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

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1 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, February 2016 Mohrenstrasse Berlin Germany Tel

2 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 Page 2 (26)

3 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 Page 3 (26)

4 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 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 Page 4 (26)

5 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 Page 5 (26)

6 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 Page 6 (26)

7 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 Page 7 (26)

8 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 Page 8 (26)

9 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 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 δ j F j N δ j F k F j = γ j=k δ j F k γ j dt + F k γ k dw (N), j LIBOR models, multi-curve extensions, callable structured derivatives Page 9 (26)

10 LIBOR dynamics, cont... also written as df k F k N δ j F j = γ j=k 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 Page 10 (26)

11 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 Page 11 (26)

12 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 Page 12 (26)

13 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 Page 13 (26)

14 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 Page 14 (26)

15 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 Page 15 (26)

16 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 Page 16 (26)

17 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 Page 17 (26)

18 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 Page 18 (26)

19 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 Page 19 (26)

20 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 Page 20 (26)

21 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 Page 21 (26)

22 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 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 δ l F l F l N δ l F l F l + α l l= j 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 Page 22 (26)

23 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 Page 23 (26)

24 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 Page 24 (26)

25 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 Page 25 (26)

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

Introduction to Financial Mathematics

Introduction to Financial Mathematics Department of Mathematics University of Michigan November 7, 2008 My Information E-mail address: marymorj (at) umich.edu Financial work experience includes 2 years in public finance investment banking