Chp.19 Term Structure of Interest Rates (II)

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

2. 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:

3. 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.

4. 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.

5. 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

6. 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.

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

8. 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.

9. 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:

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

11. 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

12. 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.

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

14. 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.

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

16. Solving the bond price PDE • At the second step

17. 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.

18. 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.

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

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

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

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

23. PDE solution(2) • Solve the second ODE with

24. PDE solution(3) • Solve the first ODE with

25. PDE solution(3)

26. PDE solution(3)

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

28. 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

29. 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

30. Vasicek Model by Expectation • Take intergral

31. Vasicek Model by Expectation • So • We have

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

33. Vasicek Model by Expectation

34. 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

35. Vasicek Model by Expectation • And variance

36. Vasicek Model by Expectation • So • Plugging the mean and variance

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

38. Vasicek Model by Expectation • In the risk-neutral measure

39. CIR Model

40. CIR Model • Guess • Take derivatives and substitue • So

41. CIR Model • Solve these ODEs • Where

42. CIR Model

43. 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.

44. Multifactor Affine Model

45. Multifactor Affine Model Where

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

47. PDE solution

48. Multifactor Affine Model

49. Multifactor Affine Model

50. Multifactor Affine Model