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Chp.19 Term Structure of Interest Rates (II)

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Chp.19 Term Structure of Interest Rates (II)

- 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:

- 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.

- 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.

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

- 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.

- Example: in a riskless economy
- With constant interest rate,

- 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.

- 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:

- Suppose there is only one state variable, r. Apply Ito’s Lemma
- Then we can get:

- 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

- 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.

A second approach is risk-neutral approach

- Define:
- We can then get
- price bonds with risk neutral probability:

- 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.

- Now we solve the PDE with boundary condition
- numerically.
- Express the PDE as
- The first step is

- At the second step

- 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.

- 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.

- The discount factor process is:
- The basic bond differential equation is:
- Method: Guess and substitute

- Guess
- Boundary condition: for any r,
so

- Boundary condition: for any r,
- The result is

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

- Substituting these derivatives into PDE
- This equation has to hold for every r, so we get ODEs

- Solve the second ODE with

- Solve the first ODE with

- Remark: the log prices and log yields are linear function of interest rates
- means the term structure is always upward sloping.

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

- 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

- Take intergral

- So
- We have

- Next we solve the discount factor process
- Plugging r

- The first integral includes a deterministic function, so gives rise to a normally distributed r.v. for
- Thus is normally distributed withmean

- And variance

- So
- Plugging the mean and variance

- Rearrange into
- Which is the same as in the PDE approach

- In the risk-neutral measure

- Guess
- Take derivatives and substitue
- So

- Solve these ODEs
- Where

- 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

- Guess
- Basically, recall that
- Use Ito’s Lemma

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.

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.

- 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 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.

- 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.

- 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: 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.

- 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.

- 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.

- 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!