Multi-dimensional Term Structure Models

Transcription

1 Multi-dimensional Term Structure Models We will focus on the affine class. But first some motivation. A generic one-dimensional model for zero-coupon yields, y(t; τ), looks like this dy(t; τ) =... dt + σ y (t, τ)dw, where W is a one-dimensional Brownian motion (under some measure). Asset Pricing II, June

2 An aside: Can we use a Prop type argument to determine σ y in terms of v (aka σ P, the ZCB price volatility) or σ (aka σ f, the forward rate volatility)? (Yes; easy.) what... means under some, say, Q? (Yes; a bit of care is needed cuase of the shift to time to maturity. See Björk s section on Musiela parametrization.) Asset Pricing II, June

3 This means that cov t ( y(t, τ i ), y(t, τ j )) = σ y (t, τ i )σ y (t, τ i ) t + O( t 2 ), where y(t, τ i ) = y(t + t, τ i ) y(t, τ i ). So cov t ( y(t, τ i ) y(t, τ j ) vart ( y(t, τ i )) var t ( y(t, τ j )) 1 for t 0, or in words (instantaneous) yield(-change)s are (conditionally) perfectly correlated. Asset Pricing II, June

4 How does this look for US data? (τ i, τ j ) (Monthly data ( = 1/12) from 1952 to Diagonal elements are standard deviations, off-diagonals are correlations.) Off-diagonal elements do not look too much like 1 s. (Not a formal test, and you could ask if = 1/12 is small, but still...) Asset Pricing II, June

5 Factor Analysis A classical statistical discipline. Not a formal model, but healthy for data analysis. Ask: What is the effective rank of the covariance matrix? A sensible measure of this is how many eigenvalues are (close to) 0? And you can answer this question in terms explained variance (cumulative sum divided by total sum of eigenvalues). Here: up to eigenvector # fraction of variance explained and we d say something like 2-3 factors, probably. Asset Pricing II, June

6 Further, there is a good chance you ll hear the words level, slope and curvature and see graphs like this one US yield changes : Level, slope and curvature (?) eigenvector(coordinate) eigenvector 3 eigenvector 1 eigenvector maturity Asset Pricing II, June

7 Affine Factor Models An n-factor model is one where r(t) = R(X(t)), dx(t) = µ(x(t))dt + Σ(X(t))dW Q (t), where X is a stochastic process whose coordinates are referred to as factors (abstract so far), and R : R n R, µ : R n R n, and Σ : R n R n n are functions. We say that the model is affine if R, each coordinate in µ, and each coordinate in ΣΣ are affine functions (of some x R n ). (Note that in general [ΣΣ ] i,j [Σ i,j ] 2.) Asset Pricing II, June

8 Or with symbols R(x) = δ 0 + }{{} δ x 1 n µ(x) = }{{} K (}{{} θ x) n n n 1 Σ(x) = }{{} Σ S(x), }{{} n n n n where everything that looks constant is, and S(x) denotes a diagonal matrix whose ith diagonal element is [S(x)] i,i = α i + β }{{} i x, 1 n and the meaning of matrix- is then obvious. (As is some positivity restriction.) Asset Pricing II, June

9 This is the most common parametrization. Too easy if it were the only one! Clear that R and µ are affine. Not hard to convince yourself that (ΣS(x)Σ ) i,j is affine in x. Less clear that the form of the volatility isn t a restriction. Duffie & Kan show that it isn t really. And now for the zero coupon bond pricing. We have (the first equality: always, the second: here) P (t, T ) = E Q t (e R T t r(u)du ) = E Q t (e R T t (δ 0+ P n i=1 δ i X i (u))du ), Asset Pricing II, June

10 Put differently P (t, T ) exp( t r(u)du) is a Q-martingale. 0 And that can be only if the Q-drift (rate) of the ZCB price is r(t) = R(X(t)). Because r(t) = R(X(t)) and X is a time-homogeneous Markov process, P (t, T ) is of the form P (t, T ) = f(x(t), T t), for some function f. This is heading towards a restriction on the drift. Asset Pricing II, June

11 Affine ZCB Price Theorem (Duffie & Kan) In an affine model, ZCB prices are of the exponentially affine form P (t, T ) = exp(a(t t) B (T t)x(t)) where the function B : R R n solves the system of ODEs db dτ = δ K B(τ) 1 2 n (Σ B(τ)) 2 i β i i=1 and the function A : R R solves the ODE da dτ = δ 0 θ K B(τ) n (Σ B(τ)) 2 i α i. i=1 Asset Pricing II, June

12 Proof: Use Martingality. Multi-dimensional Ito; careful w/ vectors and matrices. The matching principle: If a + b x = 0 for all x (in some open set), then a = 0 and b = 0. BLACKBOARD Asset Pricing II, June

13 Remarks For B we have a coupled system of n ODEs. Much better than PDEs w/ multidimensional state variables. Sometimes we can solve in closed form, sometimes we can t. Not hard numerically ( pedestrian Euler, or Runge/Kutta). Two theoretical issues (Duffie & Kan) The converse. (Proposition p. 386.) Well-definedness (admissibility) when some β s are non-0. (Theorem p. 388.) Two examples: Gaussian models and sum of independent CIRs. Answer some questions, raise others Dai & Singleton. Asset Pricing II, June

14 A 2D Gaussian model (akin to Brigo & Mercurio Section 4.2) Suppose where ( κ1 0 dx = 0 κ 2 r(t) = δ 0 + X 1 (t) + X 2 (t) ) ( X1 (t) X 2 (t) ) ( σ1 0 dt + ρσ 2 1 ρ2 σ 2 ) dw Q In this case we find (BLACKBOARD) that B i (τ) = 1 e κ iτ and that A is a rather lengthy expression that we may or may not need. κ i Asset Pricing II, June

15 The same short rate level may give different yield curves, ie. P (t, T ) f(r(t), T t). log(ptau)/maturities maturities Asset Pricing II, June

16 Quick & dirty estimation: Calibrate to yield (difference) covariance matrix. Note that with B(τ, κ) = 1 τ (B(τ, κ 1), B(τ, κ 2 )) we have cov( y(t, τ i )), y(t, τ j )) t ( B σ 2 (τ i, κ) 1 σ 1 σ 2 ρ σ 1 σ 2 ρ σ2 2 ) B(τ j, κ) With a guess of the 5 parameters (forget about δ 0 for a moment) we get a theoretical (approximate, unconditional instantaneous) covariance matrix. We may try to estimate parameters by getting as close as possible to the empirical covariance matrix. Asset Pricing II, June

17 With yields of 7 maturities, the empirical covariance matrix has effectively (6 7)/2 = 21 entries. A simple least squares fit to 50 years of US data gives (R-code and data on homepage) Parameter κ 1 κ 2 σ 1 σ 2 ρ Estimated value Asset Pricing II, June

18 And that gives a picture like this for the standard deviations (calibrate to covariance, show standard deviations and correlations in graphs) sqrt(dt) scaled standard deviation of dy(maturity) maturity Asset Pricing II, June

19 And for the correlations: correlation w/ maturity 0.25 maturity correlation correlation w/ maturity 0.5 maturity correlation w/ maturity 1 maturity correlation correlation w/ maturity 2 maturity correlation w/ maturity 5 maturity correlation correlation w/ maturity 10 maturity Asset Pricing II, June correlation correlation correlation

20 Observations: Not the worst fit, you ll ever see. We need a high negative correlation between factors to make yields as uncorrelated as they are empirically. We can use δ 0 to calibrate to today s observed yield curve as earlier. Asset Pricing II, June

21 More observations: Parameters aren t really identified; just switch indices. Proper inference: Do maximum likelihood; it s just a Gaussian first-order VAR. Problem: Factors are not observable. Solution: Invert to express in terms of yields. Problem: Parameter dependent transform Jacobian. If we want to use all observed yields, we get some kind of filtering problem. Models are affine in data not in parameters. The whole P/Q or parameter risk adjustment question pops up again with a vengeance! Asset Pricing II, June

22 In the empirical covariance matrix we averaged out any conditional information. Consistent w/ a Gaussian model; not necessarily w/ data. (recent) Litterature (links on homepage) Dai & Singleton Duffee Cheridito, Filipovic & Kimmel and much, much more... Asset Pricing II, June

23 Messing with your head, I (Ar models in the language of Dai & Singleton) Suppose that somebody (messr s Hull & White for instance) comes along with a model like this: dr(t) = (θ + u(t) ar(t))dt + σ 1 dw 1 where where dw 1 dw 2 = ρdt. du(t) = bu(t)dt + σ 2 dw 2 Looks sexy : It s Vasicek with stochastic mean reversion level and correlation. And they can even find ZCB prices. It is, however, just the Gaussian model from above in disguise! Definitely BLACKBOARD Asset Pricing II, June

24 Messing with your head, II That β s are all 0 is because we want a Gaussian model. Fair enough. But: Why is δ = (1, 1)? Why is θ = 0? Why is K diagonal? Why is Σ 1,2 = 0? Why is α = (1, 1)? Are they real restrictions or just needed for identification? Asset Pricing II, June

25 The variable X i = δ i X i has same κ i, and just scaled volatility. The variable X i = X i θ i is a Gaussian process that mean reverts to 0. Shift absorbed by δ 0. (Aside: CIR + constant isn t CIR.) If K can be diagonalized (note: K is not symmetric), say by M ie. then with Y = MX we have MKM 1 = D, dy = d(mx) = MKXdt + MΣdW = DMY dt + MΣdW and we re good. = DY dt + ΣdW, Asset Pricing II, June

26 At least K can be made lower triangular, by defining X i s in a Gaussian elimination way. We get B ODEs with a simple recursive structure. (To avoid degenerate cases, diagonal elements are non-0.) Volatility terms enter only through the symmetric matrix ΣΣ, so 3 free parameters are enough. Given some Σ, we can diagonalize ΣΣ by M and then use M to rotate W and get diagonal volatility. Here we ve actually proven Dai & Singleton s characterization (section B.1) of A 0 (m)-models. (They use Σ = I, rather than δ = (1,..., 1).) Asset Pricing II, June

27 Independent CIRs Suppose r(t) = δ 1 X 1 (t) + δ 1 X 2 (t) where the X s are independent CIR-type processes dx i (t) = κ i (θ i X i (t))dt + X i (t)dw i (t) Fits the general framework. But the ZCB price formula immediately reduces to a product of CIR-formulas. BLACKBOARD Asset Pricing II, June

28 Can we make correlated CIRs just saying dw 1 dw 2 = ρdt? Yes, but we can t solve for ZCB prices (with the ODEs here, at least), because it s not an affine model: [ΣΣ ] = ρ X 1 X2 a + b X Asset Pricing II, June

29 Making Independent CIRs Look Good Volatility is clearly stochastic: 50 years of daily changes of US 3M interest rates interest rate change year Asset Pricing II, June

30 The skeptic: Really? 50 years of daily changes us US 3M interest rates interest rate change year 50 years of US 3M interest rates interest rate year 50 years of daily changes of log s of US 3M interest rates diff(log(data[, 4])) dates[2:nobs] Asset Pricing II, June

31 Such petty details aside: We can rewrite as Longstaff/Schwartz stochastic volatility. BLACKBOARD We get a richer (state-variable dependent) conditional variance, but loose on correlation. Asset Pricing II, June