Extended Libor Models and Their Calibration

Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Haindorf, 7 Februar 2008 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 1 / 43

Overview 1 Introduction Forward Libor Models 2 Modelling Modelling under Terminal Measure Modelling under Forward Measures 3 Pricing and Calibration Pricing of Caplets Specification Analysis Calibration Procedure 4 Calibration in work Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 2 / 43

Forward Libor Models Introduction Forward Libor Models Tenor structure: 0 = T 0 < T 1 <... < T M < T M+1 with accrual periods δ i := T i+1 T i Zero coupon bonds: B k (t), t [0, T k ] with B k (T k ) = 1 Forward Libor rates: L 1 (t),..., L M (t) L k (t) = 1 ( ) Bk (t) δ k B k+1 (t) 1, t [0, T k ], k = 1,..., M Remark L 1,..., L M are based on simple compounding that is an investor receives 1$ at T k and pays 1 + δ k L k (t) at T k+1 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 3 / 43

Modelling Modelling under Terminal Measure The BGM/Jamshidian LIBOR market model dl i = M j=i+1 δ j L i L j γ i γ j 1 + δ j L j dt + L i γ i dw (M+1), where (W (M+1) (t) 0 t T M ) is a D-dimensional Wiener process under P M+1 and γ i = (γ i,1,..., γ i,d ) R D, i = 1,..., M are deterministic volatility vector functions, called factor loadings, to be determined via calibration. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 4 / 43

Modelling Modelling under Terminal Measure Scalar volatility and correlation functions A scalar volatility functions t σ i (t) := γ i (t) = d γi,k 2 (t), 0 t T i, 1 i M, k=1 and a (local) correlation functions t ϱ ij (t) := γ i(t) γ j (t) γ i (t) γ j (t), 0 t min(t i, T j ), 1 i, j M, are the only economically relevant objects! Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 5 / 43

Modelling Modelling under Terminal Measure Scalar volatility and correlation functions For any deterministic orthogonal matrix valued map Q : t Q(t) R D D the process W t = t 0 Q(u)dW u is again a standard Brownian motion in R D. Hence, taking dw t := Q d W t, yields the same Libor process with γ := γq. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 6 / 43

Modelling The Hull-White Parametrization Modelling under Terminal Measure For a fixed correlation matrix ϱ (0) := [ϱ (0) ij ] 0 i,j<m of rank D and a fixed non-negative vector Λ := [Λ i ] 0 i<m take σ i (t) := Λ i m(t), 0 t T i, ϱ ij (t) := ϱ (0) i m(t),j m(t), 0 t min(t i, T j ), 1 i, j M with m(t) := min{m : T m t}. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 7 / 43

Modelling Modelling under Terminal Measure The Hull-White Parametrization Find unit vectors e (0) i R D satisfying ϱ (0) ij = e (0) i e (0) j and set e i (t) := e (0) i m(t). The dynamics of the Hull-White LIBOR model is determined by the volatility structure γ i (t) := σ i (t)e i (t) = Λ i m(t) e (0) i m(t), 1 i M. Observation Good feature: A kind of time-shift homogeneity which is economically sensible. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 8 / 43

Modelling Modelling under Terminal Measure More flexible volatility structure Define γ i (t) = c i g(t i t)e (0) i m(t), 0 t min(t i, T j ), 1 i, j M, ϱ (0) kl = e (0) k e (0) l, 0 k, l < M with suitably parameterized function g and correlation matrix ϱ (0). Observation For c i c, model is basically time-shift homogeneous. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 9 / 43

Modelling Modelling under Terminal Measure Simple parametrization of scalar volatilities g(x) := g a,b,g (x) := g + (1 g + ax)e bx, a, b, g > 0 A typical shape of g : x g 0.5,0.4,0.6 (x) g (x) 0.7 0.8 0.9 1.0 1.1 1.2 0 2 4 6 8 10 x Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 10 / 43

Modelling Modelling under Terminal Measure Semi-parametric full rank correlation structures Consider initial correlation structures of the following form ϱ (0) ij :== min(b i, b j ), 1 i, j < M, max(b i, b j ) where the sequences i b i and i b i /b i+1 strictly increasing. are positive and Observation Implicit consequence: D = M, w.l.g. b 1 = 1. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 11 / 43

Modelling Modelling under Terminal Measure Semi-parametric full rank correlation structures The assumption that the sequence i b i b i+1 is increasing forces that for fixed p is increasing. i ϱ i,i+p = Cor( L i, L i+p ) Example Correlation between a seven and a nine year forward is higher than the correlation between a three and a five year forward. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 12 / 43

Modelling Modelling under Terminal Measure Semi-parametric full rank correlation structures Question Is min(b i, b j )/ max(b i, b j ) a correlation structure at all? Take i b i increasing with b 0 = 0, b 1 = 1 and set a i := Consider standard normal i.i.d r.v. Z k and set Then, for i j Y i := i a k Z k. k=1 Cov(Y i, Y j ) = and the correlations are ϱ Yi,Y j = b i /b j. i ak 2 = b2 i k=1 b 2 i b 2 i 1. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 13 / 43

Modelling Modelling under Terminal Measure Semi-parametric full rank correlation structures For every correlation structure b i /b j, i < j, with increasing b i and b i /b i+1 it holds [ M ] b i = exp min(l 1, i 1) l l=2 for a nonnegative numbers i, i 0, 2 i M. In particular, i := (ln b i 1 + ln b i+1 2 ln b i ), 2 i M 1, M := ln b m ln b m 1. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 14 / 43

Modelling Modelling under Terminal Measure Semi-parametric full rank correlation structures The following representation thus holds ϱ ij = min(b [ i, b j ) max(b i, b j ) = exp l 0, 2 l M. M l=i+1 min(l i, j i ) l ], From this representation we may derive conveniently various low parametric structures consistent with the structure ϱ ij = min(b i, b j ) max(b i, b j ). Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 15 / 43

Modelling Modelling under Terminal Measure Example: a two parametric structure The choice 2 =... = M 1 =: α > 0 and M =: β > 0 gives the correlation structure i+j+1 i j (β+α(m ϱ ij = e 2 )), i, j = 1,..., m. To gain parameter stability we set, η := α(m 1)(m 2)/2, ϱ := ϱ 1m = exp[ α(m 1)(m 2)/2 (m 1)β] and get a two parametric structure: ϱ ij = e i j M 1( ln ϱ +η M i j+1 M 2 ), 0 < η < ln ϱ with i, j = 1,..., M. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 16 / 43

Modelling Modelling under Terminal Measure General Libor Model under P M+1 For i = 1,..., M dl i (t) L i (t) = A(M+1) i (dt)+ Γ i dw (M+1) (t) }{{} Continuous Part + E ( ψ i (t, u) µ ν (M+1)) (dt, du) } {{ } Jump Part A i are predictable drift processes W (M+1) is a D-dimensional Brownian motion under P M+1 Γ i are predictable D-dimensional volatility processes ω µ(dt, du, ω) is a random point measure on R + E with P M+1 -compensator ν (M+1) (dt, du) Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 17 / 43

Modelling Modelling under Terminal Measure Libor Model under P M+1 We consider random point measures µ of finite activity satisfying E [ ( ψ i (t, u) µ ν (M+1)) Nt ] (dt, du) = d ψ i (s l, u l ), where (s l, u l ) R + E are jumps of µ and l=1 N t is a Poisson process with intensity λ E = R... R }{{} m u l R m is distributed with p 1 (dx 1 )... p m (dx m ) Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 18 / 43

Modelling Modelling under Terminal Measure Drift term under P M+1 The requirement that L i is a martingale under P (M+1) implies A (M+1) i M δ j L j (dt) = Γ i Γ j dt+ 1 + δ j L j j=i+1 + λ(t)dt ψ i (u, t)p(du) R m M j=i+1 ( 1 + δ ) jl j ψ i (t, u). 1 + δ j L j Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 19 / 43

Modelling Modelling under Forward Measures Dynamic of L i under P i+1 Since L i is a martingale under P i+1 dl i = Γ i dw (i+1) + ψ i (t, u)(µ ν (i+1) )(dt, du), L i where dw (i+1) = M j=i+1 E δ j L j 1 + δ j L j Γ i dt + dw (M+1) is a standard Brownian motion under P i+1 and M ( ν (i+1) (dt, du) = ν (M+1) (dt, du) 1 + δ ) jl j ψ j (t, u). 1 + δ j L j j=i+1 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 20 / 43

Modelling Modelling under Forward Measures Dynamic of L i under P i+1 The logarithmic version reads as [ Nt ] d ln(l i ) = A (i+1) (dt) + Γ i dw (i+1) + d φ i (s l, u l ) i=1 with φ i = ln(1 + ψ i ) and A (i+1) (dt) = 1 2 Γ i 2 dt ψ i (t, u)ν (i+1) (dt, du) R m Observation For i < M the new compensator ν (i+1) is not deterministic and ln(l i ) is generally not affine under P i+1. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 21 / 43

Caplet Volas Caplets Pricing and Calibration Pricing of Caplets The price of j-th caplet at time zero is given by C j (K ) = δ j B j+1 (0)E Pj+1 [(L j (T j ) K ) + ] Tenors Strikes Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 22 / 43

Pricing and Calibration Pricing of Caplets Pricing Caplets under P i+1 In terms of log-forward moneyness v = ln(k /L j (0)) C j (v) := δ j B j+1 (0)L j (0)E Pj+1 [(e X j (t) e v ) + ], with X j (t) = log(l j (t)) log(l j (0)). Define then O j (v) = E Pj+1 [(e X j (t) e v ) + ] (1 e v ) +, F{O j }(z) := R O j (v)e ivz dz = 1 Φ P j+1 (z i; T j ). z(z i) Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 23 / 43

Pricing and Calibration Specification Analysis Characteristic Function of L M under P M+1 Since d ln(l M ) = 1 2 Γ i 2 dt + Γ i dw (M+1) (t) + d [ Nt ] φ i (s l, u l ) l=1 and N t, W (M+1) and u l are mutually independent Φ PM+1 (z; T ) = Φ C P M+1 (z; T )Φ J P M+1 (z; T ), where Φ C P M+1 (z; T ) ( Φ J P M+1 (z; T )) is the c.f. of continuous (jump) part. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 24 / 43

Pricing and Calibration Specification Analysis Specification Analysis: Continuous Part For some predictable vector volatility process (v 1 (t),..., v d (t)) define Γ i = 1 r 2 SV γ i1 1 r 2 SV γ i2.. 1 r 2 SV γ id r SV β i1 v1 (t).. r SV β id vd (t), W (M+1) = W (M+1) 1 W (M+1) 2.. W (M+1) d W (M+1) 1.. W (M+1) d with mutually independent d-dimensional Brownian motions W (M+1) and W (M+1). Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 25 / 43

Pricing and Calibration Specification Analysis Specification Analysis: Continuous Part Let γ i (t) = c i g(t i t)e i, e i R d, where c i > 0 are loading factors g i ( ) is a scalar volatility function e i are unit vectors coming from the decomposition of the correlation matrix ζ ζ ij = e i e j, 1 i, j M, be a deterministic volatility structure of the input Libor market model calibrated to ATM caps and ATM swaptions. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 26 / 43

Pricing and Calibration Specification Analysis Specification Analysis: Continuous Part Define a new time independent volatility structure via β i β j = min(i,j) 1 min(i, j) k=1 1 T k Tk 0 γ i (t)γ j (t) dt. Remark The covariance of the process ξ i (t) := t 0 Γ i (t)dw (M+1) satisfies cov(ξ i (t), ξ j (t)) t 0 γ i (t)γ j (t)ds and is approximately the same as in the input LMM. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 27 / 43

Pricing and Calibration Specification Analysis Specification Analysis: Continuous Part Two possible specifications for the volatility process v Stochastic Volatility Heston Model dv k = κ k (1 v k )dt +σ k ϱ k vk dw (M+1) k +σ k (1 ϱ 2 k ) v k dv (M+1) k, Stochastic Volatility BN Model dv k = κ k v k dt + σ k ϱ k dw (M+1) k + σ k (1 ϱ 2 (M+1) k )dv k. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 28 / 43

Pricing and Calibration Specification Analysis Specification Analysis It holds Φ C P M+1 (z; T ) = Φ C D,P M+1 (z; T ) Φ C SV,P M+1 (z; T ), where Φ C D,P M+1 (z; T ) = exp ( 12 ( ) ) T θ2m (T ) z 2 + iz, θm 2 (T ) = γ M 2 dt 0 and Φ C SV,P M+1 (z; T ) = exp (A M (z; T ) + B M (z; T )) Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 29 / 43

Pricing and Calibration Specification Analysis Specification Analysis In particular and A M (z; T ) = κ { [ M 1 gm e d ]} MT σm 2 (a M + d M )T 2 ln 1 g M B M (z; T ) = (a M + d M )(1 e d MT ) σ 2 M (1 g Me d MT ) a M = κ M iϱ M ω M z d M = am 2 + ω2 M (z2 + iz) g M = a M + d M a M d M, ω M = r SV β MM σ M Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 30 / 43

Pricing and Calibration Specification Analysis Specification Analysis As can be easily seen ) A M (z; T ) lim = α M ω M (iϱ M + 1 ϱ 2 z z M T and with B M (z; T ) 1 ϱ 2 M lim = + iϱ M z z α M := κ M σ 2 M σ M Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 31 / 43

Pricing and Calibration Specification Analysis Specification Analysis Let us take φ i (u, t) = u β i, then the characteristic function of the jump part is given by ( ) Φ J P M+1 (z; T ) = exp λt (e izv 1)µ M (v) dv, R where µ M is the density of u β M (t). Observation Due to the Riemann-Lebesgue theorem Φ J P M+1 (z; T ) exp( λt ), z. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 32 / 43

Pricing and Calibration Specification Analysis Specification Analysis: Asymptotic Properties Computing sequentially L 2 := lim z log(φ PM+1 (z; T ))/z 2, we get L 1 := lim z [ log(φpm+1 (z; T ))/z (z + i)l 2 ], [ ] L 0 := lim log(φ PM+1 (z; T )) (z 2 + iz)l 2 zl 1, z L 0 = λ, L 2 = 1 2 θ2 M (T ) and Re L 1 = 1 ϱ 2 M σ M α M ω M 1 ϱ 2 M T, Im L 1 = ϱ M σ M α M ω M ϱ M T Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 33 / 43

Pricing and Calibration Calibration Procedure Parameters Estimation: Linearization Observation From the knowledge of L 1 (T ) for two different T one can reconstruct all parameters of the SV process Theorem Ψ PM+1 (z; T ) := log(φ PM+1 (z; T )) = L 2 (z 2 + iz) + L 1 z + L 0 + R 0 + R 1 (z), where R 0 = R 0 (α M, κ M, ϱ M, ω M ) is a constant not depending on λ and R 1 (z) 0, z. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 34 / 43

Pricing and Calibration Calibration Procedure Parameters Estimation: Projection Estimators We find estimates for L 2, L 1 and L 0 in the form of weighted averages L 2,U := L 1,U := L 0,U := Re( Ψ PM+1 (u))w2 U (u) du, Im( Ψ PM+1 (u))w1 U (u) du i L 2,U, Re( Ψ PM+1 (u))w0 U (u) du R 0 with ( ) Ψ PM+1 (u) := ln 1 u(u + i)f{õm}(u + i). Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 35 / 43

Pricing and Calibration Parameters Estimation Calibration Procedure The weights are given by w U 2 = U 3 w 2 (u/u), w U 1 = U 2 w 1 (u/u), w U 0 = U 1 w 0 (u/u), where 1 1 w 2 (u)du = 0, 1 1 uw 2 (u)du = 0, 1 1 u 2 w 2 (u)du = 1, 1 1 1 1 w 0 (u)du = 1, w 1 (u) du = 0, 1 1 1 1 uw 0 (u)du = 0, uw 1 (u) du = 1, 1 1 u 2 w 0 (u)du = 0. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 36 / 43

Pricing and Calibration Calibration Procedure Parameters Estimation: Jump distribution Define F{ µ M }(z) = Ψ PM+1 (z; T ) L 2 (z 2 + iz) L 1 z L 0 R 0 or equivalently µ M := F 1 [( ΨPM+1 ( ; T ) L 2 ( 2 + i ) L 1 L 0 R ) ] 0 1 [ U,U] ( ) Remark Due to lack of data and numerical errors µ M may not be a density and needs to be corrected. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 37 / 43

Pricing and Calibration Calibration Procedure Parameters Estimation: Further optimization Upon finding ( L0,U, L 1,U, L ) 2,U T := we may consider T as a final set of parameters or consider nonlinear least-squares ( σ M, ϱ M, κ M, λ ) J (T ) = N w i CM T (K i) C M (K i ) 2 i=1 and minimize J (T ) over the parametric set S R 4 taking as initial value T. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 38 / 43

Pricing and Calibration Calibration Procedure Approximative dynamics of L i under P i+1 It holds approximately dl i Γ i dw (i+1) + e u β j (µ ν (i+1) )(dt, du), L i E where dw (i+1) is a standard Brownian motion under P i+1 and ν (i+1) (dt, du) = ν (M+1) (dt, du) M j=i+1 ( 1 + δ jl j (0)e u β j 1 + δ j L j (0) ). Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 39 / 43

Pricing and Calibration Calibration Procedure Approximative dynamics of v k under P i+1 By freezing the Libors at their initial values we obtain an approximative v k dynamics ( ) ( dv k κ (i+1) k θ (i+1) ) (i+1) k v k dt + σ k vk ϱ k d W k + 1 ϱ 2 (i+1) kdw k with reversion speed parameter κ (i+1) k := and mean reversion level κ k r SV σ k ϱ k M j=i+1 θ (i+1) k := κ k κ (i+1) k δ j L j (0) 1 + δ j L j (0) β jk,. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 40 / 43

Pricing and Calibration Calibration Procedure Pricing Caplets under P M+1 The price of j-th caplet at time zero can be alternatively written as [ ] Bj+1 (T j ) C j (K ) = δ j B M+1 (0)E PM+1 B M+1 (T j ) (L j(t j ) K ) + Note that B j+1 (T j ) B M+1 (T j ) = M k=j+1 (1 + δ k L k (T j )) = M k=j+1 (1 + δ k )E ξ exp M k=j+1 ξ k ln(l k (T j )), where {ξ k } M k=j+1 are independent random variables and each ξ k takes two values 0 and 1 with probabilities 1/(1 + δ k ) and δ k /(1 + δ k ). Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 41 / 43

Pricing and Calibration Calibration Procedure Pricing Caplets under P M+1 Thus, F{O j }(z) = 1 E ξφ M+1 (z i, ξ j+1,..., ξ M ), z(z i) where Φ M+1 (z j, z j+1,..., z M ) is the joint characteristic function of (ln(l j (T j )),..., ln(l M (T j ))) under P M+1. Remark Instead of terminal measure P M+1 we could consider P l+1 with 1 < l < M + 1. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 42 / 43

Calibration in work Calibration results for 14.08.2007 Caplet volas for different caplet periods Caplet Volatilities 0.10 0.12 0.14 0.16 0.18 [ 17.5, 18 ] [ 15, 15.5 ] Caplet Volatilities 0.10 0.12 0.14 0.16 0.18 [ 12.5, 13 ] [ 10, 10.5 ] 0.02 0.04 0.06 0.08 0.10 0.02 0.04 0.06 0.08 0.10 Strikes Strikes Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 43 / 43

Bibliography Belomestny, D. and Spokoiny, V. Spatial aggregation of local likelihood estimates with applications to classification, Annals of Statistics, 2007, 35(5), 2287 2311. Belomestny, D. and Reiß, M. Spectral calibration of exponential Lévy models, Finance and Stochastics, 2006, 10(4), 449 474. Belomestny, D. and Schoenmakers, J. A jump-diffusion Libor model and its robust calibration, SFB649 Discussion Paper, 2006, 037. Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar 2008 43 / 43