16. Inﬂation-Indexed Swaps

Given a set of datesT1, . . . , TM, an Inﬂation-Indexed Swap (IIS) is a swap where, on each payment date, Party A pays Party B the inﬂation rate over a predeﬁned period, while Party B pays Party A a ﬁxed rate. The inﬂation rate is calculated as the percentage return of the CPI index over the time interval it applies to. Two are the main IIS traded in the market: the zero coupon (ZC) swap and the year-on-year (YY) swap. In a ZCIIS, at the ﬁnal timeTM, assumingTM=Myears, Party B pays Party A the ﬁxed amount M N[(1 +K)−1],(16.1) whereKandNare, respectively, the contract ﬁxed rate and nominal value. In exchange for this ﬁxed payment, Party A pays Party B, at the ﬁnal time TM, the ﬂoating amount I(TM) N−1.(16.2) I0 In a YYIIS, at each timeTi, Party B pays Party A the ﬁxed amount N ϕiK, whereϕiis the contract ﬁxed-leg year fraction for the interval [Ti−1, Ti], while Party A pays Party B the (ﬂoating) amount I(Ti) N ψi−1,(16.3) I(Ti−1) whereψiis the ﬂoating-leg year fraction for the interval [Ti−1, Ti],T0:= 0 andNis again the contract nominal value. Both ZC and YY swaps are quoted, in the market, in terms of the corre-sponding ﬁxed rateK. The ZCIIS and YYIIS (mid) ﬁxed-rate quotes in the Euro market on October 7th 2004 are shown in Figure 16.1, for maturities up to twenty years. The reference CPI is the Euro-zone ex-tobacco index.

16.1 Pricing of a ZCIIS

Standard no-arbitrage pricing theory implies that the value at timet, 0≤ t < TM, of the inﬂation-indexed leg of the ZCIIS is

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16. Inﬂation-Indexed Swaps

2.6

2.55

2.5

2.45

2.4

2.35

2.3

2.25

2.2

2.15

2.1 0

2

4

6

8

10 Maturity

12

14

16

YY rates ZC rates 18 20

Fig. 16.1.Euro inﬂation swap rates as of October 7, 2004.

T I(TM) M −n(u)du t ZCIIS(t, TM, I0, N) =N Ene−1Ft, I0

(16.4)

whereFtdenotes theσ-algebra generated by the relevant underlying pro-cesses up to timet. By the foreign-currency analogy, the nominal price of a real zero-coupon bond equals the nominal price of the contract paying oﬀ one unit of the CPI index at bond maturity, see also the general formula (2.31). In formulas, for eacht < T: ! " ! " T T −r(u)du−n(u)du tt I(t)Pr(t, T) =I(t)EreFt=Ene I(T)Ft. (16.5) Therefore, (16.4) becomes I(t) ZCIIS(t, TM, I0, N) =N Pr(t, TM)−Pn(t, TM),(16.6) I0

which at timet= 0 simpliﬁes to

ZCIIS(0, TM, N) =N[Pr(0, TM)−Pn(0, TM)].

(16.7)

Formulas (16.6) and (16.7) yield model-independent prices, which are not based on speciﬁc assumptions on the evolution of the interest rate market, but simply follow from the absence of arbitrage. This result is extremely important since it enables us to strip, with no ambiguity, real zero-coupon bond prices from the quoted prices of zero-coupon inﬂation-indexed swaps. In fact, the market quotes values ofK=K(TM) for some given maturities TM, so that equating (16.7) with the (nominal) present value of (16.1), and getting the discount factorPn(0, TM) from the current (nominal) zero-coupon

16.2 Pricing of a YYIIS

651

curve, we can solve for the unknownPr(0, TM). We thus obtain the discount 1 factor for maturityTMin the real economy: M Pr(0, TM) =Pn(0, TM)(1 +K(TM)).(16.8) Remark 16.1.1.(ZCIIS and Forward CPI). Kazziha (1999) deﬁnes the T-forward CPI at timetas the ﬁxed amountXto be exchanged at timeT for the CPII(T), for which such a swap has zero value at timet, in analogy with the deﬁnition of a forward LIBOR rate we gave in Chapter 1. From formula (16.5), we immediately obtain

I(t)Pr(t, T) =XPn(t, T). This is consistent with deﬁnition (15.1), which was directly based on the foreign-currency analogy. The advantage of Kazziha’s approach is that no foreign-currency analogy is required for the deﬁnition of the forward CPI’sIi, and the pricing system she deﬁnes is only based on nominal zero-coupon bonds and forward CPI’s. In her setting, the value at time zero of aTM-forward CPI can be obtained from the market quoteK(TM) by applying this simple formula M IM(0) =I(0)(1 +K(TM)), which is perfectly equivalent to (16.8).

16.2 Pricing of a YYIIS

Compared to that of a ZCIIS, the valuation of a YYIIS is more involved. Notice, in fact, that the value at timet < Tiof the payoﬀ (16.3) at timeTiis I(Ti) T i −n(u)du t YYIIS(t, Ti−1, Ti, ψi, N) =N ψiEne−1Ft, I(Ti−1) (16.9) which, assumingt < Ti−1(otherwise we fall back to the previous case), can be calculated as T T i −n(u)duT)duI(Ti) i−1 −n(u t i−1 N ψiEne Ene−1FTi−1Ft. I(Ti−1) (16.10) The inner expectation is nothing butZCIIS(Ti−1, Ti, I(Ti−1),1), so that we obtain ! " T i−1 −n(u)du t N ψiEne[Pr(Ti−1, Ti)−Pn(Ti−1, Ti)]Ft !T"(16.11) i−1 −n(u)du t =N ψiEne Pr(Ti−1, Ti)Ft−N ψiPn(t, Ti).

1 The real discount factors for intermediate maturities can be inferred by taking into account the typical seasonality eﬀects in inﬂation.

652

16. Inﬂation-Indexed Swaps

The last expectation can be viewed as the nominal price of a derivative paying oﬀ, in nominal units, the real zero-coupon bond pricePr(Ti−1, Ti) at time Ti−1. If real rates were deterministic, then this price would simply be the present value, in nominal terms, of the forward price of the real bond. In this case, in fact, we would have: !T" i−1 −n(u)du t Ene Pr(Ti−1, Ti)Ft=Pr(Ti−1, Ti)Pn(t, Ti−1) Pr(t, Ti) =Pn(t, Ti−1). Pr(t, Ti−1) In practice, however, real rates are stochastic and the expected value in (16.11) is model dependent. For instance, under dynamics (15.2), the forward price of the real bond must be corrected by a factor depending on both the nominal and real interest rates volatilities and on the respective correlation. This is explained in the following.

16.3 Pricing of a YYIIS with the JY Model

T Denoting byQthe nominalT-forward measure for a general maturityT n T and byEthe associated expectation, we can write: n YYIIS(t, Ti−1, Ti, ψi, N) (16.12) Ti−1 Pn(t, Ti−1)En =N ψiPr(Ti−1, Ti)Ft−N ψiPn(t, Ti). Remembering formula (3.39) for the zero-coupon bond price in the Hull and White (1994b) model:

−Br(t,T)r(t) Pr(t, T) =Ar(t, T)e , # $ 1 −ar(T−t) Br(t, T1) = −e , ar M2 P(0, T)σ r M r−2art2 Ar(t, Texp) = Br(t, T)fr( (1−, 0, t)−e)Br(t, T) M P(0, t) 4ar r (16.13)

and noting that, by the change-of-numeraire toolkit in Section 2.3, and for-Ti−1 mula (2.12) in particular, the real instantaneous rate evolves underQn according to

Ti−1 r(t)]dt+σ dW(t) dr(t) = [−ρn,rσnσrBn(t, Ti−1) +ϑr(t)−ρr,IσIσr−arr r (16.14) Ti−1Ti−1 withWraQn-Brownian motion, we have that the real bond price Ti−1 Pr(Ti−1, Ti) is lognormally distributed underQn, sincer(Ti−1) is still a normal random variable under this (nominal) forward measure. After some tedious, but straightforward, algebra we ﬁnally obtain

16.3 Pricing of a YYIIS with the JY Model 653 YYIIS(t, Ti−1, Ti, ψi, N) (16. Pr(t, Ti)C(t,Ti−1,Ti)15) =N ψiPn(t, Ti−1)e−N ψiPn(t, Ti), Pr(t, Ti−1) where 1 C(t, Ti−1, Ti) =σrBr(Ti−1, Ti)Br(t, Ti−1)ρr,IσI−σrBr(t, Ti−1) 2 ρn,rσnρn,rσn + 1 +arBn(t, Ti−1)−Bn(t, Ti−1). an+aran+ar The expectation of a real zero-coupon bond price under a nominal forward measure, in the JY model, is thus equal to the current forward price of the real bond multiplied by a correction factor, which depends on the (instan-taneous) volatilities of the nominal rate, the real rate and the CPI, on the (instantaneous) correlation between nominal and real rates, and on the (in-stantaneous) correlation between the real rate and the CPI. The exponential ofCis the correction term we mentioned above. This term accounts for the stochasticity of real rates and, indeed, vanishes for σr= 0. The value at timetof the inﬂation-indexed leg of the swap is simply obtained by summing up the values of all ﬂoating payments. We thus get I(t) YYIIS(t,T, Ψ, N) =N ψι(t)Pr(t, Tι(t))−Pn(t, Tι(t)) I(Tι(t)−1) M (16.16) Pr(t, Ti)Ti−1,Ti) C(t, +N ψiPn(t, Ti−1)e−Pn(t, Ti), Pr(t, Ti−1) i=ι(t)+1 where we setT:={T1, . . . , TM},Ψ:={ψ1, . . . , ψM}andι(t) = min{i:Ti> 2 t}, and where the ﬁrst payment after timethas been priced according to (16.6). In particular att= 0, YYIIS(0,T, Ψ, N) =N ψ1[Pr(0, T1)−Pn(0, T1)] M Pr(0, Ti)C(0,Ti−1,Ti) +N ψiPn(0, Ti−1)e−Pn(0, Ti) Pr(0, Ti−1) (16.17) i=2 M 1 +τiFn(0;Ti−1, Ti) C(0,Ti−1,Ti) =N ψiPn(0, Ti)e−1. 1 +τiFr(0;Ti−1, Ti) i=1 The advantage of using Gaussian models for nominal and real rates is clear as far as analytical tractability is concerned. However, the possibility of negative rates and the diﬃculty in estimating historically the real rate parameters led to alternative approaches. We now illustrate two diﬀerent market models that have been proposed for alternative valuations of a YYIIS and other inﬂation-indexed derivatives. 2 By deﬁnition,Tι(t)−1≤t < Tι(t).

654

16. Inﬂation-Indexed Swaps

16.4 Pricing of a YYIIS with a First Market Model

For an alternative pricing of the above YYIIS, we notice that we can change measure and, as explained in Section 2.8, re-write the expectation in (16.12) as Ti−1TiPr(Ti−1, Ti) P(t, T)E P(TT , )F=P n i−1in r −1ni t (t, Ti)EnFt Pn(Ti−1, Ti) 1 +τ TiiFn(Ti−1;Ti−1, Ti) EnFt, =Pn(t, Ti) 1 +τiFr(Ti−1;Ti−1, Ti) (16.18)

which can be calculated as soon as we specify the distribution of both forward rates under the nominalTi-forward measure. It seems natural, therefore, to resort to a LFM, which postulates the evolution of simply-compounded forward rates, namely the variables that explicitly enter the last expectation, see Section 6.3. This approach, followed by Mercurio (2005), is detailed in the following. SinceI(t)Pr(t, Ti) is the price of an asset in the nominal economy, we have that the forward CPI Pr(t, Ti) Ii(t) =I(t) Pn(t, Ti) TiTi is a martingale underQby the deﬁnition itself ofQ. Assuming lognormal n n dynamics forIi, I dIi(t) =σI,iIi(t)dW(t),(16.19) i I Ti W-Brownian motion, and whereσI,iis a positive constant andiis aQn assuming also that both nominal and real forward rates follow a LFM, the analogy with cross-currency derivatives pricing implies that the dynamics of Ti F(∙e giv Section 14.4) n;Ti−1, Ti) andFr(∙;Ti−1, Ti) underQnby (seear en n t;Ti−1, Ti) =σn,iFn(t;Ti−1dW dFn(, Ti)i(t), (16.20) r −σ dtρ σ +σ dW( dFr(t;Ti−1, Ti) =Fr(t;Ti−1, Ti)I,r,i I,i r,i r,i it), n r whereσn,iandσr,iare positive constants,WandWare two Brownian i i motions with instantaneous correlationρi, andρI,r,iis the instantaneous I r correlation betweenI(∙) andF dW(t)d(t) =ρ i r(∙;Ti−1, Ti), i.e.iWi I,r,idt. AllowingσI,i,σn,iandσr,ito be deterministic functions of time does not complicate the calculations below. We assume hereafter that such volatilities are constant for ease of notation only. In practice, however, the implications of using constant or time-dependent coeﬃcients should be carefully analyzed. See also Chapter 7 and Remark 18.0.1 below. The expectation in (16.18) can then be easily calculated with a numerical Ti3 integration by noting that, underQand conditional onFt, the pair n 3 To lighten the notation, we simply write (Xi, Yi) instead of (Xi(t), Yi(t)).

16.4 Pricing of a YYIIS with a First Market Model 655 Fn(Ti−1;Ti−1, Ti)Fr(Ti−1;Ti−1, Ti) (Xi, Yiln) = ,ln (16.21) Fn(t;Ti−1, Ti)Fr(t;Ti−1, Ti) is distributed as a bivariate normal random variable with mean vector and variance-covariance matrix, respectively, given by 2 σ)ρ σ(t)σ(t) µx,i(t)x,i(ty,ii x,i M=, V=,(16.22) Xi,YiXi,Yi2 ρ σ(t)σ(t(t µy,i(t)y,ii x,i )σy,i) where 1 2 µx,i(t) =−σ( 2n,iTi−1−t), σx,i(t) =σn,iTi−1−t, 1 2 µy,i(t) =−σ ρ 2r,i−I,r,iσI,iσr,i(Ti−1−t), σy,i(t) =σr,iTi−1−t. It is well known that the densityfXi,Yi(x, y) of (Xi, Yi) can be decomposed 4 as f(x, y) =f(x, y)f(y), Xi,YiXi|YiYi where 2 x−µx,i(t)y−µy,i(t) −ρi σy,i(t) 1σx,i(t) fXi|Yi(x, y) =√exp−2 2 2(1−ρ) ρ σx,i(t) 2π1−i i 2 1 1y−µy,i(t) fYi(y) =√exp−. (t) σy,i(t) 2π2σy,i (16.23)

The last expectation in (16.18) can thus be calculated as +∞ +∞x (1 +τiFn(t;Ti−1, Ti)e)fXi|Yi(x, y)dx −∞ fYi(y)dy y 1 +τiFr(t;Ti−1, Ti)e −∞ y−µ(t) y,i12 2 µx,i(t)+ρiσx,i(t) +σ(t)(1−ρ) +∞σ(t)x,i i y,i2 1 +τiFn(t;Ti−1, Ti)e =fYi(y)dy y 1 +τ F(t;T , T)e −∞ii r −1i 12 2 +∞ρiσx,i(t)z−σ(t)ρ x,i i 2 12 1 +τiFn(t;Ti−1, Ti)e1−z =√e dz, 2 µy,i(t)+σy,i(t)z 1 +τiFr(t;Ti−1, Ti)e2π −∞ yielding:

YYIIS(t, Ti−1, Ti, ψi, N) 12 2 +∞ρiσx,i(t)z−σ(t)ρ x,i i 2 12 1 +τiFn(t;Ti−1, Ti)e1−z 2 =N ψiPn(t, Ti)√e dz µy,i(t)+σy,i(t)z −∞1 +τiFr(t;Ti−1, Ti)e2π −N ψiPn(t, Ti). (16.24)

4 See also Appendix E for a similar calculation.

656

16. Inﬂation-Indexed Swaps

To value the whole inﬂation-indexed leg of the swap some care is needed, since we cannot simply sum up the values (16.24) of the single ﬂoating payments. In 5 fact,asnotedbySchl¨ogl(2002)inamulti-currencyversionoftheLFM,we cannot assume that the volatilitiesσI,i,σn,iandσr,iare positive constants for alli, because there exists a precise relation between two consecutive forward CPIs and the corresponding nominal and real forward rates, namely:

Ii(t) 1 +τiFn(t;Ti−1, Ti) =. Ii−1(t) 1 +τiFr(t;Ti−1, Ti)

(16.25)

Clearly, if we assume thatσI,i,σn,iandσr,iare positive constants,σI,i−1 cannot be constant as well, and its admissible values are obtained by equating the (instantaneous) quadratic variations on both sides of (16.25). However, by freezing the forward rates at their time 0 value in the diﬀusion coeﬃcients of the right-hand-side of (16.25), we can still get forward CPI volatilities that are approximately constant. For instance, in the one-factor model case,

τiFr(t;Ti−1, Ti)τiFn(t;Ti−1, Ti) σI,i−1=σI,i+σr,i−σn,i 1 +τiFr(t;Ti−1, Ti) 1 +τiFn(t;Ti−1, Ti) τiFr(0;Ti−1, Ti)τiFn(0;Ti−1, Ti) ≈σI,i+σr,i−σn,i. 1 +τiFr(0;Ti−1, Ti+) 1 τiFn(0;Ti−1, Ti)

Therefore, applying this “freezing” procedure for eachi < Mstarting from σI,M, or equivalently for eachi >2 starting fromσI,1, we can still assume that the volatilitiesσI,iare all constant and set to one of their admissible values. The value at timetof the inﬂation-indexed leg of the swap is thus given by I(t) YYIIS(t,T, Ψ, N) =N ψι(t)Pr(t, Tι(t))−Pn(t, Tι(t)) I(Tι(t)−1) M +N ψiPn(t, Ti) i=ι(t)+1 12 2 +∞ρiσx,i(t)z−σ(t)ρ x,i i 2 12 1 +τiFn(t;Ti−1, Ti)e1−z 2 ∙ √e dz−1. µy,i(t)+σy,i(t)z 1 +τiFr(t;Ti−1, Ti)e2π −∞ (16.26)

In particular att= 0, 5 See also Section 14.5.4.

16.5 Pricing of a YYIIS with a Second Market Model 657 M YYIIS(0,T, Ψ, N) =N ψ1[Pr(0, T1)−Pn(0, T1)] +N ψiPn(0, Ti) i=2 12 2 +∞ρiσx,i(0)z−σ(0)ρ 2 12 x,i i 1 +τiFn(0;Ti−1, Ti)e1−z ∙ √e dz−1 2 µy,i(0)+σy,i(0)z −∞1 +τiFr(0;Ti−1, Ti)e2π M =N ψiPn(0, Ti) i=1 12 2 +∞ρiσx,i(0)z−σ(0)ρ 1 +τ F(0;TT , )e1 2 12 x,i i i n i−1i−z 2 ∙ √e dz−1. µy,i(0)+σy,i(0)z −∞1 +τiFr(0;Ti−1, Ti)e2π (16.27)

This YYIIS price depends on the following parameters: the (instantaneous) volatilities of nominal and real forward rates and their correlations, for each payment timeTi,i= 2, . . . , M; the (instantaneous) volatilities of forward inﬂation indices and their correlations with real forward rates, again for each i= 2, . . . , M. Compared with expression (16.17), formula (16.27) looks more compli-cated both in terms of input parameters and in terms of the calculations involved. However, one-dimensional numerical integrations are not so cum-bersome and time consuming. Moreover, as is typical in a market model, the input parameters can be determined more easily than those coming from the previous short-rate approach. In this respect, formula (16.27) is preferable to (16.17). As in the JY case, valuing a YYIIS with a LFM has the drawback that the volatility of real rates may be hard to estimate, especially when resorting to a historical calibration. This is why, in the literature, a second market model has been proposed, which enables us to overcome this estimation issue. In the following section we will review this approach, which has been independently developed by Kazziha (1999), Belgrade, Benhamou and Koehler (2004) and Mercurio (2005).

16.5 Pricing of a YYIIS with a Second Market Model

Applying the deﬁnition of forward CPI and using the fact thatIiis a mar-Ti tin rQwe gale undenalso write, for, can t < Ti−1, I(Ti) Ti , Ti, ψi, N) =N ψi−1F YYIIS(t, Ti−1P(t, Ti)En t I(Ti−1) Ii(Ti) Ti =N ψiP(t, Ti)E−1Ft(16.28) n Ii−1(Ti−1) I T i(Ti−1) i =N ψiP(t, Ti )En−1Ft. Ii−1(Ti−1)

658

16. Inﬂation-Indexed Swaps

Ti s g The dynamics ofIiunderQnby (16.19) and an analogous evolutioni iven Ti−1 Ti holds forIi−1underQn. The dynamics ofIi−1underQcan be derived n by applying the change-of-numeraire toolkit in Section 2.3. We get: τiσn,iFn(t;Ti−1, Ti) I (t)σ−+dW(t), dIi−1(t) =Ii−1I,i−1ρI,n,idti−1 1 +τiFn(t;Ti−1, Ti) (16.29) I Ti whereσI,i−1is a positive constant,Wis aQ-Brownian motion with i−1n I I t)dW(t) =ρI,idt dWi−1(i, andρI,n,iis the instantaneous correlation between Ii−1(∙) andFn(∙;Ti−1, Ti). Ti The evolution ofIi−1, underQ, depends on the nominal forward rate n Fn(∙;Ti−1, Ti), so that the calculation of (16.28) is rather involved in general. To avoid unpleasant complications, like those induced by higher-dimensional integrations, we freeze the drift in (16.29) at its current time-tvalue, so that Ti Ii−1(Ti−1) conditional onFtis lognormally distributed also underQ. This n leads to Ii(T Tii−1)Ii(t)Di(t) E nFt=e , Ii−1(Ti−1)Ii−1(t) where τiσn,iFn(t;Ti−1, Ti) Di(t) =σI,i−1ρI,n,i−ρI,iσI,i+σI,i−1(Ti−1−t), 1 +τiFn(t;Ti−1, Ti) so that Ii(t)D(t) i YYIIS(t, Ti−1, Ti, ψi, N) =N ψiPn(t, Ti)e−1 Ii−1(t) Pn(t, Ti−1)Pr(t, Ti)Di(t) =N ψiPn(t, Ti)e−1. Pn(t, Ti)Pr(t, Ti−1) (16.30)

Finally, the value at timetof the inﬂation-indexed leg of the swap is Iι(t)(t) YYIIS(t,T, Ψ, N) =PN ψ (t, T−1 ι(t)n ι(t) I(Tι(t)−1) M Ii(t) Di(t) +N ψiPn(t, Ti)e−1 Ii−1(t) i=ι(t)+1 I(t) =N ψ P(t, T)−P(t, T) ι(t)r ι(t)n ι(t) I(T) ι(t)−1 M (t, Ti) Pr Di(t) +N ψiPn(t, Ti−1)e−Pn(t, Ti). Pr(t, Ti−1) i=ι(t)+1 (16.31)

In particular att= 0,