LIBOR Market Models with Stochastic Basis

Transcription

1 LIBOR Market Models with Stochastic Basis Fabio Mercurio Bloomberg, New York Swissquote Conference on Interest Rate and Credit Risk Lausanne, 28 October 2010

2 Stylized facts Before the credit crunch of 2007, interest rates in the market showed typical textbook behavior. For instance: A (canonical) floating rate bond is worth par at inception. The forward rate implied by two deposits coincides with the corresponding FRA rate. Compounding two consecutive 3m forward LIBOR rates yields the corresponding 6m forward LIBOR rate. These properties allowed one to construct a well-defined zero-coupon curve. Then August 2007 arrived, and our convictions began to waver: The liquidity crisis widened the basis between previously-near rates. Consider the following graphs...

3 USD Market quotes Overnight-indexed-swap rates and LIBORs USD 3m OIS rates vs 3m Depo rates

4 USD Market quotes Overnight-indexed-swap rates and LIBORs USD 6m OIS rates vs 6m Depo rates

5 USD Market quotes FRA rates and OIS forward rates USD 3x6 FRA vs 3x6 fwd OIS

6 USD Market quotes Interest rate swaps with different frequencies USD 5y swaps: 1m vs 3m

7 EUR Market quotes Overnight-indexed-swap rates and LIBORs EUR 3m OIS rates vs 3m Depo rates

8 EUR Market quotes Overnight-indexed-swap rates and LIBORs EUR 6m OIS rates vs 6m Depo rates

9 EUR Market quotes FRA rates and OIS forward rates EUR 3x6 FRA vs 3x6 fwd OIS

10 EUR Market quotes FRA rates and OIS forward rates EUR 6x12 FRA vs 6x12 fwd OIS

11 EUR Market quotes Interest rate swaps with different frequencies EUR 5y swaps: 3m vs 6m

12 Market segmentation of rates EUR market rates (as of 26 March 2009) (Figure graciously provided by Kirill Levin)

13 Market segmentation of rates Banks construct different zero-coupon curves for each market rate tenor: 1m, 3m, 6m, 1y,... Example: The EUR 6m curve

14 The discount curve We take the OIS zero-coupon curve, stripped from market OIS swap rates, as the discount curve: T P D (0, T ) = P OIS (0, T ) The rationale behind this is that in the interbank derivatives market, a collateral agreement (CSA) is often negotiated between the two counterparties. We assume here that the collateral is revalued daily at a rate equal to the overnight rate. If the CSA reduces the counterparty risk to zero, it makes sense to discount with OIS rates since they can be regarded as risk-free. The OIS curve can be stripped from OIS swap rates using standard (single-curve) bootstrapping methods.

15 A new definition of forward LIBOR rate: The FRA rate Given times t T 1 < T 2, the time-t FRA rate FRA(t; T 1, T 2 ) is defined as the fixed rate to be exchanged at time T 2 for the LIBOR rate L(T 1, T 2 ) so that the swap has zero value at time t. L(T 1, T 2 ) K t T 1 T 2 Under the T 2 -forward measure Q T 2 D, we immediately have FRA(t; T 1, T 2 ) = E T [ ] 2 D L(T1, T 2 ) F t, In the classic single-curve valuation, FRA rates and corresponding discount-curve forward rates coincide: [ ] 1 PD (t, T 1 ) FRA(t; T 1, T 2 ) = F D (t; T 1, T 2 ) = T 2 T 1 P D (t, T 2 ) 1

16 A new definition of forward LIBOR rate: The FRA rate In fact, in the single-curve case L(T 1, T 2 ) is defined by the classic relation [ ] 1 1 L(T 1, T 2 ) = T 2 T 1 P D (T 1, T 2 ) 1 = F D (T 1 ; T 1, T 2 ), so that E T [ ] 2 D L(T1, T 2 ) F t = FRA(t; T1, T 2 ) = E T [ ] 2 D FD (T 1 ; T 1, T 2 ) F t = FD (t; T 1, T 2 ) In our dual-curve setting, however, implying that L(T 1, T 2 ) F D (T 1 ; T 1, T 2 ) = L OIS (T 1, T 2 ) FRA(t; T 1, T 2 ) F D (t; T 1, T 2 )

17 A new definition of forward LIBOR rate: The FRA rate The FRA rate above is the natural generalization of a forward rate to the dual-curve case. In fact: 1. The FRA rate coincides with the classically-defined forward rate. 2. At its reset time T 1, the FRA rate FRA(T 1 ; T 1, T 2 ) coincides with the LIBOR rate L(T 1, T 2 ). 3. The FRA rate is a martingale under the corresponding forward measure. 4. Its time-0 value FRA(0; T 1, T 2 ) can be stripped from market data. These properties will prove to be very convenient when pricing swaps and options on LIBOR/swap rates.

18 The valuation of interest rate swaps (IRSs) (under the assumption of distinct forward and discount curves) Given times T a,..., T b, consider an IRS whose floating leg pays at each T k the LIBOR rate with tenor T k T k 1, which is set (in advance) at T k 1, i.e. τ k L(T k 1, T k ) where τ k denotes the year fraction. The time-t value of this payoff is: FL(t; T k 1, T k ) = τ k P D (t, T k )E T [ ] k D L(Tk 1, T k ) F t =: τ k P D (t, T k )L k (t) where L k (t) := FRA(t; T k 1, T k ). The swap s fixed leg is assumed to pay the fixed rate K on dates Tc S,..., Td S, with year fractions τ S j.

19 The valuation of interest rate swaps (cont d) The IRS value to the fixed-rate payer is given by IRS(t, K ) = b k=a+1 τ k P D (t, T k )L k (t) K d j=c+1 τ S j P D (t, T S j ) We can then calculate the corresponding forward swap rate as the fixed rate K that makes the IRS value equal to zero at time t. At t = 0, we get: S 0,b,0,d (0) = b k=1 τ kp D (0, T k )L k (0) d j=1 τ S j P D (0, T S j ) where L 1 (0) is the first floating payment (known at time 0).

20 The valuation of interest rate swaps (cont d) In practice, this swap rate formula can be used to bootstrap the rates L k (0). The bootstrapped L k (0) can then be used to price other swaps based on the given tenor. Swap rate Formulas OLD b k=1 τ k P(0,T k )F k (0) d j=1 τ j S P(0,Tj S ) = 1 P(0,T S d ) d j=1 τ S j P(0,T S j ) NEW b k=1 τ k P D (0,T k )L k (0) d j=1 τ j S P D (0,Tj S )

21 The valuation of caplets Let us consider a caplet paying out at time T k τ k [L(T k 1, T k ) K ] +. The caplet price at time t is given by: Cplt(t, K ; T k 1, T k ) = τ k P D (t, T k )E T { k D [L(Tk 1, T k ) K ] + } F t = τ k P D (t, T k )E T { k D [Lk (T k 1 ) K ] + } F t The FRA rate L k (t) = E T k D [L(T k 1, T k ) F t ] is, by definition, a martingale under Q T k D. Assume that L k follows a (driftless) geometric Brownian motion under Q T k D. Straightforward calculations lead to a (modified) Black formula for caplets.

22 The valuation of swaptions A payer swaption gives the right to enter at time T a = T S c an IRS with payment times for the floating and fixed legs given by T a+1,..., T b and T S c+1,..., T S d, respectively. Therefore, the swaption payoff at time T a = T S c [ Sa,b,c,d (T a ) K ] + d where K is the fixed rate and j=c+1 is τ S j P D (T S c, T S j ), S a,b,c,d (t) = b k=a+1 τ kp D (t, T k )L k (t) d j=c+1 τ S j P D (t, T S j ) Assume that S a,b,c,d is a (lognormal) martingale under the associated swap measure. We thus obtain a (modified) Black formula for swaptions.

23 The new market formulas for caps and swaptions Type Formulas OLD Cplt τ k P(t, T k ) Bl ( K, F k (t), v k Tk 1 t ) NEW Cplt τ k P D (t, T k ) Bl ( K, L k (t), v k Tk 1 t ) OLD PS NEW PS d j=c+1 τ S j P(t, T S j ) Bl ( K, S OLD (t), ν T a t ) d j=c+1 τ S j P D (t, T S j ) Bl ( K, S a,b,c,d (t), ν T a t )

24 The multi-curve LIBOR Market Model (LMM) In the classic (single-curve) LMM, one models the joint evolution of a set of consecutive forward LIBOR rates. What about our multi-curve case? When pricing a payoff depending on same-tenor LIBOR rates, it is convenient to model the FRA rates L k. This choice is also convenient in the case of a swap-rate dependent payoff. In fact, we can write: S a,b,c,d (t) = b k=a+1 τ kp D (t, T k )L k (t) d j=c+1 τ S j P D (t, T S j ) b = ω k (t)l k (t) k=a+1 ω k (t) := τ k P D (t, T k ) d j=c+1 τ S j P D (t, T S j ) But, is the modeling of FRA rates enough?

25 The multi-curve LIBOR Market Model (LMM) Alternative formulations In fact, we also need to model the OIS forward rates, k = 1,..., M: F k (t) := F D (t; T k 1, T k ) = 1 [ ] PD (t, T k 1 ) 1 τ k P D (t, T k ) Denote by S k (t) the spread S k (t) := L k (t) F k (t) By definition, both L k and F k are martingales under the forward measure Q T k D, and thus their difference S k is as well. The LMM can be extended to the multi-curve case in three different ways by: 1. Modeling the joint evolution of rates L k and F k. 2. Modeling the joint evolution of rates L k and spreads S k. 3. Modeling the joint evolution of rates F k and spreads S k.

26 A first McLMM Modeling rates F k and L k Let us consider a set of times T = {0 < T 0,..., T M } compatible with a given tenor. We assume that each rate L k (t) evolves under Q T k D according to dl k (t) = σ k (t)l k (t) dz k (t), t T k 1 Likewise, we assume that df k (t) = σ D k (t)f k(t) dz D k (t), t T k 1 The drift of X {L k, F k } under Q T j D is equal to Drift(X; Q T j D ) = d X, ln(p D(, T k )/P D (, T j )) t, dt

27 A first McLMM Dynamics under a general forward measure Proposition. The dynamics of L k and F k under Q T j D are: [ k ρ L,F k,h dl k (t) = σ k (t)l k (t) τ h DσD h (t)f ] h(t) 1 + τ h=j+1 h D j < k : F dt + dz j k h(t) (t) [ k df k (t) = σk D(t)F ρ D,D k,h k(t) τ h DσD h (t)f ] h(t) 1 + τ D h=j+1 h F dt + dz j,d k (t) h(t) { dl k (t) = σ k (t)l k (t) dz j k j = k : (t) df k (t) = σk D(t)F k(t) dz j,d k (t) [ j ρ L,F k,h dl k (t) = σ k (t)l k (t) τ h DσD h (t)f ] h(t) 1 + τ h=k+1 h D j > k : F dt + dz j k h(t) (t) [ j df k (t) = σk D(t)F ρ D,D k,h k(t) τ h DσD h (t)f ] h(t) 1 + τh DF dt + dz j,d k (t) h(t) h=k+1

28 A first McLMM Dynamics under the spot LIBOR measure The spot LIBOR measure QD T associated to times T = {T 0,..., T M } is the measure whose numeraire is BD T (t) = P D (t, T β(t) 1 ) β(t) 1 j=0 P D (T j 1, T j ), where β(t) = m if T m 2 < t T m 1, m 1. Application of the change-of numeraire technique leads to: [ k dl k (t) = σ k (t)l k (t) h=β(t) [ k df k (t) = σk D (t)f k(t) h=β(t) ρ L,F k,h τ h DσD h (t)f ] h(t) 1 + τh DF dt + dzk d h(t) (t) ρ D,D k,h τ D h σd h (t)f h(t) 1 + τ D h F h(t) ] dt + dz d,d k (t)

29 A first McLMM The pricing of caplets The pricing of caplets in our multi-curve lognormal LMM is straightforward. We get: Cplt(t, K ; T k 1, T k ) = τ k P D (t, T k ) Bl(K, L k (t), v k (t)) where Tk 1 v k (t) := t σ k (u) 2 du As expected, this formula is analogous to that obtained in the single-curve lognormal LMM. Here, we just have to replace the old forward rates with the corresponding FRA rates and use the discount factors of the OIS curve.

30 A first McLMM The pricing of swaptions Our objective is to derive an analytical approximation for the implied volatility of swaptions. To this end, we recall that S a,b,c,d (t) = b k=a+1 ω k (t)l k (t), ω k (t) = τ k P D (t, T k ) d j=c+1 τ S j P D (t, T S j ) Contrary to the single-curve case, the weights are not functions of the FRA rates only, since they also depend on discount factors calculated on the OIS curve. Therefore we can not write, under Q c,d D, ds a,b,c,d (t) = b k=a+1 S a,b,c,d (t) L k (t) σ k (t)l k (t) dz c,d k (t).

31 A first McLMM The pricing of swaptions However, we can resort to a standard approximation technique and freeze weights ω k at their time-0 value: S a,b,c,d (t) b k=a+1 ω k (0)L k (t), thus also freezing the dependence of S a,b,c,d on rates F D h. Hence, we can write: ds a,b,c,d (t) b k=a+1 ω k (0)σ k (t)l k (t) dz c,d k (t). Like in the classic single-curve LMM, we then: Match instantaneous quadratic variations Freeze FRA and swap rates at their time-0 value

32 A first McLMM The pricing of swaptions This immediately leads to the following (payer) swaption price at time 0: PS(0, K ; T a+1,..., T b, Tc+1 S,..., T d S ) d = τj S P D (0, Tj S ) Bl ( ) K, S a,b,c,d (0), V a,b,c,d, j=c+1 where the swaption volatility (multiplied by T a ) is given by V a,b,c,d = b ω h (0)ω k (0)L h (0)L k (0)ρ Ta h,k (S a,b,c,d (0)) 2 σ h (t)σ k (t) dt h,k=a+1 Again, this formula is analogous in structure to that obtained in the single-curve lognormal LMM. 0

33 A second McLMM A general framework for the single-tenor case Let us fix a given tenor x and consider a time structure T = {0 < T x 0,..., T x M x } compatible with x. Let us define forward OIS rates by Fk x (t) := F D(t; Tk 1 x, T k x ) = 1 [ PD (t, Tk 1 x ) ] τk x P D (t, Tk x) 1, k = 1,..., M x, where τk x is the year fraction for the interval (T k 1 x, T k x ], and basis spreads by S x k (t) := Lx k (t) F x k (t), k = 1,..., M x. By definition, both L x k and F k x are martingales under the forward measure Q T k x D. Hence, their difference S x k is a QT x k D -martingale as well.

34 A second McLMM A general framework for the single-tenor case We define the joint evolution of rates F k and spreads S k under the spot LIBOR measure QD T, whose numeraire is BD T (t) = P D (t, Tβ(t) 1 x ) β(t) 1 j=0 P D (Tj 1 x, T j x ), where β(t) = m if Tm 2 x < t T m 1 x, m 1, and T 1 x := 0. Our single-tenor framework is based on assuming that, under QD T, OIS rates follow general SLV processes: dfk x (t) = φf k (t, F k x (t))ψf k (V F (t)) [ k ] τh x ρ h,kφ F h (t, F h x(t))ψf h (V F (t)) 1 + τh xf h x(t) dt + dzk T (t) h=β(t) dv F (t) = a F (t, V F (t)) dt + b F (t, V F (t)) dw T (t)

35 where A second McLMM A general framework for the single-tenor case φ F k, ψf k, af and b F are deterministic functions of their respective arguments Z T = {Z1 T,..., Z M T x } is an M x -dimensional QD T -Brownian motion with instantaneous correlation matrix (ρ k,j ) k,j=1,...,mx W T is a QD T -Brownian motion whose instantaneous correlation with Zk T is denoted by ρx k for each k. The stochastic volatility V F is assumed to be a process common to all OIS forward rates. We assume that V F (0) = 1. Generalizations can be considered where each rate F x k has a different volatility process.

36 A second McLMM A general framework for the single-tenor case We then assume that also the spreads Sk x follow SLV processes. For computational convenience, we assume that spreads and their volatilities are independent of OIS rates. This implies that each Sk x is a QT D -martingale as well. Finally, the global correlation matrix that includes all cross correlations is assumed to be positive semidefinite. Remark. Several are the examples of dynamics that can be considered. Obvious choices include combinations (and permutations) of geometric Brownian motions and of the stochastic-volatility models of Hagan et al. (2002) and Heston (1993). However, the discussion that follows is rather general and requires no dynamics specification.

37 A second McLMM Caplet pricing Let us consider the x-tenor caplet paying out at time T x k τ x k [Lx k (T x k 1 ) K ]+ Our assumptions on the discount curve imply that the caplet price at time t is given by Cplt(t, K ; Tk 1 x, T k x ) = τk x P D(t, Tk x )E T k x { D [L x k (Tk 1 x ) K } ]+ F t = τk x P D(t, Tk x )E T k x { D [F x k (Tk 1 x ) + Sx k (T k 1 x ) K } ]+ F t Assume we explicitly know the Q T x k D -densities f S x k (T x k 1 ) and f F x k (T x k 1 ) (conditional on F t ) of S x k (T x k 1 ) and F x k (T x k 1 ), respectively, and/or the associated caplet prices.

38 A second McLMM Caplet pricing Thanks to the independence of the random variables Fk x(t k 1 x ) and Sx k (T k 1 x ) we equivalently have: Cplt(t, K ; T x k 1, T x k ) = = τ x k P D(t, T x k ) + + E T x k D E T x k D { [F x k (T x k 1 ) (K z)]+ F t } fs x k (T x k 1 ) (z) dz { [S x k (T x k 1 ) (K z)]+ F t } ff x k (T x k 1 ) (z) dz One may use the first or the second formula depending on the chosen dynamics for F x k and Sx k. To calculate the caplet price one needs to derive the dynamics of Fk x and V F under the forward measure Q T k x D. Notice that the Q T k x D -dynamics of Sx k and its volatility are the same as those under QD T.

39 A second McLMM Caplet pricing The dynamics of F x k and V F under Q T x k D df x k (t) = φf k (t, F x k (t))ψf k (V F (t)) dz k k (t) are given by: dv F (t) = a F (t, V F (t)) dt + b F (t, V F (t)) [ ] k τh x φf h (t, F h x(t))ψf h (V F (t))ρ x h 1 + τh xf h x(t) dt + dw k (t) h=β(t) where Zk k and W k are Q T k x D -Brownian motions. By resorting to standard drift-freezing techniques, one can find tractable approximations of V F for typical choices of a F and b F, which will lead either to an explicit density f F x k (Tk 1 x ) or to an explicit option pricing formula (on Fk x). This, along with the assumed tractability of S x k, will finally allow the calculation of the caplet price.

40 A second McLMM Swaption pricing Let us consider a (payer) swaption, which gives the right to enter at time Ta x = Tc S an interest-rate swap with payment times for the floating and fixed legs given by Ta+1 x,..., T b x and Tc+1 S,..., T d S, respectively, with T b x = T d S and where the fixed rate is K. The swaption payoff at time T x a = T S c [ Sa,b,c,d (T x a ) K ] + d j=c+1 is given by τ S j P D (T S c, T S j ), where the forward swap rate S a,b,c,d (t) is given by S a,b,c,d (t) = b k=a+1 τ k xp D(t, Tk x)lx k (t) d j=c+1 τ. j S P D (t, Tj S )

41 A second McLMM Swaption pricing The swaption payoff is conveniently priced under Q c,d D : PS(t, K ; Ta x,..., Tb x, T c+1 S,..., T d S ) d = τj S P D (t, Tj S ) E c,d { [ D Sa,b,c,d (Ta x ) K ] + } Ft j=c+1 To calculate the last expectation, we set and write: S a,b,c,d (t) = = ω k (t) := b k=a+1 b k=a+1 τ x k P D(t, T x k ) d j=c+1 τ S j P D (t, T S j ) ω k (t)l x k (t) ω k (t)f x k (t) + b k=a+1 ω k (t)s x k (t) =: F(t) + S(t)

42 A second McLMM Swaption pricing The processes S a,b,c,d, F and S are all Q c,d D -martingales. F is equal to the classic single-curve forward swap rate that is defined by OIS discount factors, and whose reset and payment times are given by Tc S,..., Td S. If the dynamics of rates Fk x are sufficiently tractable, we can approximate F(t) by a driftless stochastic-volatility process, F(t), of the same type as that of Fk x. The process S is more complex, since it explicitly depends both on OIS discount factors and on basis spreads. However, we can resort to a standard approximation and freeze the weights ω k at their time-0 value, thus removing the dependence of S on OIS discount factors.

43 A second McLMM Swaption pricing We then assume we can further approximate S with a dynamics S similar to that of Sk x, for instance by matching instantaneous variations. After the approximations just described, the swaption price becomes PS(t, K ; Ta x,..., Tb x, T c+1 S,..., T d S ) d = τj S P D (t, Tj S ) E c,d x D {[ F(T a ) + S(T a x ) K ] + } Ft j=c+1 which can be calculated exactly in the same fashion as the previous caplet price.

44 A second McLMM A tractable class of multi-tenor models Let us consider a time structure T = {0 < T 0,..., T M } and tenors x 1 < x 2 < < x n with associated time structures T x i = {0 < T x i 0,..., T x i M xi }. We assume that each x i is a multiple of the preceding tenor x i 1, and that T xn T x n 1 T x 1 = T. Forward OIS rates are defined, for each tenor x {x 1,..., x n }, by Fk x (t) := F D(t; Tk 1 x, T k x ) = 1 [ PD (t, Tk 1 x ) ] τk x P D (t, Tk x) 1, k = 1,..., M x, and basis spreads are defined by S x k (t) = FRA(t, T x k 1, T x k ) F x k (t) = Lx k (t) F x k (t), k = 1,..., M x. L x k, F x k, Sx k are martingales under the forward measure QT x k D.

45 A second McLMM A tractable class of multi-tenor models We assume that, under the spot LIBOR measure QD T, the OIS forward rates F x 1 k, k = 1,..., M 1, follow shifted-lognormal" stochastic-volatility processes [ df x 1 k (t) = σx 1 1 ] k (t)v F (t) τ x + F x 1 1 k (t) k [ ] V F (t) k h=β(t) ρ h,k σ x 1 h (t) dt + dz T k (t) dv F (t) = a F (t, V F (t)) dt + b F (t, V F (t)) dw T (t) where: For each k, σ x 1 k is a deterministic function; {Z1 T,..., Z M T 1 } is an M 1 -dimensional QD T -Brownian motion with correlations (ρ k,j ) k,j=1,...,m1 ; V F is correlated with every Zk T, dw T (t)dzk T (t) = ρx k dt, and V F (0) = 1.

46 A tractable class of multi-tenor McLMMs The dynamics of forward rates Fk x, for tenors x {x 2,..., x n }, can be obtained by Ito s lemma, noting that Fk x can be written in terms of smaller rates F x 1 k as follows: i k h=i k 1 +1 [1 + τ x 1 h F x 1 for some indices i k 1 and i k. h (t)] = 1 + τ x k F x k (t), We then assume, for each tenor x {x 1,..., x n }, the following one-factor models for the spreads: S x k (t) = Sx k (0)Mx (t), k = 1,..., M x, where, for each x, M x is a (continuous and) positive QD T -martingale independent of rates F k x and of the stochastic volatility V F. Clearly, M x (0) = 1.

47 A tractable class of multi-tenor McLMMs Rate dynamics under the associated forward measure When moving from measure QD T to measure x QT k D, the drift of a (continuous) process X changes according to Drift(X; Q T x k D ) = Drift(X; QT D ) + d X, ln(p D(, T x k )/BT D ( )) t dt Applying Ito s lemma, we get, for each x {x 1,..., x n }, [ 1 ] dfk x (t) = σx k (t)v F (t) + Fk x (t) dz k,x (t) where σk x, x {x 2,..., x n }, is a deterministic function, whose value is determined by σ x 1 h and ρ h,k, and τ x k dv F (t) = V F (t)b F (t, V F (t)) i k h=β(t) k σ x 1 h (t)ρx 1 h + a F (t, V F (t)) dt + b F (t, V F (t)) dw k,x (t) dt

48 A tractable class of multi-tenor McLMMs The above dynamics of Fk x are the simplest stochastic volatility dynamics that are consistent across different tenors x. If 3m-rates follow shifted-lognormal processes with common stochastic volatility, the same type of dynamics (modulo the drift correction in the volatility) is also followed by 6m-rates (under the respective forward measures). This allows us to price simultaneously, with the same type of formula, caps and swaptions with different tenors x. Option prices can then be calculated as suggested before. Swaption formulas can be simplified by noting that: b S(t) = ω k (t)sk x (0)Mx (t) k=a+1 M x (t) b k=a+1 ω k (0)S x k (0) = S(0)M x (t)

49 A tractable class of multi-tenor McLMMss An explicit example of rate and spread dynamics We now assume constant volatilities σ x 1 k (t) = σx 1 k and SABR dynamics for V F. This leads to the following dynamics for the x-tenor rate Fk x under x QT k [ 1 ] dfk x (t) = σx k V F (t) + Fk x (t) dz k,x (t) τ x k dv F (t) = ɛ[v F (t)] 2 i k h=β(t) k D : σ x 1 h ρx 1 h dt + ɛv F (t) dw k,x (t), with V F (0) = 1, where also σ x k is now constant and ɛ R+. We then assume that basis spreads for all tenors x are governed by the same geometric Brownian motion: M x M, dm(t) = σm(t) dz (t) where Z is a Q T k x k,x D -Brownian motion independent of Zk and W k,x and σ is a positive constant.

50 A tractable class of multi-tenor McLMMss An explicit example of rate and spread dynamics Caplet prices can easily be calculated as soon as we smartly approximate the drift term of V F. We get: Cplt(t, K ; T x k 1, T x k ) = a x k (t) S x k (t)e 1 2 σ2 T x k 1 +σ T x k 1 z ; T x k 1, T x k Cplt SABR ( t, F x k (t) + 1 τ x k ) 1 2π e 1 2 z2 dz + τk x P D(t, Tk x )(F k x (t) K )Φ( ak x ( (t)) ) + τk x P D(t, Tk x )Sx k (t)φ ak x (t) + σ Tk 1 x t, K + 1 τ x k where ( K + 1 ak x (t) := τk ln x Sk x(t) + 1 ) ( ) 2 σ2 (Tk 1 x t) / σ Tk 1 x t and the SABR parameters are σk x (corrected for the drift approximation), ɛ and ρ x k (the SABR β is here equal to 1).

51 A tractable class of multi-tenor McLMMss An explicit example of rate and spread dynamics This caplet pricing formula can be used to price caps on any tenor x. In fact, cap prices on a non-standard tenor z can be derived by calibrating the market prices of standard y-tenor caps using the formula with x = y and assuming a specific correlation structure ρ i,j. One then obtains in output the model parameters: σ x 1 k, k = 1,..., M 1 ρ x 1 k, k = 1,..., M 1 ɛ σ Finally, with these calibrated parameters one can price z-based caps, using again the caplet formula above, this time setting x = z.

52 A tractable class of multi-tenor McLMMss An explicit example of rate and spread dynamics We finally consider an example of calibration to market data of this multi-tenor McLMM. We use EUR data as of September 15th, 2010 and calibrate 6-month caps with (semi-annual) maturities from 3 to 10 years. The considered strikes range from 2% to 7%. We minimize the sum of squared relative differences between model and market prices. We assume that OIS rates are perfectly correlated with one another, that all ρ x 1 k are equal to the same ρ and that the drift of V F is approximately linear in V F. The average of the absolute values of these differences is 19bp. After calibrating the model parameters to caps with x = 6m, we can apply the same model to price caps based on the 3m-LIBOR (x = 3m), where we assume that σ 3m i k 1 = σ 3m i k for each k.

53 A tractable class of multi-tenor McLMMss An explicit example of rate and spread dynamics Figure: Absolute differences (in%) between market and model cap volatilities.

54 A tractable class of multi-tenor McLMMss An explicit example of rate and spread dynamics Figure: Absolute differences (in bp) between model-implied 3m-LIBOR cap volatilities and model 6m-LIBOR ones.

55 Conclusions We started by describing the changes in market interest rate quotes which have occurred since August We have shown how to price the main interest rate derivatives under the assumption of distinct curves for generating future LIBOR rates and for discounting. We have then shown how to extend the LMM to the multi-curve case, retaining the tractability of the classic single-curve LMM. We have finally introduced an extended LMM, where we jointly model rates and spreads with different tenors. References: Mercurio, F. (2010a) Modern LIBOR Market Models: Using Different Curves for Projecting Rates and for Discounting. International Journal of Theoretical and Applied Finance 13, Mercurio, F. (2010b) LIBOR Market Models with Stochastic Basis. Available online on the ssrn web site.