PRICING OF INFLATION-INDEXED DERIVATIVES

Transcription

1 PRICING OF INFLATION-INDEXED DERIVATIVES FABIO MERCURIO BANCA IMI, MILAN The Inaugural Fixed Income Conference, Prague, September

2 Stylized facts Inflation-indexed bonds have been issued since the 80 s, but it is only in the very last years that these bonds, and inflation-indexed derivatives in general, have become quite popular. Inflation is defined as the percentage increment of a reference index, the Consumer Price Index (CPI), which is a basket of good and services. In theory, but also in practice, inflation can become negative. Banks typically issues inflation-linked bonds, where a zero-strike floor is offered in conjunction with the pure bond (to grant positive coupons, the inflation rate is floored at zero). Floors with low strikes are the most actively traded options on inflation rates. Other extremely popular derivatives are inflation-indexed swaps. The Inaugural Fixed Income Conference, Prague, September

3 Stylized facts (cont d) Inflation-indexed derivatives require a specific model to be valued. Main references: Barone and Castagna (1997), van Bezooyen et al. (1997), Hughston (1998), Cairns (2000) and Jarrow and Yildirim (2003). Inflation derivatives are priced with a foreign-currency analogy (the pricing is equivalent to that of a cross-currency interest-rate derivative). What is typically modelled is the evolution of the instantaneous nominal and real rates and of the CPI (interpreted as the exchange rate between the nominal and real economies). The real rate one models is the rate we can lock in by suitably trading in inflation swaps. The true real rate will be only known at the end of the corresponding period (as soon as the CPI s value is known). The Inaugural Fixed Income Conference, Prague, September

4 Purpose and outline of the talk Our purpose is to price analytically, and consistently with no arbitrage, inflation-indexed swaps and options. We start by introducing the two main types of inflation swaps. We first apply the JY model and then propose two different market models. We derive closed-form formulas in all the presented cases. We then introduce inflation caps and floors, and we price them both under the JY model and under our second market model. The advantage of our market-model approach is in terms of a better understanding of the model parameters and of a more accurate calibration to market data. The Inaugural Fixed Income Conference, Prague, September

5 Some notations and definitions We denote by I(t) the CPI s value at time t. We use the subscripts n and r to denote quantities in the nominal and real economies, respectively. The zero-coupon bond prices at time t for maturity T in the nominal and real economies are denoted, respectively, by P n (t, T ) and P r (t, T ). The instantaneous forward rates at time t for maturity T are defined by f x (t, T ) = ln P x(t, T ), x {n, r} T and the corresponding instantaneous short rates by n(t) = f n (t, t), r(t) = f r (t, t). The Inaugural Fixed Income Conference, Prague, September

6 Some notations and definitions (cont d) Given the future time interval [T i 1, T i ], the related forward LIBOR rates, at time t, are F x (t; T i 1, T i ) = P x(t, T i 1 ) P x (t, T i ), x {n, r}, τ i P x (t, T i ) where τ i is the year fraction for [T i 1, T i ]. We denote by Q n and Q r the nominal and real risk-neutral measures, respectively, and by E x the expectation associated to Q x, x {n, r}. We denote by Q T n the nominal T -forward measure and by E T n the associated expectation. Finally, the σ-algebra generated by the relevant processes up to time t is denoted by F t. The Inaugural Fixed Income Conference, Prague, September

7 Historical plots of CPIs sep aug aug jul Sep Aug Aug Jul 04 Figure 1: Left: EUR CPI Unrevised Ex-Tobacco. Right: USD CPI Urban Consumers NSA. Monthly closing values from 30-Sep-01 to 21-Jul-04. The Inaugural Fixed Income Conference, Prague, September

8 Inflation-indexed swaps Given a set of dates T 1,..., T M, 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 I over the time interval [t, T ] it applies to: i(t, T ) := I(T ) I(t) 1. Two are the main IIS traded in the market: the zero coupon (ZC) swap; the year-on-year (YY) swap. The Inaugural Fixed Income Conference, Prague, September

9 Zero-coupon inflation-indexed swaps Party A N[(1 + K) M 1] 0 [ T M ] Party B N I(TM ) I 0 1 In a ZCIIS, at time T M = M years, Party B pays Party A the fixed amount N[(1 + K) M 1], where K and N are, respectively, the contract fixed rate and the contract nominal value. Party A pays Party B, at the final time T M, the floating amount [ ] I(TM ) N 1. I 0 The Inaugural Fixed Income Conference, Prague, September

10 Year-on-year inflation-indexed swaps Party A Nϕ i K 0 T 1 T 2 T i 1 T i T M Party B [ ] Nψ I(Ti ) i I(T i 1 ) 1 In a YYIIS, at each time T i, Party B pays Party A the fixed amount Nϕ i K, while Party A pays Party B the (floating) amount [ ] I(Ti ) Nψ i I(T i 1 ) 1, where ϕ i and ψ i are, respectively, the fixed- and floating-leg year fractions for the interval [T i 1, T i ], T 0 := 0 and N is again the swap nominal value. The Inaugural Fixed Income Conference, Prague, September

11 Pricing of a ZCIIS Standard no-arbitrage pricing theory implies that the value at time t, 0 t < T M, of the inflation-indexed leg of the ZCIIS is { ZCIIS(t, T M, I 0, N) = NE n e R [ ] } T M t n(u) du I(TM ) 1 F t. I 0 The foreign-currency analogy implies that, for each t < T : I(t)P r (t, T ) = I(t)E r { e R T t r(u) du Ft } = E n { e R T t n(u) du I(T ) Ft }, namely, the nominal price of a real zero-coupon bond equals the nominal price of the contract paying one unit of the CPI index at the bond maturity. We thus have: ZCIIS(t, T M, I 0, N) = N [ ] I(t) P r (t, T M ) P n (t, T M ). I 0 The Inaugural Fixed Income Conference, Prague, September

12 Pricing of a ZCIIS (cont d) The ZCIIS price is therefore model-independent: it is not based on specific assumptions on the interest rates evolution, but simply follows 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 zerocoupon inflation-indexed swaps. The market quotes values of K = K(T M ) for some given maturities T M. The ZCIIS corresponding to (T M, K(T M )) has zero value at time t = 0 if and only if N[P r (0, T M ) P n (0, T M )] = NP n (0, T M )[(1 + K(T M )) M 1] P r (0, T M ) = P n (0, T M )(1 + K(T M )) M. The Inaugural Fixed Income Conference, Prague, September

13 Pricing of a YYIIS The valuation of a YYIIS is less straightforward and, as we shall see, requires the specification of an interest rate model. The value at time t < T i of the YYIIS payoff at time T i is { YYIIS(t, T i 1, T i, ψ i, N) = Nψ i E n e R T i t n(u) du which, assuming t < T i 1, can be calculated as [ I(Ti ) I(T i 1 ) 1 Nψ i E n { e R T i 1 t n(u) du E n [ e R T i T i 1 n(u) du ( I(Ti ) I(T i 1 ) 1) FTi 1 The inner expectation is nothing but ZCIIS(T i 1, T i, I(T i 1 ), 1) Nψ i E n { e R T i 1 t n(u) du [P r (T i 1, T i ) P n (T i 1, T i )] F t }. ] } F t, ] } Ft. The Inaugural Fixed Income Conference, Prague, September

14 We thus have: Pricing of a YYIIS (cont d) YYIIS(t) = Nψ i E n { e R T i 1 t n(u) du P r (T i 1, T i ) Ft } Nψ i P n (t, T i ). This last expectation can be viewed as the nominal price of a derivative paying off P r (T i 1, T i ) at time T i 1. If real rates were deterministic: E n { e R T i 1 t n(u) du P r (T i 1, T i ) F t } = P r (T i 1, T i )P n (t, T i 1 ) = P r(t, T i ) P r (t, T i 1 ) P n(t, T i 1 ), which is the present value, in nominal terms, of the forward price of the real bond. In practice, however, real rates are stochastic and the above expected value is model dependent. The Inaugural Fixed Income Conference, Prague, September

15 The JY model Under the real-world probability space (Ω, F, P ), with associated filtration F t, Jarrow and Yildirim assumed that df n (t, T ) = α n (t, T ) dt + ς n (t, T ) dw P n (t) df r (t, T ) = α r (t, T ) dt + ς r (t, T ) dw P r (t) di(t) = I(t)µ(t) dt + σ I I(t) dw P I (t) with I(0) = I 0 > 0, and f x (0, T ) = fx mkt (0, T ), x {n, r}, where (W P n, W P r, W P I ) is a Brownian motion with correlations ρ n,r, ρ n,i and ρ r,i ; α n, α r and µ are adapted processes; ς n and ς r are deterministic functions; σ I is a positive constant. The Inaugural Fixed Income Conference, Prague, September

16 The JY model (cont d) Choosing the forward rate volatilities as ς n (t, T ) = σ n e a n(t t), ς r (t, T ) = σ r e a r(t t), where σ n σ r, a n and a r are positive constants, and using the equivalent formulation in terms of instantaneous short rates, we have the following. Proposition. The Q n -dynamics of n, r and I are, respectively, dn(t) = [ϑ n (t) a n n(t)] dt + σ n dw n (t) dr(t) = [ϑ r (t) ρ r,i σ I σ r a r r(t)] dt + σ r dw r (t) di(t) = I(t)[n(t) r(t)] dt + σ I I(t) dw I (t) where (W n, W r, W I ) is a Brownian motion with correlations ρ n,r, ρ n,i and ρ r,i, and ϑ x (t) = f x(0, t) T + a x f x (0, t) + σ2 x 2a x (1 e 2a xt ), x {n, r}. The Inaugural Fixed Income Conference, Prague, September

17 Pricing of a YYIIS: the JY model We remember that: YYIIS(t) = Nψ i E n { e R T i 1 t n(u) du P r (T i 1, T i ) Ft } Nψ i P n (t, T i ) = Nψ i P n (t, T i 1 )E T i 1 n { Pr (T i 1, T i ) Ft } Nψi P n (t, T i ), and also remember the zero-coupon bond price formula in the Hull and White (1994) model: P r (t, T ) = A r (t, T )e B r(t,t )r(t), B r (t, T ) = 1 a r [1 e a r(t t) ], A r (t, T ) = P r M } (0, T ) {B Pr M (0, t) exp r (t, T )f Mr (0, t) σ2 (1 e 2art )B r (t, T ) 2. 4a r The Inaugural Fixed Income Conference, Prague, September

18 Pricing of a YYIIS: the JY model (cont d) Since the real instantaneous rate evolves under Q T i 1 n according to dr(t) = [ ρ n,r σ n σ r B n (t, T i 1 )+ϑ r (t) ρ r,i σ I σ r a r r(t)] dt+σ r dw T i 1 r (t), r(t i 1 ) remains a normal random variable, and hence the real bond price P r (T i 1, T i ) is lognormally distributed also under Q T i 1 n. After some tedious, but straightforward, algebra we finally obtain YYIIS(t)= Nψ i P n (t, T i 1 ) P r(t, T i ) P r (t, T i 1 ) ec(t,t i 1,T i ) Nψ i P n (t, T i ), where [ ( C(t, T i 1, T i ) =σ r B r (T i 1, T i ) B r (t, T i 1 ) ρ r,i σ I 1 2 σ rb r (t, T i 1 ) + ρ n,rσ n ( 1 + ar B n (t, T i 1 ) )) ρ ] n,rσ n B n (t, T i 1 ). a n + a r a n + a r The Inaugural Fixed Income Conference, Prague, September

19 Pricing of a YYIIS: the JY model (cont d) The value at time t of 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) = Nψ ι(t) I(T ι(t) 1 ) P r(t, T ι(t) ) P n (t, T ι(t) ) + N M i=ι(t)+1 [ ψ i P n (t, T i 1 ) P ] r(t, T i ) P r (t, T i 1 ) ec(t,t i 1,T i ) P n (t, T i ), where T := {T 1,..., T M }, Ψ := {ψ 1,..., ψ M } and ι(t) = min{i : T i > t}. In particular at t = 0, YYIIS(0) = M ψ i P n (0, T i ) i=1 [ ] 1 + τi F n (0; T i 1, T i ) 1 + τ i F r (0; T i 1, T i ) ec(0,t i 1,T i ) 1. The Inaugural Fixed Income Conference, Prague, September

20 Pricing of a YYIIS: the LIBOR market model For an alternative pricing of the above YYIIS, we notice that P n (t, T i 1 )E T { i 1 n Pr (T i 1, T i ) { } } F t = Pn (t, T i )E T Pr (T i i 1, T i ) n F t P n (T i 1, T i ) { = P n (t, T i )E T i 1 + τi F n (T i 1 ; T i 1, T i ) n Ft }. 1 + τ i F r (T i 1 ; T i 1, T i ) It seems natural, therefore, to resort to a (lognormal) LIBOR model, for both nominal and real rates. Since I(t)P r (t, T i ) is the price of an asset in the nominal economy, the forward CPI I i (t) := I(t) P r(t, T i ) P n (t, T i ) is a martingale under Q T i n. The Inaugural Fixed Income Conference, Prague, September

21 Pricing of a YYIIS: the LIBOR market model (cont d) We assume: di i (t) = σ I,i I i (t) dw I i (t). where σ I,i is a positive constant and W I i is a Q T i n -Brownian motion. Assuming also that both nominal and real forward rates follow a lognormal LIBOR market model, the foreign-currency analogy implies that, under Q T i n, df n (t; T i 1, T i ) = σ n,i F n (t; T i 1, T i ) dw n i (t), df r (t; T i 1, T i ) = F r (t; T i 1, T i ) [ ρ I,r,i σ I,i σ r,i dt + σ r,i dw r i (t) ], where σ n,i and σ r,i are positive constants, Wi n and Wi r are two standard Brownian motions with instantaneous correlation ρ i, and ρ I,r,i is the instantaneous correlation between I i ( ) and F r ( ; T i 1, T i ). The Inaugural Fixed Income Conference, Prague, September

22 Pricing of a YYIIS: the LIBOR market model (cont d) The last expectation can now be easily calculated with a numerical integration by noting that, under Q T i n and conditional on F t, the pair ( (X i, Y i ) = ln F n(t i 1 ; T i 1, T i ), ln F ) r(t i 1 ; T i 1, T i ) F n (t; T i 1, T i ) F r (t; T i 1, T i ) is distributed as a bivariate normal random variable with mean vector and variance-covariance matrix respectively given by [ ] [ µx,i (t) M Xi,Y i =, V µ y,i (t) Xi,Y i = σx,i 2 (t) ρ ] iσ x,i (t)σ y,i (t) ρ i σ x,i (t)σ y,i (t) σy,i 2 (t), where µ x,i (t) = 1 2 σ2 n,i(t i 1 t), σ x,i (t) = σ n,i Ti 1 t, µ y,i (t) = [ 1 2 σ2 r,i ρ I,r,i σ I,i σ r,i ] (Ti 1 t), σ y,i (t) = σ r,i Ti 1 t. The Inaugural Fixed Income Conference, Prague, September

23 Pricing of a YYIIS: the LIBOR market model (cont d) It is well known that the joint density f Xi,Y i (x, y) can be decomposed as where f Xi Y i (x, y) = f Yi (y) = f Xi,Y i (x, y) = f Xi Y i (x, y)f Yi (y), ( x µx,i (t) σ x,i (t) ρ i y µ y,i (t) σ y,i (t) 1 σ x,i (t) 2π exp 1 ρ 2 2(1 ρ 2 i i ) [ 1 σ y,i (t) 2π exp 1 ( ) ] 2 y µy,i (t). 2 σ y,i (t) ) 2 The expectation can thus be calculated as (we set F x (t) := F x (t; T i 1, T i )) + [ 1 + ] 1 + τ i F r (t) e y (1 + τ i F n (t)e x ) f Xi Y i (x, y) dx f Yi (y) dy The Inaugural Fixed Income Conference, Prague, September

24 Pricing of a YYIIS: the LIBOR market model (cont d) The value at time t of the inflation-indexed leg of the swap is thus given by [ ] I(t) YYIIS(t, T, Ψ, N) = Nψ ι(t) I(T ι(t) 1 ) P r(t, T ι(t) ) P n (t, T ι(t) ) + N M i=ι(t)+1 [ + ψ i P n (t, T i ) 1 + τ i F n (t) e ρ iσ x,i (t)z 1 2 σ2 x,i (t)ρ2 i 1 + τ i F r (t) e µ y,i(t)+σ y,i (t)z e 1 ] 2 z2 dz 1. 2π N.B. In theory, the volatilities σ I,i can not be constant for each i, see Schlögl (2002). In practice, however, they are approximately constant. In particular at t = 0, YYIIS(0, T, Ψ, N) = = N M i=1 [ + ψ i P n (0, T i ) 1 + τ i F n (0) e ρ iσ x,i (0)z 1 2 σ2 x,i (0)ρ2 i 1 + τ i F r (0) e µ y,i(0)+σ y,i (0)z e 1 ] 2 z2 dz 1 2π The Inaugural Fixed Income Conference, Prague, September

25 Pricing of a YYIIS: the LIBOR market model (cont d) This YYIIS price depends on: the (instantaneous) volatilities of nominal and real forward rates and their correlations, for i = 2,..., M; the correlations between real forward rates and forward inflation indices, again for i=2,..., M. Compared with the JY expression, this last formula looks more complicated both in terms of input parameters and in terms of the calculations involved. However, one-dimensional numerical integrations are not so cumbersome. Moreover, as is typical in a market model, the input parameters can be determined more easily than in the previous short-rate approach. Both approaches seen so far have the drawback that the volatility of real rates may be hard to estimate. This is why we propose a second marketmodel approach, which enables us to overcome this estimation issue. The Inaugural Fixed Income Conference, Prague, September

26 Pricing of a YYIIS: a second market model Applying the definition of forward CPI and using the fact that I i is a martingale under Q T i n, we can also write, for t < T i 1, { } YYIIS(t, T i 1, T i, ψ i, N) = Nψ i P (t, T i )E T I(Ti ) i n I(T i 1 ) 1 F t { } = Nψ i P (t, T i )E T Ii (T i i ) n I i 1 (T i 1 ) 1 Ft { } = Nψ i P (t, T i )E T Ii (T i i 1 ) n I i 1 (T i 1 ) 1 Ft. We recall that, under Q T i n, di i (t) = σ I,i I i (t) dw I i (t) and that an analogous evolution holds for I i 1 under Q T i 1 n. The Inaugural Fixed Income Conference, Prague, September

27 Pricing of a YYIIS: a second market model (cont d) The dynamics of I i 1 under Q T i n are di i 1 (t) = I i 1 (t)σ I,i 1 τ i σ n,i F n (t; T i 1, T i ) 1 + τ i F n (t; T i 1, T i ) ρ I,n,i dt + σ I,i 1 I i 1 (t) dw I i 1(t), where σ I,i 1 is a positive constant, Wi 1 I is a QT i n -Brownian motion with dwi 1 I (t) dw i I(t) = ρ I,i dt, and ρ I,n,i is the instantaneous correlation between I i 1 ( ) and F n ( ; T i 1, T i ). The evolution of I i 1, under Q T i n, depends on the nominal rate F n ( ; T i 1, T i ). To avoid unpleasant calculations, we freeze the above drift at its current time-t value, so that I i 1 (T i 1 ) conditional on F t is lognormally distributed also under Q T i n. The Inaugural Fixed Income Conference, Prague, September

28 Pricing of a YYIIS: a second market model (cont d) Also the ratio I i (T i 1 )/I i 1 (T i 1 ) conditional on F t distributed under Q T i n. This leads to { } E T Ii (T i i 1 ) n Ft = I i(t) I i 1 (T i 1 ) I i 1 (t) edi(t), is lognormally where [ ] τi σ n,i F n (t; T i 1, T i ) D i (t) = σ I,i τ i F n (t; T i 1, T i ) ρ I,n,i ρ I,i σ I,i + σ I,i 1 (T i 1 t), so that YYIIS(t, T i 1, T i, ψ i, N) = Nψ i P n (t, T i ) [ ] Pn (t, T i 1 )P r (t, T i ) P n (t, T i )P r (t, T i 1 ) edi(t) 1. N.B. This pricing method is equivalent to, but independently derived from, that of Belgrade, Benhamou and Koehler (2004). The Inaugural Fixed Income Conference, Prague, September

29 Pricing of a YYIIS: a second market model (cont d) Finally, the value at time t of the inflation-indexed leg of the swap is [ ] I(t) YYIIS(t, T, Ψ, N) = Nψ ι(t) I(T ι(t) 1 ) P r(t, T ι(t) ) P n (t, T ι(t) ) + N M i=ι(t)+1 In particular at t = 0, [ ψ i P n (t, T i 1 ) P ] r(t, T i ) P r (t, T i 1 ) edi(t) P n (t, T i ). YYIIS(0, T, Ψ, N) = N M ψ i P n (0, T i ) i=1 [ ] 1 + τi F n (0; T i 1, T i ) 1 + τ i F r (0; T i 1, T i ) edi(0) 1. The Inaugural Fixed Income Conference, Prague, September

30 Pricing of a YYIIS: a second market model (cont d) This YYIIS price depends on: the (instantaneous) volatilities of forward inflation indices and their correlations; the (instantaneous) volatilities of nominal forward rates; the instantaneous correlations between forward inflation indices and nominal forward rates. This pricing formula looks pretty similar to that in the JY case and may be preferred to the one in the LIBOR market model, since it combines the advantage of a fully-analytical formula with that of a market-model approach. Moreover, the correction term D does not depend on the volatility of real rates, which is typically difficult to estimate. The only drawback is that the approximation it is based on may be rough for long maturities T i. The formula, however, is exact when the correlations ρ I,n,i are set to zero and the terms D i are simplified accordingly. The Inaugural Fixed Income Conference, Prague, September

31 Inflation-indexed caplets An Inflation-Indexed Caplet (IIC) is a call option on the inflation rate implied by the CPI index. Analogously, an Inflation-Indexed Floorlet (IIF) is a put option on the same inflation rate. In formulas, at time T i, the IICF payoff is Nψ i [ω ( )] + I(Ti ) I(T i 1 ) 1 κ, where κ is the IICF strike, ψ i is the contract year fraction for the interval [T i 1, T i ], N is the contract nominal value, and ω = 1 for a caplet and ω = 1 for a floorlet. We set K := 1 + κ. The Inaugural Fixed Income Conference, Prague, September

32 Inflation-indexed caplets (cont d) Standard no-arbitrage pricing theory implies that the value at time t T i 1 of the IICF at time T i is IICplt(t, T i 1, T i, ψ i, K, N, ω) { = Nψ i E n e R [ ( )] } + T i t n(u) du I(Ti ) ω I(T i 1 ) K F t { [ ( )] } + = Nψ i P n (t, T i )E T I(Ti ) i n ω I(T i 1 ) K Ft. The pricing of a IICF is thus similar to that of a forward-start (cliquet) option. We now derive analytical formulas both under the JY model and under our second market model approach. The Inaugural Fixed Income Conference, Prague, September

33 Inflation-indexed caplets: the JY model The assumption of Gaussian nominal and real rates leads to a CPI that is lognomally distributed under Q n. When we move to a (nominal) forward measure the type of distribution is preserved. I(T Hence, i ) I(T i 1 ) conditional on F t is lognormally distributed also under Q T i n, and the IICF price can be calculated as follows. If X is a lognormal random variable with E(X) = m and Std[ln(X)] = v, then E { ( [ω(x K)] +} = ωmφ ω ln m K + ) ( 1 2 v2 ωkφ ω ln m K ) 1 2 v2, v v where Φ denotes the standard normal distribution function. The Inaugural Fixed Income Conference, Prague, September

34 Inflation-indexed caplets: the JY model (cont d) The conditional expectation of I(T i )/I(T i 1 ) is immediately obtained through the price of a YYIIS: { } E T I(Ti ) i n Ft = P n(t, T i 1 ) P r (t, T i ) I(T i 1 ) P n (t, T i ) P r (t, T i 1 ) ec(t,t i 1,T i ). The variance of the log of the ratio can be equivalently calculated under the (nominal) risk-neutral measure. We get: { Var T i n ln I(T } i) F t = V 2 (t, T i 1, T i ), I(T i 1 ) where, setting T i := T i T i 1, V 2 (t, T i 1, T i ) = The Inaugural Fixed Income Conference, Prague, September

35 Inflation-indexed caplets: the JY model (cont d) = σ2 n 2a 3 (1 e a n T i ) 2 [1 e 2a n(t i 1 t) ] + σ2 r n σ n σ r 2ρ n,r 2a 3 r (1 e a r T i ) 2 [1 e 2a r(t i 1 t) ] a n a r (a n + a r ) (1 e a n T i )(1 e a r T i )[1 e (a n+a r )(T i 1 t) ] [ + σi T 2 i + σ2 n a 2 T i + 2 e a n T i 1 e 2a n T i 3 ] n a n 2a n 2a n [ + σ2 r a 2 T i + 2 e a r T i 1 e 2a r T i 3 ] r a r 2a r 2a r [ σ n σ r 2ρ n,r T i 1 e a n T i 1 e ar Ti + 1 ] e (an+ar) Ti a n a r a n a r a n + a r ] + 2ρ n,i σ n σ I a n [ T i 1 e a n T i a n 2ρ r,i σ r σ I a r [ T i 1 e a r T i a r ]. The Inaugural Fixed Income Conference, Prague, September

36 Inflation-indexed caplets: the JY model (cont d) We finally have: IICplt(t, T i 1, T i, ψ i, K, N, ω) [ P n (t, T i 1 ) P r (t, T i ) = ωnψ i P n (t, T i ) P n (t, T i ) P r (t, T i 1 ) ec(t,t i 1,T i ) Φ ω ln P n(t,t i 1 )P r (t,t i ) KP n (t,t i )P r (t,t i 1 ) + C(t, T i 1, T i ) V 2 (t, T i 1, T i ) V (t, T i 1, T i ) KΦ ω ln P n(t,t i 1 )P r (t,t i ) KP n (t,t i )P r (t,t i 1 ) + C(t, T i 1, T i ) 1 2 V 2 ] (t, T i 1, T i ), V (t, T i 1, T i ) which is clearly of a Black and Scholes type. The Inaugural Fixed Income Conference, Prague, September

37 Inflation-indexed caplets: a market model We now try and calculate the IICF price under a market model. To this end, we apply the tower property of conditional expectations to get IICplt(t, T i 1, T i, ψ i, K, N, ω) } E T i = Nψ i P n (t, T i )E T n {[ω(i(t i ) KI(T i 1 ))] + F Ti 1 i n F t I(T i 1 ), where we assume that I(T i 1 ) > 0. Sticking to a market-model approach, the calculation of the inner expectation depends on whether we model forward rates or the forward CPIs. We here follow our second market-model approach, since it allows us to derive a simpler formula with less input parameters. The Inaugural Fixed Income Conference, Prague, September

38 Inflation-indexed caplets: a market model (cont d) Assuming again that, under Q T i n, di i (t) = σ I,i I i (t) dw I i (t) and remembering that I(T i ) = I i (T i ), we have: { [ω(i(ti ) KI(T i 1 ))] + } F Ti 1 E T i n = E T { i n [ω(ii (T i ) KI(T i 1 ))] + } F Ti 1 = ωi i (T i 1 )Φ ω ln I i(t i 1 ) KI(T i 1 ) σ2 I,i (T i T i 1 ) σ I,i Ti T i 1 ωki(t i 1 )Φ ω ln I i(t i 1 ) KI(T i 1 ) 1 2 σ2 I,i (T i T i 1 ). σ I,i Ti T i 1 The Inaugural Fixed Income Conference, Prague, September

39 Inflation-indexed caplets: a market model (cont d) By definition of I i 1, the IICF price thus becomes ωnψ i P n (t, T i )E T I i i (T i 1 ) n I i 1 (T i 1 ) Φ ω ln I i(t i 1 ) KI i 1 (T i 1 ) σ2 I,i (T i T i 1 ) σ I,i Ti T i 1 KΦ ω ln I i(t i 1 ) KI i 1 (T i 1 ) 1 2 σ2 I,i (T i T i 1 ) F t σ I,i Ti T i 1. Remembering the dynamics of I i 1 under Q T i n, and freezing again the drift at its time-t value, we have that under Q T i n : ln I ( i(t i 1 ) I i 1 (T i 1 ) F t N ln I i(t) I i 1 (t) + D i(t) 1 ) 2 V i 2 (t), Vi 2 (t). where V 2 i (t) := (σ2 I,i 1 + σ2 I,i 2ρ I,iσ I,i 1 σ I,i )(T i 1 t). Therefore, The Inaugural Fixed Income Conference, Prague, September

40 Inflation-indexed caplets: a market model (cont d) IICplt(t, T i 1, T i, ψ i, K, N, ω) = ωnψ i P n (t, T i ) I i(t) I i 1 (t) edi(t) Φ ω ln I i(t) KI i 1 (t) + D i(t) V2 i (t) V i (t) KΦ ω ln I i(t) KI i 1 (t) + D i(t) 1 2 V2 i (t), V i (t) where V i (t) := V 2 i (t) + σ2 I,i (T i T i 1 ), and I i(t) I i 1 (t) = 1+τ if n (t;t i 1,T i ) 1+τ i F r (t;t i 1,T i ). This price depends on the volatilities of the two forward inflation indices and their correlation, the volatility of nominal forward rates, and the correlations between forward inflation indices and nominal forward rates. The Inaugural Fixed Income Conference, Prague, September

41 Inflation-indexed caps An inflation-indexed cap is a stream of inflation-indexed caplets. An analogous definition holds for an inflation-indexed floor. Given the set of dates T 0, T 1,..., T M, with T 0 = 0, a IICapFloor pays off, at each time T i, 1,..., M, Nψ i [ω ( )] + I(Ti ) I(T i 1 ) 1 κ, where κ is the IICapFloor strike, ψ i are the contract year fractions for the intervals [T i 1, T i ], 1,..., M, N is the contract nominal value, ω = 1 for a cap and ω = 1 for a floor. We again set K := 1 + κ, T := {T 1,..., T M } and Ψ := {ψ 1,..., ψ M }. The Inaugural Fixed Income Conference, Prague, September

42 Inflation-indexed caps (cont d) Sticking to our market model, from the caplet pricing formula we get: M IICapFloor(0, T, Ψ, K, N, ω) = ωn ψ i P n (0, T i ) i=1 1 + τ if n (0; T i 1, T i ) 1 + τ i F r (0; T i 1, T i ) edi(0) Φ ω ln 1+τ if n (0;T i 1,T i ) K[1+τ i F r (0;T i 1,T i )] + D i(0) V2 i (0) V i (0) KΦ ω ln 1+τ if n (0;T i 1,T i ) K[1+τ i F r (0;T i 1,T i )] + D i(0) 1 2 V2 i (0). V i (0) This price depends on the volatilities of forward inflation indices and their correlations, the volatilities of nominal forward rates, and the instantaneous correlations between forward inflation indices and nominal forward rates. The Inaugural Fixed Income Conference, Prague, September

43 Conclusions The pricing of inflation-indexed derivatives requires the modelling of both nominal and real rates and of the reference consumer price index. The foreign-currency analogy allows one to view real rates as the rates in a foreign economy and to treat the CPI as an exchange rate. Assuming a Gaussian distribution for both instantaneous (nominal and real) rates, as in the Jarrow and Yildirim (2003) model, we have derived analytical formulas for inflation-indexed swaps and caps. We have also proposed two alternative market-model approaches leading to analytical formulas with easier-to-estimate input parameters. Our market-model approach allows for extensions based on forward volatility uncertainty in the spirit of Brigo, Mercurio and Rapisarda (2004) or Gatarek (2003). The Inaugural Fixed Income Conference, Prague, September