1. The Term Structure Models

2. 即期利率、貼現函數與遠期利率定義 ：到期期限為m期的零息債券在第t期(或稱當期)的即期利率the spot rates。
：到期期限為m期面額為$1的零息債券在第t期的貼現函數(the discount functions)。
：遠期利率，或稱為未來m年後的一年期利率。(以下下標t均省略)

3. 即期利率、貼現函數與遠期利率間之關係 即期利率、貼現函數與遠期利率互有緊密關聯，三者中只要知道其中任何一項利率，即可據以推導其餘二項利率
這三種利率通常都不是可直接觀察到
目的是建立一平滑的利率期限結構

4. 即期利率與貼現函數的關聯
間斷複利
連續複利

5. 即期利率與遠期利率的關聯
此一關聯性是應用rational expectation hypothesis,較長期的即期利率是目前短期利率與未來短期利率的平均值進行推導遠期利率。
間斷複利
連續複利

6. Interest Rate Models
利率模型包括兩大類：均衡模型(equilibrium models)以及無套利模型(no-arbitrage models)
Equilibrium models usually start with assumptions about economic variables and derive a process for the short rate. They then explore what the process for r implies about bond prices and option prices.
In a one-factor equilibrium model, the process for r involves only one source of uncertainty.
A no-arbitrage model is a model designed to be exactly consistent with today’s term structure of interest rates.

7. Interest Rate Models
均衡模型,例如the Merton model, Vasicek model, Cox-Ingersoll-Ross (CIR) model, Health-Jarrow-Merton model etc.
無套利模型,例如Ho-Lee model, Black-Derman-Toy model etc.

8. Stochastic Processes of Interest Rates --- Example 1
If we assume that the continuously compounded zero-coupon bond yield y(t) for a given maturity t jumps continuously with mean zero and standard deviation of one, we would write:
dy = dz

9. Stochastic Processes of Interest Rates --- Example 2
We can scale the jumps of interest rates by multiplying the noise generating Wiener process by a scale factor. In the case of interest rates, this scale factor is the “volatility” of the stochastic process driving interest rate movements.

10. Stochastic Processes of Interest Rates --- Example 3
We can incorporate interest rate cycles to interest rate model by introducing some form of drift in interest rates over time.
This formula assumes that on average, the change interest rates over a given instant will be the change given by the function , with random shocks in the amount of

11. Stochastic Processes of Interest Rates --- Example 4
Example 1 to example 3 represent a random walk of interest rates, with or without a drift term.
The best way to build the interest rate cycle into the random movement of interest rates is to assume that the interest rate drifts back to some long-term level (mean-reversion).
This is the Ornstein-Uhlenbeck process.

12. Ito’s Lemma --- One Random Variable
Once a term structure model has been chosen, we need to be able to draw conclusions about how securities, whose values depend on interest rates, should move around.
If a given interest rate y is assumed to be the random factor that determines the price of a zero-coupon bond P, it is logical to ask how the zero-coupon bond price moves as time passes and y varies,或可表示為P(y, t)。

13. Ito’s Lemma --- Two Random Variable
If the zero-coupon bond price depended on two random variables, x and y, then we would be dealing with a “two-factor” term structure model.
The instantaneous correlation between the two Wiener processes which provide the random shocks to s and y m is reflected in the formula for the random movement of P by the instantaneous correlation coefficient rho.

14. Using Ito’s lemma to build a term structure model
Step 1: Make an assumption about the stochastic process that interest rates follow.
Step 2: Use Ito’s lemma to specify how zero-coupon bond prices move.
Step 3: Impose the constraint that there not be riskless arbitrage in the bond market.
Step 4: Solve the resulting partial differential equation for zero-coupon bond prices.
Step 5: Examine whether this implies reasonable or unreasonable conditions in the market.
Step 6: If the economic implications are reasonable, proceed to value other securities with the model.

15. The Merton Term Structure Model
Step 1: Assume one-factor model and
If a given interest rate r is assumed to be the random factor that determines the price of a zero-coupon bond P, it is logical to ask how the zero-coupon bond price moves as time passes and y varies, or P(r,t,T).
Step 2: By Ito’s Lemma

16. The Merton Term Structure Model (Continued)
Step 3: Construct a portfolio W with two bonds. (Assume a parallel shift in interest rates.)

17. The Merton Term Structure Model (Continued)
The proper ratio w for zero interest rate risk is
Since the portfolio is risk-free, the instantaneous return on the portfolio should be
The market price of risk
This is the fixed income equivalent of the Sharpe ratio.

18. The Merton Term Structure Model (Continued)
Substitute , and eliminate the dt from both sides, rearrange, and divide by interest rate volatility to get (Sharpe ratio)
The market price of risk
This is the fixed income equivalent of the Sharpe ratio.
Rearranging and choosing bond one and dropping the subscript one means that the yield y must satisfy the following

19. The Merton Term Structure Model (Continued)
Step 3: The boundary condition is
Let time to maturity as , and guess that P has the solution
The boundary condition implies that

20. The Merton Term Structure Model (Continued)
Step 4: We need to solve for the forms of the functions F and G.
代入微分方程並除上P，整理後可得
整理為

21. The Merton Term Structure Model (Continued)
For the above relation be held, it needs
解微分方程如下

22. The Merton Term Structure Model (Continued)

23. The Merton Term Structure Model (Continued)
The solution to the equation is:

24. The Merton Term Structure Model (Continued)
收益曲線一有hump，且當到期期限大到某一點時，收益率呈負值。

25. The Merton Term Structure Model (Continued)
It is a simple analytical formula;
Zero-coupon bond prices are a quadratic function of time to maturity;
Yields turn negative (and zero-coupon bond prices rise above one) beyond a certain point;
If interest rate volatility is zero, zero-coupon bond yields are constant for all maturities and equal to r.

26. The Ho-Lee Model
Ho and Lee (1986) extended the Merton model to fit a given initial yield curve perfectly in a discrete time framework.
It can be derived by the equivalent model using continuous time and the no arbitrage approach.
The Ho-Lee model is a binomial version of the Vasicek model without mean reversion, in which the one-period interest rate is assumed to have a deterministic drift.
Assume that the short rate r has a time dependent drift term:

27. The Ho-Lee Model (Continued)
Guess the solution P is as follows.

28. The Ho-Lee Model (Continued)

29. The Extended Merton Model (Continued)
The function a(s) is chosen such that the theoretical zero-coupon yield to maturity y and the actual zero-coupon yield are exactly the same.
The extension term’s magnitude, therefore, must offset the negative interest zero-coupon bond yields that would otherwise be predicted by the model.

30. The Vasicek Model
Both the Merton model and the Ho and Lee model are based on an assumption about random interest rate movements that zero-coupon bond yields will be negative at every single instant in time for long maturities beyond a critical maturity.
The Ho and Lee model offsets the negative yields with an extension factor that must grow larger and larger as maturities lengthen.
Vasicek (1977) proposed a model that avoids the certainty of negative yields and eliminates the need for a potentially infinitely large extension factor.

31. The Vasicek Model (Continued)
r is the instantaneous short rate of interest;
is the speed of mean reversion;
is the long-term expected value for r, and;
is the instantaneous standard deviation of r;
Z is the standard Wiener process

32. The Vasicek Model (Continued)
The speed of mean reversion is positive.
The drift term in the Vasicek model pulls the short rate r back toward
When , r tends to drift upward.
When , r tends to drift downward.

33. The Vasicek Model (Continued)
Guess the zero-coupon bond has the solution.
代入微分方程並除上P

34. The Vasicek Model (Continued)
This relationship must hold for all values of r, so the coefficient
of r must be zero:

35. The Vasicek Model (Continued)

36. The Vasicek Model (Continued)

37. The Vasicek Model (Continued)
The yield to maturity is positive for almost all realistic sets of parameter values.
Because r(s) is normally distributed, there is a positive probability that r(s) can be negative.
The magnitude of this theoretical problem with the Vasicek model depends on the level of interest rates and the parameters chosen.
In general, it should be a minor consideration for most applications.
Very low interest rates in Japan in the last decade, with short rates often under 0.02%, did lead to high probabilities of negative rates using both the Vasicek and extended Vasicek models when sigma was set to match obserable prices of caps and floors.

38. The Vasicek Model (Continued)
The Vasicek model allows us to calculate the expected value and variance of the short rate at any time in the future s from the perspective of current time t.

39. The Extended Vasicek-Hull and White Model
Hull and White(1990) model is also referred to as the extended Vasicek model.
This model allows the market price of risk term to drift over time, instead of assuming constant as in the Vasicek model.
Hull and White bridged the gap between the observable yield curve and the theoretical yield curve implied by the Vasicek model by extending the theoretical yield curve to fit the actual market data.

40. The Extended Vasicek Model (Continued)
Hull and White use the time-dependent drift term in interest rates, theta, where theta in turn depends on a time-dependent market price of risk.

41. The Alternative One-Factor Models
The CIR Model
The Dothan Model
The Longstaff Model
The Black, Derman, and Toy Model
The Black and Karasinski Model

42. The CIR Model
The model has been proposed by Cox, Ingersoll, and Roll (1985).
CIR assumes that the short-term interest rate is the single stochastic factor driving interest rate movements, and that the variance of the short rate of interest is proportional to the level of interest rates.
The stochastic movements in the short rate take the form:

43. The CIR Model (Continued)
The value of a zero-coupon bond with maturity takes the form:

44. The CIR Model (Continued)
Advantages:
1. The interest rate volatility is higher in periods of high interest rates than it is in periods or low interest rates.
2. This varying-volatility has the highly desirable property of preventing negative short rates of interest.
Disadvantages:
1. It is difficult to estimate parameters for the CIR model.
2. It fails to provide a good description of the Treasury market.
3. It may not fit some yield curve shapes where instantaneous forward rates turn negative.

45. The Dothan Model
Dothan (1978) provides a model of short rate movements where the short-term risk-less rate of interest r follows a geometric Wiener process.
The short rate has a lognormal distribution and will therefore always be positive.
The Dothan model lacks the “mean reversion” term which causes interest rate cycles. This term’s omission makes the Dothan model much less realistic than the CIR or Vasicek models.

46. The Longstaff Model
Longstaff(1989) proposes a model in which the variance of the short rate is proportional to the level of the short rate, like the CIR model, and the mean reversion of the short rate is a function of its square root:
The resulting pricing equation for zero-coupon bonds is

47. The Black, Derman, and Toy Model
Black, Derman, and Toy(1990) suggest another model which avoids the problem of negative interest rates and allows for time dependent parameters.
This model combines the ability to fit the observable yield curve with the non-negative restriction on interest rates and the ability to model the volatility curve observable in the market. However, it lacks tractable analytical solutions.

48. The Black and Karasinski Model
Black and Karasinski (1991) further refine the Black, Derman, and Toy approach with the explicit incorporation of time-dependent mean reversion:

49. Two-Factor Interest Rate Models
The Brennan and Schwartz Model
The Two-Factor CIR Model
The Two-Factor Vasicek Model
The Longstaff and Schwartz Stochastic Volatility Model

50. The Brennan and Schwartz Model
Brennan and Schwartz (1979) introduced a two-factor model where both a long-term rate and a short-term rate follow a joint Gauss-Markov processes.
The long-term rate is defined as the yield on a consol (perpetual) bond.
Brennan and Schwartz assume that the log of the short rate has the following stochastic process:

51. The Two-Factor CIR Model
Chen and Scott (1992) derive a two-factor model in which the nominal rate of interest I is the sum of two independent variables , both of which follow the stochastic process specified by CIR:
Chen and Scott show that the price of a discount bond in this model is
where A and B have the same definition as in the CIR model, with the addition of appropriate subscripts.

52. The Two-Factor Vasicek Model
Hull and White (1993) show that there is a similar extension for the Vasicek model when the nominal interest rate I is the sum of two factors
The value of a zero-coupon bond with maturity tau is simply the product of two factors which have exactly the same functional form as the single factor Vasicek model:
Both stochastic factors are assumed to follow stochastic processes identical to the normal Vasicek model:

53. The Longstaff and Schwartz Stochastic Volatility Model
Longstaff and Schwartz (1992) propose a model in which two stochastic factors, which are assumed to be uncorrelated, drive interest rate movements. The factors x and y are assumed to follow the stochastic processes:

54. Chen’s Three-Factor Term Structure Model
As factors are added to a term structure model, it becomes more realistic and more complex. Chen (1994) introduces a three-factor term structure model where the short-term rate of interest is random and mean-reverting around a level which is also random.