A new approach to LIBOR modeling application of affine processes Antonis Papapantoleon FAM – TU Vienna Special Semester on Stochastics with Emphasis on Finance RICAM, Linz, Austria, 4 December 2008 Antonis Papapantoleon (TU Vienna) Affine LIBOR model 1 / 27
1 Interest rate markets 2 LIBOR model: Axioms 3 LIBOR and Forward price model 4 Affine processes 5 Affine LIBOR model 6 Example: Γ-OU Martignales 7 Summary and Outlook 1 / 31
Interest rate markets Interest rates – Notation B ( t , T ): time- t price of a zero coupon bond for T , i.e. B ( T , T ) = 1; L ( t , T ): time- t forward LIBOR for [ T , T + δ ]: � � L ( t , T ) = 1 B ( t , T ) B ( t , T + δ ) − 1 δ F ( t , T , U ): time- t forward price for T and U : F ( t , T , U ) = B ( t , T ) B ( t , U ) “Master” relationship B ( t , T ) F ( t , T , T + δ ) = B ( t , T + δ ) = 1 + δ L ( t , T ) (1) 2 / 31
Interest rate markets Interest rates evolution Evolution of interest rate term structure, 2003 – 2004 (picture: Th. Steiner) 3 / 31
Interest rate markets Calibration problems 30.0 28.0 26.0 24.0 22.0 20.0 18.0 16.0 14.0 12.0 0 10.0 2 2.5 4.0 4 6.0 6 8.0 8 10.0 10 Maturity (in years) Strike rate (in %) Implied volatilities are constant neither across strike nor across maturity 1 Variance scales non-linearly over time (see e.g. Skovmand) 2 4 / 31
LIBOR model: Axioms LIBOR model: Axioms Economic thought dictates that LIBOR rates should satisfy: Axiom 1 The LIBOR rate should be positive, 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 describe the empirical evidence adequately. Models should be calibrated to liquid products (caps, atm swaptions). What axioms do the existing models satisfy? 5 / 31
LIBOR and Forward price model LIBOR models I (Sandmann et al, Brace et al, . . . , Eberlein & ¨ Ozkan) Ansatz: model the LIBOR rate as the exponential of a semimartingale H : �� t � t � λ ( s , T k ) dH T k +1 L ( t , T k ) = L (0 , T k ) exp b ( s , T k ) ds + , (2) s 0 0 where b ( s , T k ) ensures that L ( · , T k ) ∈ M ( P T k +1 ). H has the P T k +1 -canonical decomposition � t � t � √ c s dW T k +1 x ( µ H − ν T k +1 )( ds , dx ) , H T k +1 = + (3) s t 0 0 R where the P T k +1 -Brownian motion is � � � t N δ l L ( t − , T l ) √ c s ds , � W T k +1 = W T ∗ − 1 + δ l L ( t − , T l ) λ ( t , T l ) (4) t t 0 l = k +1 6 / 31
LIBOR and Forward price model LIBOR models II and the P T k +1 -compensator of µ H is � � N δ l L ( t − , T l ) � � � e λ ( t , T l ) x − 1 ν T k +1 ( ds , dx ) = ν T ∗ ( ds , dx ) . + 1 1 + δ l L ( t − , T l ) l = k +1 7 / 31
LIBOR and Forward price model LIBOR models II and the P T k +1 -compensator of µ H is � � N δ l L ( t − , T l ) � � � e λ ( t , T l ) x − 1 ν T k +1 ( ds , dx ) = ν T ∗ ( ds , dx ) . + 1 1 + δ l L ( t − , T l ) l = k +1 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 / 31
LIBOR and Forward price model LIBOR models II and the P T k +1 -compensator of µ H is � � N δ l L ( t − , T l ) � � � e λ ( t , T l ) x − 1 ν T k +1 ( ds , dx ) = ν T ∗ ( ds , dx ) . + 1 1 + δ l L ( t − , T l ) l = k +1 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 / 31
LIBOR and Forward price model LIBOR models III: Remedies 1 “Frozen drift” approximation Brace et al, Schl¨ ogl, Glassermann et al . . . replace the random terms by their deterministic initial values: δ l L ( t − , T l ) δ l L (0 , T l ) 1 + δ l L ( t − , T l ) ≈ (5) 1 + δ l L (0 , T l ) ( + ) deterministic characteristics � closed form pricing ( − ) “ad hoc” approximation, no error estimates, compounded error . . . 2 Strong Taylor approximation approximate the LIBOR rates in the drift/compensator by L ( t , T l ) ≈ L (0 , T l ) + Y ( t , T l ) + (6) where Y is the (scaled) exponential transform of H ( Y = L og e H ) theoretical foundation, error estimates, simpler equations for MC Siopacha and Teichmann; Hubalek, Papapantoleon & Siopacha 10 / 31
LIBOR and Forward price model Forward price model I (Eberlein & ¨ Ozkan, Kluge) Ansatz: model the forward price as the exponential of a semimartingale H : �� t � t � λ ( s , T k ) dH T k +1 F ( t , T k ) = F (0 , T k ) exp b ( s , T k ) ds + , (7) s 0 0 where b ( s , T k ) ensures that F ( · , T k ) = 1 + δ L ( · , T k ) ∈ M ( P T k +1 ). H has the P T k +1 -canonical decomposition � t � t � √ c s dW T k +1 x ( µ H − ν T k +1 )( ds , dx ) , H T k +1 = + (8) s t 0 0 R where the P T k +1 -Brownian motion is � � � t N √ c s ds , � W T k +1 = W T ∗ − λ ( t , T l ) (9) t t 0 l = k +1 11 / 31
LIBOR and Forward price model Forward price model II and the P T k +1 -compensator of µ H is � � N � ν T k +1 ( ds , dx ) = exp ν T ∗ ( ds , dx ) . x λ ( t , T l ) l = k +1 Consequences: 1 the model structure is preserved; 2 caps, swaptions and multi-LIBOR products priced in closed form. So what is wrong? 12 / 31
LIBOR and Forward price model Forward price model II and the P T k +1 -compensator of µ H is � � N � ν T k +1 ( ds , dx ) = exp ν T ∗ ( ds , dx ) . x λ ( t , T l ) l = k +1 Consequences: 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! 13 / 31
LIBOR and Forward price model Forward price model II and the P T k +1 -compensator of µ H is � � N � ν T k +1 ( ds , dx ) = exp ν T ∗ ( ds , dx ) . x λ ( t , T l ) l = k +1 Consequences: 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 14 / 31
Affine processes Affine processes I An affine process in the spirit of Duffie et al (henceforth DFS) is a time-homogeneous, stochastically continuous, Markov process X with � 0 × R n ⊆ R d , with X 0 = x ∈ D , such that the state space D = R m moment generating function has the form � � E x [exp � u , X t � ] = exp φ t ( u ) + � ψ t ( u ) , x � ; (10) for all u ∈ C d or R d where the expectation is finite. We assume that X is regular (and conservative). Lemma (Flow property) The functions φ and ψ satisfy the flow equations : φ t + s ( u ) = φ t ( u ) + φ s ( ψ t ( u )) (11) ψ t + s ( u ) = ψ s ( ψ t ( u )) for all suitable u and t , s > 0 . 15 / 31
Affine processes Affine processes II Theorem (DFS, Main characterization) If X is a regular affine process, then φ and ψ solve the generalized Riccati equations ∂ ∂ t φ t ( u ) = F ( ψ t ( u )) , φ 0 ( u ) = 0 (12) ∂ ∂ t ψ t ( u ) = R ( ψ t ( u )) , ψ 0 ( u ) = u , (13) where F and R are defined via F ( u ) = ∂ � t =0 φ t ( u ) (14) � ∂ t R ( u ) = ∂ � t =0 ψ t ( u ) (15) � ∂ t Moreover, . . . 16 / 31
Affine processes Affine processes III Theorem (continued) . . . the functions F and R are of L´ evy–Khintchine form: � � a � � � e � z , u � − 1 − � u , h ( z ) � F ( u ) = � b , u � − c + 2 u , u + m ( dz ) D � � α i � � � e � z , u � − 1 − � u , h ( z ) � R i ( u ) = � β i , u � − γ i + 2 u , u + µ i ( dz ) D where ( a , b , c , m , α i , β i , γ i , µ i ) 1 ≤ i ≤ d are admissible parameters. X is a Feller process, the generator A has affine form, the semimartingale characteristics (conservative case) have affine form, etc. Conversely, to each set of admissible parameters corresponds a regular affine process on D with generator A . 17 / 31
Affine processes Affine processes IV 1 Affine processes on R : the admissibility conditions yield � F ( u ) = − c + bu + a 2 u 2 + � e zu − 1 − uh ( z ) � m ( dz ) R R ( u ) = β u , for a , c ∈ 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 � � e zu − 1 � F ( u ) = − c + bu + m ( dz ) D � R ( u ) = − γ + β u + α 2 u 2 + � e zu − 1 − uh ( z ) � µ ( dz ) , D for b , c , α, γ ∈ R � 0 and β ∈ R . There exist affine process on R � 0 which are not OU process. 18 / 31
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