LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives

Weierstrass Institute for Applied Analysis and Stochastics LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives John Schoenmakers 9th Summer School in Mathematical Finance Cape Town, 18-20 February 2016 Mohrenstrasse 39 10117 Berlin Germany Tel. +49 30 20372 0 www.wias-berlin.de 18-20.02.2016

Table of contents 1 Crash course classical LIBOR modeling LIBOR dynamics Pricing LIBOR derivatives Caps and Caplets Swaps and Swaptions 2 Stochastic volatility LIBOR models with displacement Single stochastic volatility LIBOR model with displacement LIBOR model with expiry-wise stochastic vol. and displacement LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 2 (26)

Crash course classical LIBOR modeling Time tenor structure T : 0 = T 0 < T 1 < < T N corresponding zero bond processes B i (t), 0 t T i, i = 1,...,N with B i (T i ) = 1 a.s. F i (t) is the at time t contracted effective forward rate over the period [T i 1,T i ] which has to be paid (settled) at T i : ( ) Bi 1 (t) 1 F i (t) = B i (t) 1 with δ i := T i T i 1 (1) δ i LIBOR dynamics to be derived from an arbitrage free bond system of the form: db i B i = µ i dt + η i dw η i R m, W R m, with b.c. B i (T i ) = 1 in the so called real world measure P (say). LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 3 (26)

Crash course classical LIBOR modeling, cont. The solution has the following representation (Check with Ito s formula!): [ t B i (t) = B i (0)exp µ i ds 1 t t ] η i 2 ds + η i dw. 0 2 0 0 Let us sort out a particular bond B k. Then the no-arbitrage principle (NAP) dictates that: There must exist an equivalent measure P (k) P such that all ratios B i /B k i = 1,...,N are martingales with respect to P (k). LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 4 (26)

Crash course classical LIBOR modeling, cont. Theorem: The NAP is fulfilled when there exists a scalar process r and a so called market price of risk process λ in R m such that µ i = r + η i λ almost surely, (2) and where the µ i and η i satisfy certain integrability conditions. LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 5 (26)

Crash course classical LIBOR modeling, cont. Sketch of proof: By Straightforward Itô calculus we have Lemma For Ito processes It holds that d(x/y ) X/Y dx X = µ X dt + η X dw dy Y = µ Y dt + η Y dw = (µ X µ Y η Y (η X η Y ))dt + (η X η Y ) dw LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 6 (26)

Crash course classical LIBOR modeling, cont. It then follows by Lemma 1 that d B i B k = B i B k ((µ i µ k η k (η i η k ))dt + (η i η k ) dw) by (2) = B i (((η i η B k ) λ η k (η i η k ))dt + (η i η k ) dw) k ( ) (λ η k )dt + dw }{{} = B i B k (η i η k ) equivalent to a st. Brownian motion W (k) under P (k) = B i B k (η i η k ) dw (k) (3) since on a finite time interval the process (λ η k )ds +W t is equivalent to a standard m-dimensional Brownian motion W (k). Thus, (due to the integrability conditions) the B i /B k are martingales under P (k). Remark Condition (2) is also necessary for NAP (however the proof is harder). LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 7 (26)

Crash course classical LIBOR modeling: LIBOR dynamics By considering (3) for i = k 1 we obtain from (1) df k = 1 d B k 1 δ k B k (1) again = = 1 δ k B k 1 B k with the newly introduced volatility process and where (η k 1 η k ) dw (k) 1 δ k (1 + δ k F k )(η k 1 η k ) dw (k) =: F k γ k dw (k) (4) γ k := 1 + δ kf k δ k F k (η k 1 η k ), dw (k) = (λ η k )dt + dw. (5) In particular, F k is a martingale under P (k) and, moreover, if one takes a model with deterministic s γ k (s), F k is a lognormal process under P (k) (!) LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 8 (26)

LIBOR dynamics, cont. Alternatively, if we consider (3) for k = N we get in particular that dw (N) = (λ η N )dt + dw by (5) = dw (k) + (η k η N )dt = dw (k) + N ( ) η j 1 η j dt j=k+1 = dw (k) N δ j F j + γ j dt. (6) j=k+1 1 + δ j F j We so arrive at the dynamics of all LIBORs F k, i = 2,...,N, under the terminal bond measure P (N) by combining (4) and (6): df k = F k γ k dw (k) = ( ) = F k γ k dw (N) N δ j F j γ j dt j=k+1 1 + δ j F j N δ j F k F j = γ j=k+1 1 + δ j F k γ j dt + F k γ k dw (N), j LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 9 (26)

LIBOR dynamics, cont... also written as df k F k N δ j F j = γ j=k+1 1 + δ j F k γ j dt + γ k dw (N), k = 2,...,N, j the general (Brownian motion or Wiener based) LIBOR model in the terminal measure P (N). Key of LIBOR modeling: In stead of specifying the dynamics of the bonds B i directly via µ i and η i, one specifies the LIBOR dynamics through the γ i, assuming that there exists some consistent bond system on the background!! LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 10 (26)

Pricing LIBOR derivatives Pricing a cash-flow C Tk at tenor T k : Choose a bond numéraire B l that is still alive at time T k, so k l N. Let P (l) be a measure such that all processes t B i (t)/b l (t) are martingales on the interval 0 t T i T l. Then, no-arbitrage pricing theory, the price at time t < T k is given by [ ] C t := B k (t)e (k) CTk = B B k (T k ) k (t)e (k) [C Tk ] in the measure P (k), [ ] = B l (t)e (l) CTk in the measure P (l), k l N. (7) B l (T k ) LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 11 (26)

Pricing LIBOR derivatives: Caps and Caplets An interest rate cap with strike level K with respect to a loan over period [T p,t q ] yields effectively cash-flows (F i (T i 1 ) K) + δ i at the dates T i, i = p + 1,...,q. Valuation: (starting in the terminal measure): Cap p,q (0) = B N (0)E (N) q i=p+1 (F i (T i 1 ) K) + δ i B N (T i ) q = B N (0)E (N) (F i(t i 1 ) K) + δ i i=p+1 B N (T i ) by measure transformation (cf. (7) q = B i (0)E (i) (F i(t i 1 ) K) + δ i i=p+1 B i (T i ) q = i=p+1 B i (0)E (i) (F i (T i 1 ) K) + δ i q =: i=p+1 Caplet i (0) i.e. SUM of caplets. LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 12 (26)

Caps and Caplets, cont. Most important feature: When the γ k are deterministic, then by (4) [ F k (T k 1 ) = F k (0)exp 1 Tk 1 γ 2 k 2 Tk 1 ] (s)ds + γ k (s) dw (k) (s) 0 0 and since W (k) is a Wiener process under P (k), lnf k (T k 1 ) distribution = lnf k (0) 1 Tk 1 γ 2 k 2 Tk 1 (s)ds + ζ γ k 2 ds 0 0 with ζ N (0,1) (standard normal scalar r.v.) Usually one then defines the Black volatility σk Black 1 := T k 1 Tk 1 0 γ k 2 ds to write lnf k (T k 1 ) distribution = lnf k (0) 1 ( ) 2 σk Black Tk 1 + σk Black ζ T 2 k 1 and each Caplet price follows from Black s 76 formula! This is the Black-Scholes formula with interest rate parameter zero. LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 13 (26)

Caps and Caplets, Black s 76 Let B (S 0,r,T,K,σ) be the standard BS formula for a stock call option with strike K, maturity T, interest rate r, initial stock price S, and volatility σ. We recall that ( ln S 0 K B (S 0,r,T,K,σ) = S 0 N + ( r + σ 2 /2 ) ) ( T ln σ Ke rt S 0 K N + ( r σ 2 /2 ) ) T T σ. T (8) Then ( ) Caplet k (0) = δ k B k (0)B F k (0),0,T k 1,K,σk Black. LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 14 (26)

Swaps and Swaptions Swap contract: Contract which exchanges floating LIBOR against a fixed rate K over a period [T p,t q ]. Net value at t < T p : Swap p,q (t) = B p (t) B q (t) K q δ j B j (t) j=p+1 The swap rate is that rate K which makes the contract neutral: where S p,q (t) := is called the annuity numéraire B p(t) B q (t) q j=p+1 δ jb j (t) =: B p(t) B q (t) A p,q (t) A p,q (t) = q δ j B j (t) (9) j=p+1 LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 15 (26)

Swaps and Swaptions, cont. Swaption contract over [T p,t q ]: Contract that gives the right to enter into a swap at time T p with strike K over de period [T p,t q ]. vfill Valuation: The swap will be exercised at T p only if its value is non-negative, ( ) Swaption p,q (0) = B N (0)E (N) Swapp,q (T p ) + B N (T p ) = B N (0)E (N) ( Bp (T p ) B q (T p ) K q j=p+1 δ jb j (T p ) B N (T p ) = B N (0)E (N) ( B p (T p ) B N (T p ) B q(t p ) B N (T p ) K q B j (T p ) δ j j=p+1 B n (T p ) ) + ) + where B k (T p ) B N (T p ) = B k(t p ) B k+1 (T p ) BN 1(T p ) B N (T p ) = N ( 1 + δr F r (T p ) ), p k < N. r=k+1 LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 16 (26)

Swaps and Swaptions, cont. So Swaption p,q (0) = ( N B N (0)E (N) ( 1 + δr F r (T p ) ) N ( 1 + δr F r (T p ) ) q N ( K δ j 1 + δr F r (T p ) )) + r=p+1 r=q+1 j=p+1 r= j+1 Rather complicated payoff in terms of the LIBORs! = Monte Carlo simulation gives an unbiased estimation! = Unfortunately, too slow for calibration purposes! LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 17 (26)

Swaps and Swaptions, cont. Alternative representation: Annuity measure P p,q, is the measure such that all B i /A p,q with A p,q defined by (9) are P p,q -martingales: ( Bp Swaption p,q (0) = B N (0)E (N) (T p ) B q (T p ) K q j=p+1 δ ) + jb j (T p ) B N (T p ) ( = B N (0)E (N) A p,q (T p ) S ) p(t p,q ) K + B N (T p ) transformation to annuity numéraire measure = A p,q (0)E p,q ( S p (T p,q ) K ) +. Note that S p,q (t) = B p(t) B q (t) A p,q (t) is a martingale under the annuity measure P p,q. Thus, for some volatility process σ p,q (t), we have a swap model where W p,q is Brownian motion under P p,q. ds p,q S p,q = σ p,q dw p,q, (10) LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 18 (26)

Swaps and Swaptions, cont. Swap Market Model: Swap rate model (10) with deterministic volatility process t σ p,q (t) THUS: In a Swap Market Model we have, in distribution lns p,q (T p ) = lns p,q (T p ) 1 ( ) 2 σp,q Black Tp + σp,q Black ζ T p 2 with ζ N (0,1) (standard normal scalar r.v.), and σp,q Black 1 Tp := σ p,q 2 ds. T p 0 That is, we have Black s 76 formula again: (cf. caplet derivation) ( ) Swaption p,q (0) = A p,q (0)B S p,q (0),0,T p,k,σp,q Black, where and B given by (8). LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 19 (26)

Swaps and Swaptions, cont. Problem: A LIBOR Market Model with γ k deterministic would imply that σ p,q is not deterministic and the other way around. So LIBOR Market Models and Swap Market Models are mutually inconsistent! Although... they are consistent in a good approximation... = Swaption approximation formulas for LIBOR market models based on frozen drifts (Hull & White, Rebonato, Jamshidian, Brigo & Mercurio,... Important drawback of Market Models based on deterministic volatility: They cannot capture volatility smiles! LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 20 (26)

Single stochastic volatility LIBOR model with displacement Given a pair of independent Brownian motions W = (W N,Ŵ N ) R m R m a stochastic volatility LIBOR model with displacement is defined by specifying the noise Γ j (dw N,dŴ N ) = F j + α j V β j dw N + F j + α j γ j dŵ N F j F j connected with a single volatility process dv = κ (ϑ V )dt + V ( ) σ dw N + σ dw, where W is an m-dimensional Wiener process independent of (W N,Ŵ N ). LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 21 (26)

Single stochastic volatility LIBOR model with displacement Hence in the terminal measure we have by some algebra df j F j = Γ j (dw N,dŴ N N δ ) l F l Γ l= j+1 1 + δ l F l Γ j dt l = F j + α j V β j dw N + F j + α j γ j dŵ N F j F j N δ l F l F l + α l l= j+1 1 + δ l F l F l N δ l F l F l + α l l= j+1 1 + δ l F l F l V βl Fj + α j V β j dt F j γ l Fj + α j γ j dt, F j yielding the Single stochastic volatility LIBOR model with displacement: df N j = F j + α j l= j+1 δ l (F l + α l ) 1 + δ l F l ( γl γ j +V β l β j ) dt + V β j dw N + γ j dŵ N (11) LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 22 (26)

Single stochastic volatility, some facts on cap/swaptions pricing After freezing the drift, the log(f j + α j ) have affine dynamics For swaptions an affine approximation for log(s p,q + α p,q ) will be derived As a result: cap(let)s and swaptions can be priced approximately by Fourier based methods The details will be given later in the context of multi-curve modeling LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 23 (26)

LIBOR model with expiry-wise stochastic vol. and displacement Ladkau, S., and Zhang (2013) study an extension where each LIBOR has his own volatility df N j = F j + α j l= j+1 in terms of the terminal measure δ l (F l + α l ) 1 + δ l F l ( γ j γ l + β j β l Vj V l ) dt + V j β j dw N + γ j dŵ N β, γ are deterministic, drift is a direct consequence of no-arbitrage principles, and dv j = κ j (θ j V j )dt + ( V j σ j dw N + σ j dw N) LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 24 (26)

LIBOR model expiry-wise stochastic vol. and displacement, cont. Advantage: Much more flexibility for calibration Drawback: Cap and swaption approximation formulas require beyond the usual LIBOR freezing additional approximations in the drift: V l V = l EV j V j EV l EV = l θ l V j V j EV j V j EV j EV j θ j so that V l V j θl θ j V j yielding an affine structure after freezing. Relevance: If the calibration set of caps/swaptions is relatively large, the better calibration fit justifies the somewhat less accurate cap/swaption approximation procedures LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 25 (26)

Calibration to cap/strike matrix for the single stoch. vol. model Cap(0; 2) Cap(0; 10) Cap(0; 24) Price 0.00 0.05 0.10 0.15 0.20 Price 0.00 0.05 0.10 0.15 0.20 Price 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 Strike 0.00 0.05 0.10 0.15 Strike 0.00 0.05 0.10 0.15 Strike LIBOR models, multi-curve extensions, callable structured derivatives 18-20.02.2016 Page 26 (26)