1. Advanced Term Structure Carnegie Mellon University Fall 2004

2. Structure of the course First 2 weeks: Term structure theory First 2 weeks: Term structure theory  Use Professor Shreve’s notes, Chapter 10 Remaining 5 weeks: Term structure practice Remaining 5 weeks: Term structure practice  No text; just readings and lecture notes  At the beginning we’ll refer to last semester’s text  Students (in groups of size 1 or 2) will implement HJM models

3. Grades There will be regular homework assignments. There will be regular homework assignments. There will be a final examination. There will be a final examination. The homework assignments (together) will count about as much as the final. The homework assignments (together) will count about as much as the final.

4. Miscellaneous items: I plan to give lectures in NYC on September 13 and on October 25 I plan to give lectures in NYC on September 13 and on October 25 We need to agree on meeting times with the TA (Sean). We need to agree on meeting times with the TA (Sean).

5. Recall some fundamental ideas: In a market with one or more securities trading, we always require the existence of an equivalent martingale measure (for which all discounted security prices are martingales). We do this to avoid arbitrage. In a market with one or more securities trading, we always require the existence of an equivalent martingale measure (for which all discounted security prices are martingales). We do this to avoid arbitrage. Remember that two measures are “equivalent” provided they have the same null sets. Remember that two measures are “equivalent” provided they have the same null sets.

6. In such a market If we also suppose that there is only one equivalent martingale measure then: If we also suppose that there is only one equivalent martingale measure then:  Prices (of these securities and of derivative securities) must be equal to their expected discounted values (or there will be arbitrage).

7. Term structure models have many traded securities A term structure model must describe the prices of a large number of securities: A term structure model must describe the prices of a large number of securities:  For each future date T, model must produce today’s (“date t”) price for the pure discount bond maturing at date T  Having lots of securities makes “no arbitrage” harder to achieve.

8. Recall (last spring) Section 6.5 of the text introduced terminology and notations for interest rate models. Section 6.5 of the text introduced terminology and notations for interest rate models. It also discussed some specific models in detail. It also discussed some specific models in detail. Here’s a quick review: Here’s a quick review:

9. Review 1 Section 6.5 introduced models of the form Section 6.5 introduced models of the form Where is a Brownian motion under a risk-neutral probability measure Where is a Brownian motion under a risk-neutral probability measure r is called the “spot rate” or “short rate”. r is called the “spot rate” or “short rate”.

10. Review 2 The Discount Factor is given by: The Discount Factor is given by: And the money-market price process  is given by  (t)=1/D(t) (which is the same as the above except for the “minus” sign in the exponent). And the money-market price process  is given by  (t)=1/D(t) (which is the same as the above except for the “minus” sign in the exponent).

11. Review 3 Let B(t,T) denote the date-t price of a pure- discount bond (having no default risk) maturing (and paying $1) at date T. Since discounted securities prices are martingales, D(t)B(t,T) must be a martingale, so we have: Let B(t,T) denote the date-t price of a pure- discount bond (having no default risk) maturing (and paying $1) at date T. Since discounted securities prices are martingales, D(t)B(t,T) must be a martingale, so we have: D(t)B(t,T)= {D(T)B(T,T)|F t }= {D(T)|F t }, where denotes the (conditional) expected value under the martingale measure. D(t)B(t,T)= {D(T)B(T,T)|F t }= {D(T)|F t }, where denotes the (conditional) expected value under the martingale measure.

12. Review 4 Divide both sides by D(t); this yields (when the smoke clears): Divide both sides by D(t); this yields (when the smoke clears):

13. Review 5 Two choices for  and  : dr(t)=(a(t)-b(t)r(t))dt+  (t) (t) dr(t)=(a(t)-b(t)r(t))dt+  (t) (t)  This is the Hull-White model dr(t)=(a - b r(t))dt+  (r(t)) 0.5 (t) dr(t)=(a - b r(t))dt+  (r(t)) 0.5 (t)  This is the Cox-Ingersoll-Ross model.

14. Review 6 Fix a date T, and consider the price of the maturity-T pure discount bond. For the above models, it turns out that we can find a function f(t,r) such that the bond prices are given by (for fixed T): Fix a date T, and consider the price of the maturity-T pure discount bond. For the above models, it turns out that we can find a function f(t,r) such that the bond prices are given by (for fixed T):  B(t,T)=f(t,r(t))  How? Hull-White case: we “guess” the form of f: Hull-White case: we “guess” the form of f:  f(t,r)=exp(-rC(t,T)-A(t,T)), which works. The yield is then Y(t,T)= -log(f(t,r))/(T-t) = The yield is then Y(t,T)= -log(f(t,r))/(T-t) = (rC(t,T)+A(t,T))/(T-t), an affine function of r

15. Review 7 The CIR model is similar. We get (for a, b, and  constant): The CIR model is similar. We get (for a, b, and  constant): f(t,r)=exp(-rC(t,T)-A(t,T)). The form of C(t,T) and of A(t,T) are a bit different. f(t,r)=exp(-rC(t,T)-A(t,T)). The form of C(t,T) and of A(t,T) are a bit different. This is again an affine function of r, so the yields are also affine functions. This is again an affine function of r, so the yields are also affine functions.

16. Limitations of these models Models generated by specifying the behavior of the “spot rate” or “short rate,” are called “spot-rate” or “short-rate” models. Models generated by specifying the behavior of the “spot rate” or “short rate,” are called “spot-rate” or “short-rate” models. These two are both “one-factor” models. We’ll see later that these models do not capture the “correlations” in the motion of the yield curve. These two are both “one-factor” models. We’ll see later that these models do not capture the “correlations” in the motion of the yield curve.

17. “Calibration” Recall that in general we need the martingale measure P* to be equivalent to the “true” measure P. What does this mean? Recall that in general we need the martingale measure P* to be equivalent to the “true” measure P. What does this mean?  The arguments in chapter 4 of the text (in particular equation 4.8.3) can be extended to show that the quadratic variation of a stochastic integral can be computed exactly (for a given sample path) by breaking the interval into small pieces, squaring the change in the process over each piece, adding these up, and taking the limit as the intervals get smaller.

18. In order to have an equivalent measure … Suppose we “know” what’s possible in the real world; i.e., we know the null-sets of the real world measure P. Suppose we “know” what’s possible in the real world; i.e., we know the null-sets of the real world measure P. Suppose we have a model and a martingale measure for this model. Suppose we have a model and a martingale measure for this model. What restriction do we have to impose so that we get to be equivalent to P? What restriction do we have to impose so that we get to be equivalent to P?

19. A necessary condition: If we define a set of paths by specifying restrictions * on the quadratic variation of these paths, then this set should have positive probability under (the model) if and only if it has positive probability under (the real-world probability) P. If we define a set of paths by specifying restrictions * on the quadratic variation of these paths, then this set should have positive probability under (the model) if and only if it has positive probability under (the real-world probability) P. This holds for “covariation” as well. This holds for “covariation” as well. (*) these must be “suitably measurable” (*) these must be “suitably measurable”

20. An example For the Hull-White model we have For the Hull-White model we have Hence the quadratic variation is Hence the quadratic variation is For each t>0 the left side is a random variable, but the right side is a number. Hence (under the model), for each t we know that [r,r](t) is, with probability 1, equal to that number. For each t>0 the left side is a random variable, but the right side is a number. Hence (under the model), for each t we know that [r,r](t) is, with probability 1, equal to that number.

21. This we can check! You can compute (in principle, and very nearly in practice) the quadratic variation of an observed sample path. And by the observation on the previous slide, it should be equal to the integal on the right-hand side. You can compute (in principle, and very nearly in practice) the quadratic variation of an observed sample path. And by the observation on the previous slide, it should be equal to the integal on the right-hand side. Now an optimistic view is that the observed quadratic variation tells you the function  (.). But what it really tells you is the function  (.) in the past, and we usually need to know it in the future. Now an optimistic view is that the observed quadratic variation tells you the function  (.). But what it really tells you is the function  (.) in the past, and we usually need to know it in the future. Recall: If an event occurs with probability 1, it should occur EVERY time you run the experiment Recall: If an event occurs with probability 1, it should occur EVERY time you run the experiment

22. And a stronger result for the CIR model: dR(t)=(a-bR(t))dt+  (R(t)) 0.5  dW * (t), so dR(t)=(a-bR(t))dt+  (R(t)) 0.5  dW * (t), so We re-write this as We re-write this as

23. This gives us a test of the model Note that the left hand side depends on t, while the right side doesn’t. Note that the left hand side depends on t, while the right side doesn’t. We can check to see if, for an observed sample path, the left side is constant. We can check to see if, for an observed sample path, the left side is constant. It is constant with probability 1 under the “model measure”. So, if the model is correct, it must constant be under the physical measure as well (with probability 1). It is constant with probability 1 under the “model measure”. So, if the model is correct, it must constant be under the physical measure as well (with probability 1).

24. Remarks (by Steve Shreve): “The issue of calibration of these models … is not discussed in this text.” “The issue of calibration of these models … is not discussed in this text.” “The primary shortcoming of one-factor models is that they cannot capture complicated yield curve behavior; they tend to produce parallel shifts in the yield curve, but not changes in its slope or curvature.” “The primary shortcoming of one-factor models is that they cannot capture complicated yield curve behavior; they tend to produce parallel shifts in the yield curve, but not changes in its slope or curvature.”

25. In this course we shall Discuss how to choose good models Discuss how to choose good models Study the estimation of parameters for models Study the estimation of parameters for models Introduce multi-factor models which can model changes of slope, curvature, etc. of yields Introduce multi-factor models which can model changes of slope, curvature, etc. of yields Develop numerical methods for valuing securities Develop numerical methods for valuing securities Discusse other “practical” issues (AAA subs, credit risk, …) Discusse other “practical” issues (AAA subs, credit risk, …)

26. We’ll consider two classes of models 1) Affine term structure models 1) Affine term structure models  Basic paper: Duffie and Kan, 1994 2) HJM term structure models 2) HJM term structure models  Basic Papers: Heath, Jarrow, Morton 1990- 1992 (All affine term structure models are in the HJM class of models, but most HJM models are not affine.) (All affine term structure models are in the HJM class of models, but most HJM models are not affine.) We’ll first introduce the HJM model framework We’ll first introduce the HJM model framework

27. Generalities about T.S. models Two types of securities Two types of securities  Bonds (One for each maturity T in the future)  Bank account Bank account: Bank account:  1 “share” corresponds to $1 deposited at date 0  Price, , of one “share” grows at interest rate:

28. Bond prices B(t,T) Bond prices B(t,T)  B(t,T) = price at date t of default-risk-free pure- discount-bond paying $1 at T  We always have B(T,T)=1  Prices will always be semi-martingales; for one-factor models driven by Brownian motion we have  Note: The values of  and  can depend on “other information” known at time t.  No arbitrage: we want B(t,T)/  (t) to be a martingale for each T.

29. Applying Ito’s lemma to B(t,T)/  (t) and using the fact that  has bounded variation: Applying Ito’s lemma to B(t,T)/  (t) and using the fact that  has bounded variation:

30. Under an equivalent martingale measure The coefficient of “dt” would have to be 0 The coefficient of “dt” would have to be 0 This means that  (t,T)=r(t). This means that  (t,T)=r(t). Or, if we wanted to change measures, we’d need Or, if we wanted to change measures, we’d need We’d need the same change of measure to work for every T! (strong restriction!) We’d need the same change of measure to work for every T! (strong restriction!)

31. Prices of bonds Assume that P is the equivalent mart. meas. Assume that P is the equivalent mart. meas. Thus for each T, is a martingale Thus for each T, is a martingale Moreover, B(T,T)=1 Moreover, B(T,T)=1 Hence Hence So: So:

32. Calibration to initial term structure: Required for term structure models At date 0 we can observe (in the market) prices for pure discount bonds at various maturities. At date 0 we can observe (in the market) prices for pure discount bonds at various maturities. A spot-rate model must be “calibrated” to reproduce this term structure (to avoid arbitrage between “market” and “model”). A spot-rate model must be “calibrated” to reproduce this term structure (to avoid arbitrage between “market” and “model”). It should also be calibrated to the prices of liquid instruments such as caps, floors and swaptions. It should also be calibrated to the prices of liquid instruments such as caps, floors and swaptions.

34. U.S. Treasury instruments

36. For spot rate models For the Hull-White model, the functions a and b must be chosen to: For the Hull-White model, the functions a and b must be chosen to:  Make initial prices for bonds of all maturities agree with market prices.  Match some “liquidly traded” option prices (caps, floors, swaptions)  This can be quite complicated

37. Full term structure (HJM) models Forward rates: Forward rates:  Suppose pdb’s trade; can go long or short  Then: can arrange at date t for a loan from date T 1 to date T 2 as follows:  Purchase one pdb maturing at date T 1  Sell B(t,T 1 )/B(t,T 2 ) shares of pdb maturing at T 2  Net cash flow at date t is 0  Net cash flow at date T 1 is $1  Net cash flow at date T 2 is -$B(t,T 1 )/B(t,T 2 )

38. What is the continuously compounded rate? What is the continuously compounded rate?  Rate, r 0, must satisfy  Solve to get  That’s the “constant rate” for loans from T 1 to T 2  The instantaneous forward rate at time T 1 is defined to be the limit of this r 0 as T 2 -> T 1, i.e.  Integration gives us:  It’s not surprising that r(t)=f(t,t) if there’s continuity

39. HJM one-factor model for term structure evolution Hence: Hence:

41. Ito’s lemma yields Ito’s lemma yields If we change measure, we make If we change measure, we make a martingale. a martingale.

42. Substituting this into the previous equation we see that Substituting this into the previous equation we see that As observed earlier, the coefficient of “dt” must be r(t), so we must have As observed earlier, the coefficient of “dt” must be r(t), so we must have

43. Remarks on HJM The right side of the last equation apparently depends on both t and T, while the left side doesn’t. In order for there to be an equivalent martingale measure the right hand side must not depend on T. The right side of the last equation apparently depends on both t and T, while the left side doesn’t. In order for there to be an equivalent martingale measure the right hand side must not depend on T. If P happened to be a martingale measure we’d have found  =0; i.e., If P happened to be a martingale measure we’d have found  =0; i.e., Differentiating w.r.t. T we get Differentiating w.r.t. T we get

44. Remarks continued Under the martingale measure we have Under the martingale measure we have so the evolution (under the martingale measure, i.e., the measure we use for valuation) depends only on  and not on . so the evolution (under the martingale measure, i.e., the measure we use for valuation) depends only on  and not on .

45. Still more remarks … Remark: “It is customary in the literature to write rather than and. rather than Remark: “It is customary in the literature to write rather than and. rather than so that P is the symbol used for the risk- neutral measure and no reference is ever made to the market measure …” so that P is the symbol used for the risk- neutral measure and no reference is ever made to the market measure …”.