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

1. 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:

2. 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.

3. Some famous term structure models
1.Vasicek Model:
Vasicek model is similar to AR(1) model.
2.CIR Model
The square root terms captures the fact that higher interest rate seem to be more volatile, and keeps the interest rate from zero.

4. 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

5. 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.

6. expectation approach Example: in a riskless economy
With constant interest rate,

7. 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.

8. 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:

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

10. 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

11. 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.

12. Risk-Neutral Approach
A second approach is risk-neutral approach
Define:
We can then get
price bonds with risk neutral probability:

13. 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.

14. Solving the bond price PDE numerically
Now we solve the PDE with boundary condition
numerically.
Express the PDE as
The first step is

15. Solving the bond price PDE
At the second step

16. 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.

17. 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.

18. Vasicek Model The discount factor process is:
The basic bond differential equation is:
Method: Guess and substitute

19. PDE solution:(1)
Guess
Boundary condition: for any r, so
The result is

20. PDE solution:(1)
To substitute back to PDE ,we first calculate the partial derivatives given

21. PDE solution:(1) Substituting these derivatives into PDE
This equation has to hold for every r, so we get ODEs

22. PDE solution(2)
Solve the second ODE with

23. PDE solution(3)
Solve the first ODE with

24. PDE solution(3)

25. PDE solution(3)

26. PDE solution(4)
Remark: the log prices and log yields are linear function of interest rates
means the term structure is always upward sloping.

27. 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

28. 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

29. Vasicek Model by Expectation
Take intergral

30. Vasicek Model by ExpectationSo
We have

31. Vasicek Model by Expectation
Next we solve the discount factor process
Plugging r

32. Vasicek Model by Expectation

33. 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 with mean

34. Vasicek Model by Expectation
And variance

35. Vasicek Model by ExpectationSo
Plugging the mean and variance

36. Vasicek Model by Expectation
Rearrange into
Which is the same as in the PDE approach

37. Vasicek Model by Expectation
In the risk-neutral measure

38. CIR Model

39. CIR Model
Guess
Take derivatives and substitue So

40. CIR Model
Solve these ODEs
Where

41. CIR Model

42. Multifactor Affine Models
Vasicek Model and CIR model are special cases of affine models (Duffie and Kan , 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.

43. Multifactor Affine Model

44. Multifactor Affine Model
Where

45. PDE solution
Guess
Basically, recall that
Use Ito’s Lemma

46. PDE solution

47. Multifactor Affine Model

48. Multifactor Affine Model

49. Multifactor Affine Model

50. Multifactor Affine Model
Rearrange we get the ODEs for Affine Model

51. Bibliography and Comments
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).

52. Bibliography and Comments
Constantinides(1992)
Nonlinear Model based on CIR Model,
Analytical solution.
Allows for both signs of term premium.

53. 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.

54. 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).

55. 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.

56. 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.

57. 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.

58. 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.

59. 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.

60. 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.

61. 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.

62. Thanks!