Extended Libor Models and Their Calibration

Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November 2007 1 / 30

Overview 1 Introduction Forward Libor Models 2 Modelling Modelling under Terminal Measure Modelling under Forward Measures 3 Pricing of Caplets Specification Analysis Calibration Procedure 4 Calibration in work Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November 2007 2 / 30

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 Vienna, 16 November 2007 3 / 30

Modelling Modelling under Terminal Measure 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 Vienna, 16 November 2007 4 / 30

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 Vienna, 16 November 2007 5 / 30

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 Vienna, 16 November 2007 6 / 30

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 Vienna, 16 November 2007 7 / 30

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 Vienna, 16 November 2007 8 / 30

Caplet Volas Caplets 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 Vienna, 16 November 2007 9 / 30

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 Vienna, 16 November 2007 10 / 30

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 Vienna, 16 November 2007 11 / 30

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 Vienna, 16 November 2007 12 / 30

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 Vienna, 16 November 2007 13 / 30

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 Vienna, 16 November 2007 14 / 30

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 Vienna, 16 November 2007 15 / 30

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 Vienna, 16 November 2007 16 / 30

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 Vienna, 16 November 2007 17 / 30

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 Vienna, 16 November 2007 18 / 30

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 Vienna, 16 November 2007 19 / 30

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 Vienna, 16 November 2007 20 / 30

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 Vienna, 16 November 2007 21 / 30

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 Vienna, 16 November 2007 22 / 30

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 Vienna, 16 November 2007 23 / 30

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 Vienna, 16 November 2007 24 / 30

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 Vienna, 16 November 2007 25 / 30

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 Vienna, 16 November 2007 26 / 30

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 Vienna, 16 November 2007 27 / 30

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 Vienna, 16 November 2007 28 / 30

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 Vienna, 16 November 2007 29 / 30

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 Vienna, 16 November 2007 30 / 30

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 Vienna, 16 November 2007 30 / 30