PRICING FINANCIAL CONTRACTS ON INFLATION FABIO MERCURIO BANCA IMI, MILAN http://www.fabiomercurio.it Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 1
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 goods and services. • Denoting by I ( t ) the CPI’s value at time t , the inflation rate over the time interval [ t, T ] is therefore: i ( t, T ) := I ( T ) I ( t ) − 1 . • In theory, but also in practice, inflation can become negative. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 2
Stylized facts (cont’d) Historical plots of CPI’s 116 190 115 188 114 186 113 184 112 182 111 180 110 178 109 176 30−sep−01 31−aug−02 31−aug−03 31−jul−04 30−Sep−01 31−Aug−02 31−Aug−03 31−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. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 3
Stylized facts (cont’d) • Banks are used to issue inflation-linked bonds, where a zero-strike floor is offered in conjunction with the “pure” bond. • To grant positive coupons, the inflation rate is typically floored at zero. • Accordingly, floors with low strikes are the most actively traded options on inflation rates. • Other extremely popular derivatives are inflation-indexed swaps. • Two are the main inflation-indexed swaps traded in the market: – the zero coupon (ZC) swap; – the year-on-year (YY) swap. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 4
The related literature • Inflation-indexed derivatives require a specific model to be valued. • Main references: Barone and Castagna (1997), van Bezooyen et al. (1997), Hughston (1998), Kazziha (1999), Cairns (2000), Jamshidian (2002), Jarrow and Yildirim (2003), Korn and Kruse (2003), Belgrade et al. (2004), Mercurio (2005), Kruse and N¨ ogel (2006) and Mercurio and Moreni (2006). • Inflation derivatives are priced with a foreign-currency analogy (the pricing is equivalent to that of a cross-currency interest-rate derivative). • In a short rate approach, one models the evolution of the instantaneous nominal and real rates and of the CPI (interpreted as the “exchange rate” between the nominal and real economies). • Recent approaches are based on market models, where one models forward CPI indices and nominal rates. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 5
Zero-coupon inflation-indexed swaps N [(1 + K ) M − 1] Party A ✻ ✲ T M 0 ❄ � � Party B I ( T M ) N − 1 I 0 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 ( T M ) � N − 1 . I 0 Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 6
Year-on-year inflation-indexed swaps Nϕ i K Party A ✻ ✲ 0 T 1 T 2 T i − 1 T i T M ❄ Party B � � I ( T i ) Nψ 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 ( T i ) � 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. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 7
ZCIIS and YYIIS rates Both ZC and YY swaps are quoted, in the market, in terms of the corresponding fixed rate K . 2.6 2.55 2.5 2.45 Swap rates (in %) 2.4 2.35 2.3 2.25 2.2 2.15 YY rates ZC rates 2.1 0 2 4 6 8 10 12 14 16 18 20 Maturity Figure 2: Euro inflation swap rates as of October 7, 2004. The reference CPI is the Euro-zone ex-tobacco index. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 8
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 � I ( T i ) �� + � Nψ i ω 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 . Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 9
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 ) � I ( T i ) �� � �� + � = Nψ i P n ( t, T i ) E T i ω I ( T i − 1 ) − K � F t n �� � �� + � � I i ( T i ) = Nψ i P n ( t, T i ) E T i ω I i − 1 ( T i − 1 ) − K � F t , n where we define the T i -forward CPI by I i ( t ) := I ( t ) P r ( t, T i ) P n ( t, T i ) . Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 10
A market model with stochastic volatility We assume that, under a reference measure Q : • Nominal rates F i are lognormally distributed with constant volatilities; • Forward CPI’s I i follow Heston-like dynamics with a common volatility process V ( t ) : i dZ Q ,F dF i ( t ) /F i ( t ) =( . . . ) dt + σ F i V ( t ) dZ Q ,I � d I i ( t ) / I i ( t ) =( . . . ) dt + σ I i i � V ( t ) dW Q , dV ( t ) = α ( θ − V ( t )) dt + ǫ V (0) = V 0 , where σ I i , σ F i , α , θ , ǫ and V 0 are positive constants, and 2 αθ > ǫ to ensure positiveness of V . We allow for correlations between Brownian motions Z Q ,F , Z Q ,I , W Q . i i Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 11
A market model with stochastic volatility (cont’d) We take Q = Q 0 , where Q 0 is the spot LIBOR measure corresponding to the numeraire β ( t ) � B d ( t ) = P ( t, β ( t )) [1 + τ l F l ( t )] , β ( t ) = T j if T j − 1 < t ≤ T j . l =1 By definition of B d and the change-of-measure technique, we have, under Q 0 , i τ l F l ( t ) 1 + τ l F l ( t ) dt + dZ 0 ,F � dF i ( t ) /F i ( t ) = σ F σ F l ρ F − ( t ) i i,l i l = β ( t )+1 i τ l F l ( t ) � l ρ F,I 1 + τ l F l ( t ) dt + dZ 0 ,I � V ( t ) σ I σ F d I i ( t ) / I i ( t ) = − ( t ) i i l,i l = β ( t )+1 dZ 0 ,F dZ 0 ,F dZ 0 ,I dZ 0 ,F ( t ) = ρ F,I ( t ) = ρ F i,l dt, l,i dt i l i l Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 12
The pricing of caplets The price at time t ≤ T j − 1 of the j -th caplet, is, under the measure Q T j , � + � I j ( T j ) T j IICplt j ( t, K ) = P ( t, T j ) E I j − 1 ( T j − 1 ) − K t � + ∞ ( e s − e k ) + q j = P ( t, T j ) t ( s ) ds −∞ K ) and q j t ( s ) ds = Q T j { ln [ I j ( T j ) / I j − 1 ( T j − 1 )] ∈ [ s, s + ds ] |F t } . where k = ln( Remark. Instead of having a payoff depending on a single asset S ( t ) , as it is for standard or cliquet options (paying off [ S ( T j ) /S ( T j − 1 ) − K ] + in T j ), here the payoff depends on the ratio between two different assets at two different times. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 13
The pricing of caplets (cont’d) Following Carr and Madan (1999), we rewrite the caplet price in term of its (renormalized) Fourier transform: � + ∞ IICplt j ( t, e k ) = P ( t, T j )e − ηk e − isk ψ j t ( η, s ) ds 2 π −∞ � + ∞ = P ( t, T j )e − ηk e − isk ψ j Re t ( η, s ) ds π 0 φ j t ( u − ( η + 1) i ) ψ j t ( η, u ) = ( η + iu )( η + 1 + iu ) where the only unknown is the conditional characteristic function φ j t ( · ) of ln ( I j ( T j ) / I j − 1 ( T j − 1 )) , and where η ∈ R + is used to ensure L 2 -integrability when k → −∞ . Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 14
Derivation of the characteristic function Our objective is now to find an explicit formula for φ j t . Setting Y j ( t ) := ln I j ( T j ) , we recall that, by definition of characteristic function and the Markov property: � � I j ( Tj ) iu ln T j φ j I j − 1( Tj − 1) t ( u ) = E e = H ( V ( t ) , Y j ( t ) , Y j − 1 ( t ) , F 1 ( t ) , . . . , F j ( t )) . t Applying the Feynman-Kaˇ c theorem, H can then be found by solving a related PDE. Remark. In the general case, due to the unpleasant presence of drift terms � of type V ( t ) F l ( t ) / (1 + τ l F l ( t )) , there are no a priori reasons for the PDE to be explicitly solvable. In the following, we thus investigate a particular case allowing for an explicit solution. Stochastic processes: Theory and Applications, Bressanone, 16 July 2007 15
Recommend
More recommend
