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Advanced Term Structure

Advanced Term Structure. Carnegie Mellon University Fall 2004. Structure of the course. First 2 weeks: Term structure theory Use Professor Shreve’s notes, Chapter 10 Remaining 5 weeks: Term structure practice No text; just readings and lecture notes

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Advanced Term Structure

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  1. Advanced Term Structure Carnegie Mellon University Fall 2004

  2. Structure of the course • First 2 weeks: Term structure theory • Use Professor Shreve’s notes, Chapter 10 • 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 a final examination. • 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 • 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. • 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: • 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: • 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. • It also discussed some specific models in detail. • Here’s a quick review:

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

  10. Review 2 • The Discount Factor is given by: • And the money-market price process b is given by b(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: • D(t)B(t,T)= {D(T)B(T,T)|Ft}= {D(T)|Ft}, 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):

  13. Review 5 Two choices for b and g: • dr(t)=(a(t)-b(t)r(t))dt+s(t) (t) • This is the Hull-White model • dr(t)=(a - b r(t))dt+s(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): • B(t,T)=f(t,r(t)) • How? • 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) = (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 s 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. • 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. • 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? • 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 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?

  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. • This holds for “covariation” as well. • (*) these must be “suitably measurable”

  20. An example • For the Hull-White model we have • 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.

  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. • Now an optimistic view is that the observed quadratic variation tells you the function s(.). But what it really tells you is the function s(.) 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

  22. And a stronger result for the CIR model: • dR(t)=(a-bR(t))dt+(R(t))0.5 sdW*(t), so • 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. • 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).

  24. Remarks (by Steve Shreve): • “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.”

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

  26. We’ll consider two classes of models • 1) Affine term structure models • Basic paper: Duffie and Kan, 1994 • 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.) • We’ll first introduce the HJM model framework

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

  28. 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 m and r can depend on “other information” known at time t. • No arbitrage: we want B(t,T)/b(t) to be a martingale for each T.

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

  30. Under an equivalent martingale measure • The coefficient of “dt” would have to be 0 • This means that m(t,T)=r(t). • 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!)

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

  33. U.S. Treasury instruments

  34. For spot rate models • 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

  35. Full term structure (HJM) models • Forward rates: • Suppose pdb’s trade; can go long or short • Then: can arrange at date t for a loan from date T1 to date T2 as follows: • Purchase one pdb maturing at date T1 • Sell B(t,T1)/B(t,T2) shares of pdb maturing at T2 • Net cash flow at date t is 0 • Net cash flow at date T1 is $1 • Net cash flow at date T2 is -$B(t,T1)/B(t,T2)

  36. What is the continuously compounded rate? • Rate, r0, must satisfy • Solve to get • That’s the “constant rate” for loans from T1 to T2 • The instantaneous forward rate at time T1 is defined to be the limit of this r0 as T2-> T1, i.e. • Integration gives us: • It’s not surprising that r(t)=f(t,t) if there’s continuity

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

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

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

  40. 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. • If P happened to be a martingale measure we’d have found =0; i.e., Differentiating w.r.t. T we get

  41. Remarks continued • Under the martingale measure we have so the evolution (under the martingale measure, i.e., the measure we use for valuation) depends only on s and not on a.

  42. Still more remarks … • 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 …” .

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