugging the mean and variance

### Vasicek Model by Expectation

• Rearrange into

• Which is the same as in the PDE approach

### Vasicek Model by Expectation

• In the risk-neutral measure

### CIR Model

• Guess

• Take derivatives and substitue

• So

### CIR Model

• Solve these ODEs

• Where

### Multifactor Affine Models

• Vasicek Model and CIR model are special cases of affine models (Duffie and Kan 1996, Dai and Singleton 1999).

• Affine Models maintain the convenient form that the log bond prices are linear functions of state variables(The short rate and conditional variance be linear functions of state variables).

• More state variables, such as long interest rates, term spread, (volatility),can be added as state variable.

Where

### PDE solution

• Guess

• Basically, recall that

• Use Ito’s Lemma

### Multifactor Affine Model

Rearrange we get the ODEs for Affine Model

• The choice between discrete and continuous time is just for convenience. Campbell, Lo and MacKinlay(1997) give a discrete time treatment, showing that the bond prices are also linear in discrete time two parameters square root model.

• In addition to affine, there are many other kinds of term structure models, such as Jump, regime shift model, nonlinear stochastic volatility model, etc. For the details, refer to Lin(2002).

Constantinides(1992)

• Nonlinear Model based on CIR Model,

• Analytical solution.

• Allows for both signs of term premium.

### Risk-neutral method

The risk-neutral probability method rarely make reference to the separation between drifts and market price of risk. This was not a serious problem for the option pricing, since volatility is more important.

However, it is not suitable for the portfolio analysis and other uses. Many models imply high and time-varying market price of risk and conditional Sharpe ratio.

Duffee(1999) and Duarte(2000) started to fit the model to the empirical facts about the expected returns in term structure models.

### Term Structure and Macroeconomics

• In finance, term structure models are often based on AR process.

• In macroeconomics, the interest rates are regressed on a wide variety of variables, including lagged interest rate, lagged inflation, output, unemployment, etc.

• This equation is interpreted as the decision-making rule for the short rate.

• Taylor rule(Taylor,1999), monetary VAR literature (Eichenbaum and Evans(1999).

### The criticism of finance model

• The criticism of term structure model in finance is hard when we only use one factor model.

• Multifactor models are more subtle.

• But if any variable forecasts future interest rate, it becomes a state variable, and should be revealed by bond yields.

• Bond yields should completely drive out other macroeconomic state variables as interest rate forecasters.

• But in fact, it is not.

### High-frequency research

• Balduzzi,Bertola and Foresi (1996), Piazzesi(2000) are based on diffusions with rather slow-moving state variable. The one-day ahead densities are almost exactly normal.

• Johannes(2000) points out the one day ahead densities have much fatter tails than normal distribution. This can be modeled by fast-moving state variables. Or, it is more natural to think of a jump process.

### Other Development

• All the above mentioned models describe the bond yields as a function of state variables.

• Knez, Litterman and Scheinkman(1994) make a main factor analysis on the term structure and find that most of the variance of yields can be explained by three main factors, level, slope, hump. It is done by a simple eigenvalue decomposition method.

### Remark

• Remark: This method is mainly used in portfolio management, for example, to realize the asset immunation of insurance fund.

• It is a good approximation, but just an approximation. The remaining eigenvalues are not zero. Then the maximum likelihood method is not suitable, maybe GMM is better.

• The importance of approximation depends on how you use the model, if you want to find some arbitrage opportunity, it has risk. The deviation from the model is at best a good Sharpe ratio but K factor model can not tell you how good.

### Possible Solution

• Different parameters at each point in time (Ho and Lee 1986). It is useful, but not satisfactory.

• The whole yield curve as a state variable, Kennedy(1994), SantaClara and Sornette(1999) may be the potential way.

### Market Price of Risk

• The market price of interest rate risk reflects bond the market price of real interest rate change and the market price of inflation.

• The relative contribution is very important for the nature of risk.

• If the real interest rate is constant and nominal rates change with inflation, the short term bonds are safest long term investment.

### Market Price of Risk

• If the inflation is constant and nominal rates change with the real rate, the long term bonds are safest long term investment.

• Little work is done on the separation of interest rate premia between real and inflation premium components. Buraschi and Jiltsov(1999) is one recent effort.

• Thanks!