Affine term structures for interest rate models

Transcription

1 Stefan Tappe Albert Ludwig University of Freiburg, Germany UNSW-Macquarie WORKSHOP Risk: modelling, optimization and inference Sydney, December 7th, 2017

2 Introduction Affine processes in finance: R = a + d b i Y i i=1 [ ( P(t, T ) = E exp T t and ) ] R s ds F t. Affine realizations for HJM interest rate models: r = ψ + d Y i λ i. i=1 Connection between these two concepts.

3 Related literature Affine processes in finance (among others): Duffie & Kan (1996). Filipović (2001). Duffie, Filipović & Schachermayer (2003). Filipović & Mayerhofer (2009). Affine realizations for HJM models (among others): Björk & Svensson (2001). Björk & Landén (2002). Filipović & Teichmann (2003, 2004). Tappe (2010, 2012). This talk is mainly based on these two papers: Tappe (2016) and Tappe (2017).

4 Multi factor models of bond prices Let Y be a R d -valued Markov process. We define the short rate as R = a + We define the bond prices as d b i Y i. i=1 [ ( T ) ] P(t, T ) = E exp R s ds F t. t

5 HJM type models of bond prices Let the forward rates be given by a HJM model t t f (t, T ) = f (0, T ) + α HJM (s, T )ds + σ(s, T )dw s. 0 0 Here the drift is given by the HJM drift condition α HJM (t, T ) = n k=1 We define the bond prices as T σ k (t, T ) σ k (t, s)ds. t ( T ) P(t, T ) = exp f (t, s)ds. t

6 Relation between these two approaches Suppose that R F = R HJM, where Rt HJM Then we have P F (t, T ) = P HJM (t, T ). Indeed, for each T R + we have = f (t, t). P F (, T ), PHJM (, T ) M loc, B B and these two processes have the same terminal value at T. Here the process B denotes the bank account ( t ) B t = exp R s ds, t R +. 0

7 Affine factor models Recall that the short rate is defined as R = a + d b i Y i. i=1 Moreover, the bond prices are given by [ ( T ) ] P(t, T ) = E exp R s ds F t. t The model has an affine term structure if ( P(t, T ) = exp A(T t) d i=1 ) B i (T t)yt i. Note that A (0) = a and B i (0) = b i for i = 1,..., d.

8 The HJMM equation Recall the Heath-Jarrow-Morton (HJM) model t t f (t, T ) = f (0, T ) + α HJM (s, T )ds + σ(s, T )dw s. 0 0 Musiela parametrization r t (x) := f (t, t + x). Consider the Heath-Jarrow-Morton-Musiela (HJMM) equation { drt = ( d dx r t + α HJM (r t ) ) dt + σ(r t )dw t (1) r 0 = h 0. This is a stochastic partial differential equation (SPDE). The state space H consists of functions h : R + R.

9 Affine realizations Let I H be a subset, and let V H be a subspace V = λ 1,..., λ d. Affine realization: For each h 0 I there are a curve ψ : R + H and a R d -valued Markov Y process such that r (h 0) = ψ + d Y i λ i. i=1 This means that the foliation (M t ) t R+ given by M t = ψ(t) + V, t R + is invariant for the HJMM equation (1). Affine state process: Y is a R m + R d m -valued affine process.

10 Path of the solution process on an invariant foliation

11 The singular set We assume that H = G V, and accordingly I = I V. For h 0 = g 0 + v 0 I the following are equivalent: 1 We have ψ(t) = g 0 for all t R +. 2 The foliation (M t ) t R+ only consists of a single leaf. 3 The affine space {g 0 } V is invariant for (1). 4 We have β(g 0 ) V, where β = d dx + α HJM. Hence, we define the singular set S D( d dx ) as S := β 1 (V ).

12 From HJMM to factor models Suppose the HJMM equation (1) has an affine realization. Let h 0 I S be arbitrary; it is of the form h 0 = g 0 + d c i λ i. i=1 Let l L(H, R d ) be such that l V = κ λ and G = ker(l). We obtain the R d -valued state process Y with dy t = l(β(g 0 + d j=1 Y t j λ j ))dt +l(σ(g 0 + d j=1 Y t j λ j ))dw t Y 0 = c.

13 The corresponding factor model We have an affine term structure ( P(t, T ) = exp A(T t) The mappings A and B are given by A(x) = B i (x) = x 0 x 0 g 0 (η)dη, d i=1 ) B i (T t)yt i. λ i (η)dη for i = 1,..., d.

14 From affine factor models to HJMM Now consider an affine term structure ( d ) P(t, T ) = exp A(T t) B i (T t)yt i. i=1 Here Y is a R d -valued Markov process satisfying { dyt = b(y t )dt + ρ(y t )dw t Y 0 = c. We define the quantities λ i = B i for i = 1,..., d, V = λ 1,..., λ d. Let l L(H, R d ) be such that: l V = κ λ. H = G V, where G = ker(l).

15 The corresponding HJMM model Then we have the HJMM model { drt = ( d dx r t + α HJM (r t ) ) dt + σ(r t )dw t r 0 = h 0. The initial curve h 0 and the volatility σ are given by h 0 = A + d c i λ i, i=1 σ(h) = ρ(l(h)) λ, h H.

16 One factor models Let R be a Markov process with values in R + or R. Set [ ( T P(t, T ) = E exp R s ds) ] t F t. Filipović (2001): The following statements are equivalent: 1 We have an affine term structure ( ) P(t, T ) = exp A(T t) B(T t)r t. 2 R is an affine process.

17 One-dimensional realizations for the HJMM equation Consider the HJMM equation { drt = ( d dx r t + α HJM (r t ) ) dt + σ(r t )dw t r 0 = h 0. Tappe (2016): The following statements are equivalent: 1 The HJMM equation has a one-dimensional affine realization. 2 The HJMM equation has a one-dimensional affine realization with affine state processes. In this case, we can choose the short rate R as state process and obtain the following models: Hull-White extension of the Vasi cek model σ(h) = c e γx. Ho-Lee model σ(h) = c. Hull-White extension of the CIR model σ(h) = ρ l(h).

18 The Cox-Ingersoll-Ross model For b, c R + consider the CIR short rate model { drt = (b γr t )dt + ρ R t dw t R 0 = c. We have an affine term structure ( ) P(t, T ) = exp A(T t) B(T t)r t. The functions A and B are given by A(x) = 2b ( ) ρ 2 ln 2γ exp((β + γ)x/2), (β + γ)(exp(βx) 1) + 2β 2(exp(βx) 1) B(x) = (β + γ)(exp(βx) 1) + 2β, where β := γ 2 + 2ρ 2. Note that A = b B.

19 The corresponding HJMM equation Then we have the HJMM equation { drt = ( d dx r t + α HJM (r t ) ) dt + σ(r t )dw t r 0 = h 0. We set λ := B and V := λ +. Let l L(H, R d ) be given by l(h) = h(0). Noting that Λ = B, we obtain h 0 = bλ + cλ, σ(h) = h(0) λ, h H.

20 Plots for ρ = 1 and γ =

21 The Cox-Ingersoll-Ross model Let σ : H H be of the form σ(h) = ρ h(0) λ, where: 1 ρ > 0 is a constant. 2 λ H satisfies the Riccati ODE d dx λ + ρ2 λλ + γλ = 0 for some γ R. The primitive Λ H is given by Λ(x) = 2(exp(x γ 2 + 2ρ 2 ) 1) ( γ 2 + 2ρ 2 + γ)(exp(x γ 2 + 2ρ 2 ) 1) + 2 γ 2 + 2ρ 2. Moreover, the function λ H is given by λ = 1 ρ2 2 Λ2 γλ.

22 Affine realization with affine state processes Affine realization generated by λ + and I = {h D(d/dx) : h(0) 0 and h (0) + γh(0) > 0}. The singular set is given by I S = Λ, λ +. Let (b, c) R + R + be the vector such that h 0 = bλ + cλ I S. Then we have r = bλ + Rλ, where { drt = (b γr t )dt + ρ R t dw t R 0 = c.

23 Multi dimensional models Let Y be a R d -valued Markov process, and define R = a + We define the bond prices as d b i Y i. i=1 [ ( T P(t, T ) = E exp R s ds) ] t F t. Assume we have an affine term structure ( P(t, T ) = exp A(T t) d i=1 ) B i (T t)yt i. Filipović (2001): Y does not need to be an affine process.

24 Counter example for HJMM models Consider the HJMM equation { drt = ( d dx r t + α HJM (r t ) ) dt + σ(r t )dw t r 0 = h 0. Assume that σ(h) = Φ(h)λ, where: 1 Φ : H R is Lipschitz continuous. 2 λ is given by λ(x) = e γx, x R +. Affine realization generated by V = λ, Λ = λ, λ 2. For h 0 = c1 we obtain the process Y satisfying the SDE ( a dy t = 1 Yt 1 + a 2 Yt 2 ) Φ 2 (c1 + Yt 1 λ + Yt 2 λλ) + b 1 Yt 1 + b 2 Yt 2 dt ( Φ(c1 + Y 1 + t λ + Yt 2 ) λλ) dw 0 t.

25 A two-dimensional example Let σ : H H be of the form σ(h) = ρ l(h) λ, where: 1 ρ > 0 is a constant. 2 λ H is given by λ = e γ for some γ > 0. 3 l H is a functional satisfying l(λ) = 1 and l(λ 2 ) = 0. There is κ H satisfying κ(λ) = 0 and κ(λ 2 ) = 1. Affine realization generated by λ + λ 2 with I = {h D(d/dx) : l(h) 0 and l(h + γh) > 0}.

26 The singular set and initial curves If l(1) > 0, then we have I S = 1 l(1)λ κ(1)λ 2 + and I S = {a1 + bλ + cλ 2 : a R +, b l(1)a and c R}. If l(1) = 0, then we have I S = 1 κ(1)λ 2 and I S = 1, λ 2 λ +. If l(1) < 0, then we have I S = l(1)λ + κ(1)λ and I S = {a1 + bλ + cλ 2 : a R, b l(1)a and c R}. In any case, we have I S 1 l(1)λ κ(1)λ 2 and I S 1 l(1)λ κ(1)λ 2 λ + λ 2.

27 The state processes Let (a, b, c) R R + R be such that h 0 = a(1 l(1)λ κ(1)λ 2 ) + bλ + cλ 2 I S. Then the solution to (1) with r 0 = h 0 is given by r = a(1 l(1)λ κ(1)λ 2 ) + Y 1 λ + Y 2 λ 2. The R + R-valued affine process Y satisfies the SDE [ ( ) ( ρ 2 l(1) γ dy t = aγ + γ 0 2κ(1) 2γ Y 0 = + ( ρ Y 1 t 0 ( b c ). ) dw t ρ2 γ ) ( Y 1 t Y 2 t ) ] dt

28 The bond prices We have an affine term structure ( ) P(t, T ) = exp A(T t) B 1 (T t)yt 1 B 2 (T t)yt 2. The functions A and B 1, B 2 are given by A(x) = a (x l(1) 1 e γx γ B 1 (x) = 1 e γx, γ B 2 (x) = 1 e 2γx. 2γ κ(1) 1 e 2γx 2γ ),

29 References I Björk, T. & Landén, C. (2002): On the construction of finite dimensional realizations for nonlinear forward rate models. Finance Stoch. 6(3), Björk, T. & Svensson, L. (2001): On the existence of finite dimensional realizations for nonlinear forward rate models. Math. Finance 11(2), Duffie, D., Filipović, D. & Schachermayer, W. (2003): Affine processes and applications in finance. Ann. Appl. Probab. 13(3), Duffie, D. & Kan, R. (1996): A yield-factor model of interest rates. Math. Finance 6(4), Filipović, D. (2001): A general characterization of one factor affine term structure models. Finance Stoch. 5(3),

30 References II Filipović, D. & Mayerhofer, E. (2009): Affine diffusion processes: Theory and applications. Radon Series Comp. Appl. Math. 8, Filipović, D. & Teichmann, J. (2003): Existence of invariant manifolds for stochastic equations in infinite dimension. J. Funct. Anal. 197(2), Filipović, D. & Teichmann, J. (2004): On the geometry of the term structure of interest rates. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2041), Tappe, S. (2010): An alternative approach on the existence of affine realizations for HJM term structure models. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466(2122),

31 References III Tappe, S. (2012): Existence of affine realizations for Lévy term structure models. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468(2147), Tappe, S. (2016): Affine realizations with affine state processes for stochastic partial differential equations. Stochastic Process. Appl. 126(7), Tappe, S. (2017): Time-homgeneous state processes for stochastic partial differential equations. Preprint.