A new approach to LIBOR modeling

Transcription

1 A new approach to LIBOR modeling Antonis Papapantoleon FAM TU Vienna Based on joint work with Martin Keller-Ressel and Josef Teichmann Istanbul Workshop on Mathematical Finance Istanbul, Turkey, 18 May 2009 Antonis Papapantoleon (TU Vienna) A new approach to LIBOR modeling 1 / 32

2 Outline of the talk 1 Interest rate markets 2 LIBOR model: Axioms 3 LIBOR and Forward price model 4 Affine processes 5 Affine martingales 6 Affine LIBOR model 7 Example: CIR martingales 8 Summary and Outlook 1 / 36

3 Interest rate markets Interest rates Notation B(t, T ): time-t price of a zero coupon bond for T ; B(T, T ) = 1; L(t, T ): time-t forward LIBOR for [T, T + δ]; L(t, T ) = 1 ( ) B(t, T ) δ B(t, T + δ) 1 F (t, T, U): time-t forward price for T and U; F (t, T, U) = B(t,T ) B(t,U) Master relationship F (t, T, T + δ) = B(t, T ) = 1 + δl(t, T ) (1) B(t, T + δ) 2 / 36

4 Interest rate markets Interest rates evolution Evolution of interest rate term structure, (picture: Th. Steiner) 3 / 36

5 Interest rate markets Calibration problems Strike rate (in %) Maturity (in years) 0 1 Implied volatilities are constant neither across strike nor across maturity 2 Variance scales non-linearly over time (see e.g. D. Skovmand) 4 / 36

6 LIBOR model: Axioms LIBOR model: Axioms Economic thought dictates that LIBOR rates should satisfy: Axiom 1 The LIBOR rate should be non-negative, i.e. L(t, T ) 0 for all t. Axiom 2 The LIBOR rate process should be a martingale under the corresponding forward measure, i.e. L(, T ) M(P T +δ ). Practical applications require: Models should be analytically tractable ( fast calibration). Models should have rich structural properties ( good calibration). What axioms do the existing models satisfy? 5 / 36

7 LIBOR and Forward price model LIBOR models I (Sandmann et al, Brace et al,..., Eberlein & Özkan) Ansatz: model the LIBOR rate as the exponential of a semimartingale H: ( t t ) L(t, T k ) = L(0, T k ) exp b(s, T k )ds + λ(s, T k )dh T k+1 s, (2) 0 0 where b(s, T k ) ensures that L(, T k ) M(P Tk+1 ). H has the P Tk+1 -canonical decomposition t H T t k+1 t = cs dw T k+1 s + x(µ H ν T k+1 )(ds, dx), (3) 0 0 R where the P Tk+1 -Brownian motion is W T k+1 t = W T t ( t N ) δ l L(t, T l ) δ l L(t, T l ) λ(t, T l) cs ds, (4) l=k+1 6 / 36

8 LIBOR and Forward price model LIBOR models II and the P Tk+1 -compensator of µ H is ν T k+1 (ds, dx) = ( N l=k+1 ) δ l L(t, T l ) ( ) e λ(t,t l )x ν T (ds, dx). 1 + δ l L(t, T l ) 7 / 36

9 LIBOR and Forward price model LIBOR models II and the P Tk+1 -compensator of µ H is ν T k+1 (ds, dx) = ( N l=k+1 ) δ l L(t, T l ) ( ) e λ(t,t l )x ν T (ds, dx). 1 + δ l L(t, T l ) Consequences for continuous semimartingales: 1 caplets can be priced in closed form; 2 swaptions and multi-libor products cannot be priced in closed form; 3 Monte-Carlo pricing is very time consuming coupled high dimensional SDEs! 8 / 36

10 LIBOR and Forward price model LIBOR models II and the P Tk+1 -compensator of µ H is ν T k+1 (ds, dx) = ( N l=k+1 ) δ l L(t, T l ) ( ) e λ(t,t l )x ν T (ds, dx). 1 + δ l L(t, T l ) Consequences for continuous semimartingales: 1 caplets can be priced in closed form; 2 swaptions and multi-libor products cannot be priced in closed form; 3 Monte-Carlo pricing is very time consuming coupled high dimensional SDEs! Consequences for general semimartingales: 1 even caplets cannot be priced in closed form! 2 ditto for Monte-Carlo pricing. 9 / 36

11 LIBOR and Forward price model LIBOR models III The equation for the dynamics yield the following matrix for the dependence structure... L(t, Ti 1 ) L(t, T N 2 ) L(t, T N 2 ) L(t, T N 1 ) L(t, T N 1 ) L(t, T N 1 )... L(t, T N ) L(t, T N ) L(t, T N ) L(t, T N )... L(t, T i ) L(t, T N 3 ) L(t, T N 2 ) L(t, T N 1 ) L(t, T N ) Bottom line: LIBOR rates we wish to simulate. 10 / 36

12 LIBOR and Forward price model LIBOR models IV: Remedies 1 Frozen drift approximation Brace et al, Schlögl, Glassermann et al,... replace the random terms by their deterministic initial values: δ l L(t, T l ) 1 + δ l L(t, T l ) δ ll(0, T l ) 1 + δ l L(0, T l ) (5) (+) deterministic characteristics closed form pricing ( ) ad hoc approximation, no error estimates, compounded error... 2 Log-normal and/or Monte Carlo methods best log-normal approximation (e.g. Schoenmakers) interpolations and predictor-corrector MC methods Joshi and Stacey (2008): overview paper 11 / 36

13 LIBOR and Forward price model LIBOR models V: Remedies 3 Strong Taylor approximation approximate the LIBOR rates in the drift by L(t, T l ) L(0, T l ) + Y (t, T l ) + (6) where Y is the (scaled) exponential transform of H (Y = Loge H ) theoretical foundation, error estimates, simpler equations for MC Siopacha and Teichmann; Hubalek, Papapantoleon & Siopacha Difference in implied vols between full SDE vs frozen drift and full SDE vs strong Taylor. 12 / 36

14 LIBOR and Forward price model Forward price model I (Eberlein & Özkan, Kluge) Ansatz: model the forward price as the exponential of a semimartingale H: ( t t ) F (t, T k ) = F (0, T k ) exp b(s, T k )ds + λ(s, T k )dh T k+1 s, (7) 0 0 where b(s, T k ) ensures that F (, T k ) = 1 + δl(, T k ) M(P Tk+1 ). H has the P Tk+1 -canonical decomposition t H T t k+1 t = cs dw T k+1 s + x(µ H ν T k+1 )(ds, dx), (8) 0 0 R where the P Tk+1 -Brownian motion is W T k+1 t = W T t ( t N ) λ(t, T l ) cs ds, (9) 0 l=k+1 13 / 36

15 LIBOR and Forward price model Forward price model II and the P Tk+1 -compensator of µ H is ( ) N ν T k+1 (ds, dx) = exp x λ(t, T l ) ν T (ds, dx). Consequences: l=k+1 1 the model structure is preserved; 2 caps, swaptions and multi-libor products priced in closed form. So, what is wrong? 14 / 36

16 LIBOR and Forward price model Forward price model II and the P Tk+1 -compensator of µ H is ( ) N ν T k+1 (ds, dx) = exp x λ(t, T l ) ν T (ds, dx). Consequences: l=k+1 1 the model structure is preserved; 2 caps, swaptions and multi-libor products priced in closed form. So, what is wrong? Negative LIBOR rates can occur! 15 / 36

17 LIBOR and Forward price model Forward price model II and the P Tk+1 -compensator of µ H is ( ) N ν T k+1 (ds, dx) = exp x λ(t, T l ) ν T (ds, dx). Consequences: l=k+1 1 the model structure is preserved; 2 caps, swaptions and multi-libor products priced in closed form. So, what is wrong? Negative LIBOR rates can occur! Aim: design a model where the model structure is preserved and LIBOR rates are positive. Tool: Affine processes on R d / 36

18 Affine processes Affine processes I Let X = (X t ) 0 t T be a conservative, time-homogeneous, stochastically continuous Markov process taking values in D = R d 0 ; and (P x) x D a family of probability measures on (Ω, F), such that X 0 = x, P x -a.s. for every x D. Setting { I T := u R d [ : E x e u,x T ] } <, for all x D, (10) we assume that (i) 0 I T ; (ii) the conditional moment generating function of X t under P x has exponentially-affine dependence on x; i.e. there exist functions φ t (u) : [0, T ] I T R and ψ t (u) : [0, T ] I T R d such that E x [ exp u, Xt ] = exp ( φ t (u) + ψ t (u), x ), (11) for all (t, u, x) [0, T ] I T D. 17 / 36

19 Affine processes Affine processes II The process X is a regular affine process in the spirit of Duffie, Filipović & Schachermayer (2003). Using Theorem 3.18 in Keller-Ressel (2008) F (u) := t t=0+ φ t (u) and R(u) := t t=0+ ψ t (u) (12) exist for all u I T and are continuous in u. Moreover, F and R satisfy Lévy Khintchine-type equations: ( F (u) = b, u + e ξ,u 1 ) m(dξ) (13) and R i (u) = β i, u + D αi 2 u, u ( + e ξ,u 1 u, h i (ξ) ) µ i (dξ), (14) D where (b, m, α i, β i, µ i ) 1 i d are admissible parameters. 18 / 36

20 Affine processes Affine processes III The time-homogeneous Markov property of X implies: [ E x exp u, Xt+s ] ( F s = exp φt (u) + ψ t (u), X s ), (15) for all 0 t + s T and u I T. Lemma (Flow property) The functions φ and ψ satisfy the semi-flow equations: φ t+s (u) = φ t (u) + φ s (ψ t (u)) ψ t+s (u) = ψ s (ψ t (u)) (16) with initial condition for all suitable 0 t + s T and u I T. φ 0 (u) = 0 and ψ 0 (u) = u, (17) 19 / 36

21 Affine processes Affine processes IV 1 Affine processes on R: the admissibility conditions yield F (u) = bu + a ( 2 u2 + e zu 1 uh(z) ) m(dz) R(u) = βu, for a R 0 and b, β R. Every affine process on R is an Ornstein Uhlenbeck (OU) process. 2 Affine processes on R 0 : the admissibility conditions yield ( F (u) = bu + e zu 1 ) m(dz) D R(u) = βu + α ( 2 u2 + e zu 1 uh(z) ) µ(dz), D R for b, α R 0 and β R. There exist affine process on R 0 which are not OU process, e.g. CIR. 20 / 36

22 Affine martingales Affine LIBOR model: martingales 1 Idea: 1 insert an affine process in its moment generating function with inverted time; the resulting process is a martingale; 2 if the affine process is positive, the martingale is greater than one. Theorem The process M u = (M u t ) 0 t T defined by M u t = exp (φ T t (u) + ψ T t (u), X t ), (18) is a martingale. Moreover, if u I T R d 0 then M t 1 a.s. for all t [0, T ], for any X 0 R d / 36

23 Affine martingales Affine LIBOR model: martingales 1 Proof. Using (17) and (15), we have that: [ ] [ ] E x M u T F t = Ex exp u, XT F t = exp ( φ T t (u) + ψ T t (u), X t ) = Mt u. Regarding M u t 1 for all t [0, T ]: note that if u I T R d 0, then M u t = E x [ exp u, XT Ft ] 1. (19) 22 / 36

24 Affine martingales Affine LIBOR model: martingales 1 Example (Lévy process) Consider a Lévy subordinator, then Mt u = exp (φ T t (u) + ψ T t (u), X t ) = exp ((T t)κ(u) + u X t ) 1 = exp(t κ(u)) exp (u X t tκ(u)) M, (20) which is a martingale 1 for u R d / 36

25 Affine LIBOR model Affine LIBOR model: Ansatz Consider a discrete tenor structure 0 = T 0 < T 1 < T 2 < < T N ; discounted bond prices must satisfy: B(, T k ) B(, T N ) M(P T N ), for all k {1,..., N 1}. (21) Ansatz We model quotients of bond prices using the martingales M: B(t, T 1 ) B(t, T N ) = Mu 1 t (22) B(t, T N 1 ) B(t, T N ). = M u N 1 t, (23) with initial conditions: B(0,T k ) B(0,T N ) = Mu k 0, for all k {1,..., N 1}. 24 / 36

26 Affine LIBOR model Affine LIBOR model: initial values Proposition Let L(0, T 1 ),..., L(0, T N ) be a tenor structure of non-negative initial LIBOR rates; let X be an affine process starting at the canonical value 1. 1 If γ X := sup u IT R d E 1[ e u,x T ] > B(0,T 1) >0 B(0,T N ), then there exists a decreasing sequence u 1 u 2 u N = 0 in I T R d 0, such that M u k 0 = B(0, T k), for all k {1,..., N}. (24) B(0, T N ) In particular, if γ X =, then the affine LIBOR model can fit any term structure of non-negative initial LIBOR rates. 2 If X is one-dimensional, the sequence (u k ) k {1,...,N} is unique. 3 If all initial LIBOR rates are positive, the sequence (u k ) k {1,...,N} is strictly decreasing. 25 / 36

27 Affine LIBOR model Affine LIBOR model: forward prices Forward prices have the following form B(t, T k ) B(t, T k+1 ) = B(t, T k) B(t, T N ) B(t, T N ) B(t, T k+1 ) = Muk t M u k+1 t ( = exp φ TN t(u k ) φ TN t(u k+1 ) Now, φ t ( ) and ψ t ( ) are order-preserving, i.e. + ψ TN t(u k ) ψ TN t(u k+1 ), X t ). (25) u v φ t (u) φ t (v) and ψ t (u) ψ t (v). Consequently: positive initial LIBOR rate yields positive LIBOR rates for all times. 26 / 36

28 Affine LIBOR model Affine LIBOR model: forward measures Forward measures are related via: or equivalently: dp Tk Ft = F (t, T k, T k+1 ) dp Tk+1 F (0, T k, T k+1 ) = B(0, T k+1) B(0, T k ) Mu k t M u k+1 t (26) dp Tk+1 Ft = B(0, T N) dp TN B(0, T k+1 ) B(t, T k+1) = B(0, T N) B(t, T N ) B(0, T k+1 ) Mu k+1 t. (27) Hence, we can easily see that B(, T k ) B(, T k+1 ) = Muk M u k+1 M(P T k+1 ), for all k {1,..., N 1}. (28) 27 / 36

29 Affine LIBOR model Affine LIBOR model: dynamics under forward measures The moment generating function of X t under any forward measure is [ E PTk+1 e vx t ] u = M k+1 0 E PTN [M u k+1 t e vxt ] (29) ( ( = exp φ t ψtn t(u k+1 ) + v ) φ t (ψ TN t(u k+1 )) Denote by Mu k t M u k+1 t + ψ t ( ψtn t(u k+1 ) + v ) ψ t (ψ TN t(u k+1 )), x ). = e A k+b k X t ; the moment generating function is [ E PTk+1 e v(a k +B ] k X t) = B(0, T N) B(0, T k+1 ) ( exp vφ TN t(u k ) + (1 v)φ TN t(u k+1 ) ( + φ t vψtn t(u k ) + (1 v)ψ TN t(u k+1 ) ) (30) + ψ t ( vψtn t(u k ) + (1 v)ψ TN t(u k+1 ) ), x ). 28 / 36

30 Affine LIBOR model Affine LIBOR model: caplet pricing We can re-write the payoff of a caplet as follows (here K := 1 + δk): δ(l(t k, T k ) K) + = (1 + δl(t k, T k ) 1 + δk) + ( M u k ) + T = k K = M u k+1 T k ( e A k+b k X Tk K) +. (31) Then we can price caplets by Fourier-transform methods: [ C(T k, K) = B(0, T k+1 )E PTk+1 δ(l(tk, T k ) K) +] = KB(0, T k+1) K iv R Λ A k +B k X Tk (R iv) dv (32) 2π (R iv)(r 1 iv) where Λ Ak +B k X Tk is given by (30). R 29 / 36

31 Example: CIR martingales CIR martingales The Cox-Ingersoll-Ross (CIR) process is given by dx t = λ (X t θ) dt + 2η X t dw t, X 0 = x R 0, (33) where λ, θ, η R 0. This is an affine process on R 0, with [ E x e ux t ] ) = exp (φ t (u) + x ψ t (u), (34) where φ t (u) = λθ 2η log ( 1 2ηb(t)u ) and ψ t (u) = a(t)u 1 2ηb(t)u, (35) with b(t) = { t, if λ = 0 1 e λt λ, if λ 0, and a(t) = e λt. 30 / 36

32 Example: CIR martingales CIR martingales: closed-form formula I Definition A random variable Y has location-scale extended non-central chi-square distribution, Y LSNC χ 2 (µ, σ, ν, α), if Y µ σ NC χ 2 (ν, α) Then we have that and hence log X t X t ( B(t, Tk ) B(t, T k+1 ) ( P TN LSNC χ 2 0, ηb(t), λθ η, xa(t) ), ηb(t) ( P Tk+1 LSNC χ 2 0, ηb(t) ζ(t, T N ), λθ η, ) xa(t), ηb(t)ζ(t, T N ) ) ( PTk+1 LSNC χ 2 A k, B kηb(t) ζ(t, T N ), λθ η, ) xa(t). ηb(t)ζ(t, T N ) 31 / 36

33 Example: CIR martingales CIR martingales: closed-form formula II ( ) Then, denoting by M = log B(Tk,T k ) B(T k,t k+1 ) the log-forward rate, we arrive at: [ ( ) ] + C(T k, K) = B(0, T k+1 ) E PTk+1 e M K ] } = B(0, T k+1 ) {E PTk+1 [e M 1 {M log K} K P Tk+1 [M log K] ( ) ( ) log K = B(0, T k ) χ 2 Ak log K ν,α 1 K χ 2 Ak ν,α σ 2, 1 σ 2 (36) where K = K B(0, T k+1 ) and χ 2 ν,α(x) = 1 χ 2 ν,α(x), with χ 2 ν,α(x) the non-central chi-square distribution function, and ν = λθ η, σ 1,2 = B kηb(t k ) ζ 1,2, α 1,2 = xa(t k) ηb(t k )ζ 1,2, ζ 1 = 1 2ηb(T k )ψ TN T k (u k ), ζ 2 = 1 2ηb(T k )ψ TN T k (u k+1 ). 32 / 36

34 Example: CIR martingales CIR martingales: volatility surface Example of an implied volatility surface for the CIR martingales. 33 / 36

35 Example: CIR martingales Γ-OU martingales: volatility surface Example of an implied volatility surface for the Γ-OU martingales. 34 / 36

36 Summary and Outlook Summary and Outlook 1 We have presented a LIBOR model that is very simple (Axiom 0!), and yet... captures all the important features... especially positivity and analytical tractability. 2 Future work: thorough empirical analysis extensions: multiple currencies, default risk 3 M. Keller-Ressel, A. Papapantoleon, J. Teichmann (2009) A new approach to LIBOR modeling. Preprint, arxiv/ / 36

37 Summary and Outlook Summary and Outlook 1 We have presented a LIBOR model that is very simple (Axiom 0!), and yet... captures all the important features... especially positivity and analytical tractability. 2 Future work: thorough empirical analysis extensions: multiple currencies, default risk 3 M. Keller-Ressel, A. Papapantoleon, J. Teichmann (2009) A new approach to LIBOR modeling. Preprint, arxiv/ Thank you for your attention! 36 / 36