Finite dimensional realizations of HJM models

Finite dimensional realizations of HJM models Tomas Björk Stockholm School of Economics Camilla Landén KTH, Stockholm Lars Svensson KTH, Stockholm UTS, December 2008, 1

Definitions: p t (x) : Price, at t of zero coupon bond maturing at t + x, r t (x) : Forward rate, contracted at t, maturing at t + x R t : Short rate. r t (x) = log p t(x) x p t (x) = e x 0 r t(s)ds R t = r t (0). 2

Heath-Jarrow-Morton-Musiela Idea: Model the dynamics for the entire forward rate curve. The yield curve itself (rather than the short rate R) is the explanatory variable. Model forward rates. Use observed forward rate curve as initial condition. Q-dynamics: dr t (x) = α t (x)dt + σ t (x)dw t, r 0 (x) = r0 (x), x W : d-dimensional Wiener process One SDE for every fixed x. 3

Theorem: (HJMM drift Condition) The following relations must hold, under a martingale measure Q. α t (x) = x x r t(x) + σ t (x) σ t(s)ds. 0 Moral: Volatility can be specified freely. The forward rate drift term is then uniquely determined. 4

The Interest Rate Model r t = r t ( ), σ t (x) = σ(r t, x) Heath-Jarrow-Morton-Musiela equation: dr t = µ 0 (r t )dt + σ(r t )dw t µ 0 (r t, x) = x x r t(x) + σ(r t, x) σ(r t, s)ds 0 The HJMM equation is an infinite dimensional SDE evolving in the space H of forward rate curves. 5

Sometimes you are lucky! Example: σ(r, x) = σe ax In this case the HJMM equation has a finite dimensional state space realization. We have in fact: r t (x) = B(t, x)z t A(t, x) where Z solves the one-dimensional SDE dz t = {Φ(t) az t } dt + σdw t Furthermore the state process Z can be identified with the short rate R = r(0). (A, B and Φ are deterministic functions) 6

A Hilbert Space Definition: For each (α, β) R 2, the space H α,β is defined by where where H α,β = {f C [0, ); f < } f 2 = β n n=0 0 [ f (n) (x) ] 2 e αx dx f (n) (x) = dn f dt n (x). We equip H with the inner product (f, g) = β n n=0 0 f (n) (x)g (n) (x)e αx dx 7

Properties of H Proposition: The following hold. The linear operator is bounded on H F = x H is complete, i.e. it is a Hilbert space. The elements in H are real analytic functions on R (not only on R + ). NB: Filipovic and Teichmann! 8

Stratonovich Integrals Definition The Stratonovich integral t 0 X s dy s is defined as t 0 X s dy s = t 0 X sdy s + 1 2 X, Y t X, Y t = t 0 dx sdy s, Proposition: For any smooth F we have df (t, Y t ) = F t dt + F y dy t 9

Stratonovich Form of HJMM dr t = µ(r t )dt + σ(r t ) dw t where µ(r t ) = µ 0 (r t ) 1 2 d σ, W dt Main Point: Using the Stratonovich differential we have no Itô second order term. Thus we can treat the SDE above as the ODE dr t dt = µ(r t) + σ(r t ) v t where v t = white noise. 10

Natural Questions What do the forward rate curves look like? What is the support set of the HJMM equation? When is a given model (e.g. Hull-White) consistent with a given family (e.g. Nelson-Siegel) of forward rate curves? When is the short rate Markov? When is a finite set of benchmark forward rates Markov? When does the interest rate model admit a realization in terms of a finite dimensional factor model? If there exists an FDR how can you construct a concrete realization? 11

Finite Dimensional Realizations Main Problem: When does a given interest rate model possess a finite dimensional realisation, i.e. when can we write r as z t = η(z t )dt + δ(z t ) dw (t), r t (x) = G(z t, x), where z is a finite-dimensional diffusion, and or alternatively G : R d R + R G : R d H H = the space of forward rate curves 12

Examples: σ(r, x) = e ax, σ(r, x) = xe ax, σ(r, x) = e x2, σ(r, x) = log σ(r, x) = ( 1 1 + x 2 ) 0 e s r(s)ds x 2 e ax., Which of these admit a finite dimensional realisation? 13

Earlier literature Cheyette (1996) Bhar & Chiarella (1997) Chiarella & Kwon (1998) Inui & Kijima (1998) Ritchken & Sankarasubramanian (1995) Carverhill (1994) Eberlein & Raible (1999) Jeffrey (1995) All these papers present sufficient conditions for existence of an FDR. 14

Present paper We would like to obtain: Necessary and sufficient conditions. struc- A better understanding of the deep ture of the FDR problem. A general theory of FDR for arbitrary infinite dimensional SDEs. We attack the general problem by viewing it as a geometrical problem. 15

Invariant Manifolds Def: Consider an interest rate model dr t = µ(r t )dt + σ(r t ) dw t on the space H of forward rate curves. A manifold (surface) G H is an invariant manifold if P -a.s. for all t > 0 r 0 G r t G 16

Main Insight There exists a finite dimensional realization. iff There exists a finite dimensional invariant manifold. 17

Characterizing Invariant Manifolds Proposition: (Björk-Christensen) Consider an interest rate model on Stratonovich form dr t = µ(r t )dt + σ(r t ) dw t A manifold G is invariant under r if and only if µ(r) T G (r), σ(r) T G (r), at all points of G. Here T G (r) is the tangent space of G at the point r G. 18

Main Problem Given: An interest rate model on Stratonovich form dr t = µ(r t )dt + σ(r t ) dw t An inital forward rate curve r 0 : x r 0 (x) 19

Question: When does there exist a finite dimensional manifold G, such that and r 0 G µ(r) T G (r), σ(r) T G (r), A manifold satisfying these conditions is called a tangential manifold. 20

Abstract Problem On the Hilbert space H, we are given two vector fields f 1 (r) and f 2 (r). We are also given a point r 0 H. Problem: When does there exist a finite dimensional manifold G H such that We have the inclusion r 0 G For all points r G we have the relations f 1 (r) T G (r), f 2 (r) T G (r) We call such a G an tangential manifold. 21

Easier Problem On the space H, we are given one vector field f 1 (r). We are also given a point r 0 H. Problem: When does there exist a finite dimensional manifold G H such that We have the inclusion r 0 G We have the relation f 1 (r) T G (r) Answer to Easy Problem: ALWAYS! 22

Proof: Solve the ODE dr t dt = f 1(r t ) with initial point r 0. Denote the solution at time t by e f 1t r 0 Then the integral curve { e f 1t r 0 ; t R } solves the problem, i.e. G = { e f 1t r 0 ; t R } 23

Furthermore, the mapping where G : R G G(t) = e f 1t r 0 parametrizes G. We have G = Im[G] Thus we even have a one dimensional coordinate system for G, given by ϕ : G R ϕ = G 1 24

Back to original problem: We are given two vector fields f 1 (r) and f 2 (r) and a point r 0 H. Naive Conjecture: There exists a two-dimensional tangential manifold, which is parametrized by the mapping where G : R 2 X G(s, t) = e f 2s e f 1t r 0 Generally False! Argument: If there exists a 2-dimensional manifold, then it should also be parametrized by H(s, t) = e f 1s e f 2t r 0 Moral: We need some commutativity. 25

Lie Brackets Given two vector fields f 1 (r) and f 2 (r), their Lie bracket [f 1, f 2 ] is a vector field defined by [f 1, f 2 ] = (Df 2 )f 1 (Df 1 )f 2 where D is the Frechet derivative (Jacobian). Fact: e f 1h e f 2h r 0 e f 2h e f 1h r 0 [f 1, f 2 ]h 2 Fact: If G is tangential to f 1 and f 2, then it is also tangential to [f 1, f 2 ]. 26

Definition: Given vector fields f 1 (r),..., f n (r), the Lie algebra {f 1 (r),..., f n (r)} LA is the smallest linear space of vector fields, containing f 1 (r),..., f n (r), which is closed under the Lie bracket. Conjecture: f 1 (r),..., f n (r) generates a finite dimensional tangential manifold iff dim {f 1 (r),..., f n (r)} LA < 27

Frobenius Theorem: Given n independent vector fields f 1,..., f n. There will exist an n-dimensional tangential manifold iff span {f 1,..., f n } is closed under the Lie-bracket. Corollary: Given n vector fields f 1,..., f n. Then there exists exists a finite dimensional tangential manifold iff the Lie-algebra {f 1,..., f n } LA generated by f 1,..., f n has finite dimension at each point. The dimension of the manifold equals the dimension of the Lie-algebra. 28

Proposition: Suppose that the vector fields f 1,..., f n are independent and closed under the Lie bracket. Fix a point r 0 X. Then the tangential manifold is parametrized by where G : R n G G(t 1,..., t n ) = e f nt n... e f 2t 2 e f 1t 1 r 0 29

Main result Given any fixed initial forward rate curve r 0, there exists a finite dimensional invariant manifold G with r 0 G if and only if the Lie-algebra is finite dimensional. L = {µ, σ} LA Given any fixed initial forward rate curve r 0, there exists a finite dimensional realization if and only if the Lie-algebra L = {µ, σ} LA is finite dimensional. The dimension of the realization equals dim {µ, σ} LA. 30

Deterministic Volatility σ(r, x) = σ(x) Consider a deterministic volatility function σ(x). Then the Ito and Stratonovich formulations are the same: where dr = {Fr + S} dt + σdw F = x x, S(x) = σ(x) σ(s)ds. 0 The Lie algebra L is generated by the two vector fields µ(r) = Fr + S, σ(r) = σ 31

Proposition: There exists an FDR iff σ is quasi exponential, i.e. of the form σ(x) = n i=1 where p i is a polynomial. p i (x)e α ix 32

Constant Direction Volatility σ(r, x) = ϕ(r)λ(x) Theorem Assume that ϕ (r)(λ, λ) 0. Then the model admits a finite dimensional realization if and only if λ is quasi-exponential. The scalar field ϕ(r) can be arbitrary. Note: The degenerate case ϕ(r) (λ, λ) 0 corresponds to CIR. 33

Short Rate Realizations Question: When is a given forward rate model realized by a short rate model? r(t, x) = G(t, R t, x) dr t = a(t, R t )dt + b(t, R t ) dw Answer: There must exist a 2-dimensional realization. (With the short rate R and running time t as states). Proposition: The model is a short rate model only if dim {µ, σ} LA 2 Theorem: The model is a generic short rate model if and only if [µ, σ] //σ 34

All short rate models are affine Theorem: (Jeffrey) Assume that the forward rate volatitliy is of the form σ(r t, x) Then the model is a generic short rate model if and only if σ is of the form σ(r, x) = c (Ho-Lee) σ(r, x) = ce ax (Hull-White) σ(r, x) = λ(x) ar + b (CIR) (λ solves a certain Ricatti equation) Slogan: Ho-Lee, Hull-White and CIR are the only generic short rate models. 35

Constructing an FDR Problem: Suppose that there actually exists an FDR, i.e. that dim {µ, σ} LA <. How do you construct a realization? Good news: There exists a general and easy theory for this, including a concrete algorithm. See Björk & Landen (2001). 36

Example: Deterministic Direction Volatility Model: σ i (r, x) = ϕ(r)λ(x). Minimal Realization: dz 0 = dt, dz 1 0 = [c 0Z 1 n + γϕ 2 (G(Z))]dt + ϕ(g(z))dw t, dzi 1 = (c i Zn 1 + Zi 1 1 )dt, i = 1,..., n, dz 2 0 = [d 0Z 2 q + ϕ 2 (G(Z))]dt, dzj 2 = (d j Zq 2 + Zj 1 2 )dt, j = 1,..., q. 37

Stochastic Volatility Forward rate equation: dr t = µ 0 (r t, y t )dt + σ(r t, y t )dw t, dy t = a(y t )dt + b(y t ) dv t Here W and V are independent Wiener and y is a finite dimensional diffusion living on R k. µ 0 = x r t(x) + σ(r t, y t, x) x 0 σ(r t, y t, s)ds Problem: When does there exist an FDR? Good news: This can be solved completely using the Lie algebra approach. See Björk- Landen-Svensson (2002). 38

Point Process Extensions Including a driving point process leads to hard problems. More precisely The equivalence between existence of an FDR and existence of an invariant manifold still holds. The characterization of an invariant manifold as a tangential manifold is no longer true. This is because a point process act globally whereas a Wiener process act locally, thereby allowing differential calculus. Including a driving point process requires, for a general theory, completely different arguments. The picture is very unclear. 39

Point Processes: Special Cases Chiarella & Nikitopoulos Sklibosios (2003) Sufficient Conditions Tappe (2007) Necessary Conditions using Lie algebra techinques. Elhouar (2008) Wiener driven models with point process driven volatilities using Lie algebra techinques. 40

Björk, T. & Christensen, B.J. (1999) Interest rate dynamics and consistent forward rate curves. Mathematical Finance, 9, No. 4, 323-348. Björk, T. & Gombani A. (1997) Minimal realization of interest rate models. Finance and Stochastics, 3, No. 4, 413-432. Björk, T. & Svensson, L. (2001) On the existence of finite dimensional nonlinear realizations for nonlinear forward rate rate models. Mathematical Finance, 11, 205-243. Björk, T. (2001) A geometric view of interest rate theory. In Option Pricing, Interst Rates and Risk Mangement. Cambridge University Press. Björk, T. & Landen C. (2001) On the construction of finite dimensional nonlinear realizations for nonlinear forward rate models. Finance and Stochastics. Björk, T. & Landen C. & Svenssom, L. (2002) On finite Markovian realizations for stochastic volatility forward rate models. Proc. Royal Soc. Filipovic, D. & Teichmann, J. (2001) Finite dimensional realizations for stochastic equations in the HJM framework. Journal of Functional Analysis. 41

Earlier literature Cheyette, O. (1996) Markov representation of the Heath- Jarrow-Morton model. Working paper. BARRA Inc, Berkeley. Bhar, R. & Chiarella, C. (1997) Transformation of Heath-Jarrow-Morton models to markovian systems. European Journal of Finance, 3, No. 1, 1-26. Chiarella, C & Kwon, K. (1998) Forward rate dependent Markovian transformations of the Heath-Jarrow- Morton term structure model. Finance and Stochastics, 5, 236-257. Inui, K. & Kijima, M. (1998) A markovian framework in multi-factor Heath-Jarrow-Morton models. JFQA 333 no. 3, 423-440. Ritchken, P. & Sankarasubramanian, L. (1995) Volatility structures of forward rates and the dynamics of the term structure. Mathematical Finance, 5, no. 1, 55-72. Carverhill, A. (1994) When is the spot rate Markovian? Mathematical Finance,, 305-312. Eberlein, E. & Raible, S. (1999) Term structure models driven by general Levy processes. Mathematical Finance, 9, No 1, 31-53. Jeffrey, A. (1995) Single factor Heath-Jarrow-Morton term structure models based on Markovian spot interest rates. JFQA 30 no.4, 619-642. 42