Rational Models for Inflation-Linked Derivatives
Henrik Dam Andrea Macrina David Skovmand David Sloth
q-fin.PR q-fin.MF 60J25, 60H30, 91G20, 91G30
  

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MengeROS: a Crowd Simulation Tool for Autonomous Robot Navigation
    Anoop Aroor1, Susan L. Epstein1,2 and Raj Korpan1

Department of Computer Science

The Graduate Center1 and Hunter College2 of The City University of New York

New York, NY 10065

aaroor@gradcenter.cuny.edu, susan.epstein@hunter.cuny.edu, rkorpan@gradcenter.cuny.edu

    December 30, 2023
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We construct models for the pricing and risk management of inflation-linked derivatives. The models are rational in the sense that linear payoffs written on the consumer price index have prices that are rational functions of the state variables. The nominal pricing kernel is constructed in a multiplicative manner that allows for closed-form pricing of  vanilla inflation products suchlike zero-coupon swaps, year-on-year swaps, caps and floors,  and the exotic limited-price-index swap. We study the conditions necessary for the multiplicative nominal pricing kernel to give rise to short rate models for the nominal interest rate process. The proposed class of pricing kernel models retains the attractive features of a nominal multi-curve interest rate model, such as closed-form pricing of nominal swaptions, and it isolates the so-called inflation convexity-adjustment term arising from the covariance between the underlying stochastic drivers. We conclude with examples of how the model can be calibrated to EUR data.[The authors are grateful to L. P. Hughston and to participants of the JAFEE 2016 Conference (Tokyo, August 2016), the CFE 2016 Congress (Sevilla, December 2016), 2016 QMF Conference (Sydney, December 2016), and of the London Mathematical Finance Seminar held at King's College London (30 November 2017) for comments and suggestions. The authors especially acknowledge high-quality feedback provided by two anonymous reviewers.]

 


Keywords: 
Inflation-linked derivatives, rational term structure models, convexity adjustment, calibration, pricing kernels, year-on-year swap, limited price index.

 


AMS subject classification: 60J25, 60H30, 91G20, 91G30.





§ INTRODUCTION
 
The inflation market has grown in the aftermath of the 2008 financial crisis. Central banks have been conducting aggressive quantitative easing to keep inflation off the cliff of deflation, and the ensuing fears have driven hedging needs. As a consequence, the market for trading inflation has soared to the point where standard inflation derivatives are now cleared on the London Clearing House (LCH) in numbers exceeding 100 bn EUR measured by notional outstanding value in early 2017. As this number only counts linear derivatives, the total market size is likely much larger. Among the products cleared one finds the Year-on-Year swap (YoY swap), swapping annual inflation against a fixed strike, and the Zero-Coupon swap (ZC swap), which swaps cumulative inflation against a fixed strike at maturity.

 Among the OTC-traded nonlinear derivatives, the most important is arguably the YoY cap/floor, which is in principle a portfolio of calls (caplets) or puts (floorlets) with equal strike on YoY inflation. Another significant derivative is the ZC cap/floor, which is simply a call/put on the ZC swap rate. The derivatives market is dwarfed in size by the market for inflation-linked bonds. These bonds are typically government-issued debt where the principal is linked to the consumer price index (CPI) or similar. The bonds often have an embedded YoY floor protecting the principal from being adjusted downwards by deflation. Limited Price Index  (LPI) products come with both a lower and upper bound on the principal adjustment creating a path-dependent collar on inflation. Despite its exotic nature LPIs have been in high demand by pension funds. All products should ideally be priced in a consistent manner using a tractable arbitrage-free model. Cap/floor products display volatility skews and non-flat term structures of volatility, both of which the model also should be able to capture. Besides, the model should yield closed-form solutions for the price of the most traded derivatives, here the YoY and the ZC cap/floor.

<cit.> develops a general arbitrage-free theory of interest rates and inflation in the case where the consumer price index and the real and nominal interest rate systems are jointly driven by a multi-dimensional Brownian motion. This approach is based on a foreign exchange analogy in which the CPI is treated as a foreign exchange rate, and the “real” interest rate system is treated as if it were the foreign interest rate system associated with the foreign currency. The often-cited work by <cit.> makes use of such a setup. They consider a three-factor model (i.e., driven by three Brownian motions) in which the CPI is modelled as a geometric Brownian motion, with deterministic time-dependent volatility and the two interest rate systems are treated as extended Vasicek-type (or Hull-White) models. Similar to <cit.>,  <cit.>  use a short-rate approach where the nominal and the inflation rates are both modelled by Hull-White processes while discarding the idea of a real economy. A GBM-based model for the CPI provides the baseline framework for how one might understand implied volatility in such a market, but any GBM model for the CPI does not, by construction, reproduce volatility smiles.


Further development of inflation models has paralleled that of interest rates models. For example inflation counterparts to the nominal LIBOR Market Model, see for example <cit.>, have been studied in <cit.>, <cit.>, and <cit.>. While these models can reproduce smiles—augmented with stochastic volatility or jumps—they rely on numerically intensive algorithms or approximations for the pricing of ZC cap/floors, in particular. One may say similarly of the models by <cit.>, <cit.>, and <cit.> who in a similar manner use forward inflation, or in the case of <cit.> the forward inflation swap rate, as the model primitive. <cit.> builds an inflation counterpart to the nominal model of <cit.> and <cit.>.  One also finds <cit.> in the local volatility context, <cit.> extending the GBM methodology with <cit.> stochastic volatility, and <cit.>   adding stochastic volatility to the <cit.> model.
Our work is inspired by the approach to nominal term structure of interest rates based on the so-called rational models. This choice is motivated by the success of the rational model framework as documented in the comprehensive empirical study of <cit.> who demonstrate that linear-rational models perform as well or better than similar affine term structure models. Furthermore, the rational model framework has been extended to model multiple nominal curves and credit risk in <cit.> and <cit.>; it is this approach we follow. This framework allows for analytical expressions for swaptions, which is not the case for affine term structure models. In this paper, we demonstrate how rational models for inflation are constructed, which retain the tractability of the nominal counterpart and can price, in closed-form, all the relevant derivatives suchlike YoY and ZC cap/floors and LPI swaps.
  
   In Section 2, we first present the model in full generality. Following <cit.>, we study the conditions for a short rate model representation to be obtained. In Section 3 we derive option pricing formulae under different assumptions in the driving process, and in Section 4 we end with an example that shows how the model can be simultaneously calibrated to inflation derivatives and a multiple-curve nominal interest rate market.


§ RATIONAL TERM STRUCTURES
 
We adopt the pricing kernel approach, which was pioneered  by 
<cit.>, <cit.>, <cit.>  and <cit.>—for a good summary see <cit.> and, for a more recent account, <cit.>.  <cit.> propose pricing kernel models to construct so-called curve-conversion factor processes, which link distinct yield-curves in a consistent arbitrage-free manner, and which give rise to the across-curve pricing formula for consistent valuation and hedging of financial instruments across curves. Applications include the pricing of inflation-linked and hybrid fixed-income securities. A property of the pricing kernel approach is the ease with which the pricing and hedging of multiple currencies can be handled. This is the property one benefits from when considering inflation-linked pricing, and nominal and real economies are introduced in analogy to domestic and foreign economies. Compared to the classical approach, in order to allow for negative short rates, we relax the paradigm  and consider  general semimartingale dynamics for the pricing kernels. The approach taken next is one where the existence of a pricing kernel model is postulated and its dynamics are modelled. It is via the pricing kernel that no-arbitrage price processes of tradable assets are generated by imposing that the asset price process, when multiplied by the pricing kernel process, be a martingale with respect to the probability measure the pricing kernel dynamics are produced. This no-arbitrage notion is one presented in textbooks suchlike, e.g., <cit.> and <cit.>.


 §.§ General model
 
We model a financial market by a filtered probability space (Ω,,,(_t)_0≤ t), where  denotes the real probability measure and (_t)_0≤ t the market filtration satisfying the usual conditions. A finite time horizon is considered, i.e., a time line 0≤ t≤ T< <∞, throughout. 

[Pricing kernel]
We call a stochastic process (π_t)_0≤ t with π_0=1 a pricing kernel if it is a strictly positive, , semimartingale such that π_t has  finite expectation for all t≥ 0.  

Let  ℒ_T^1(μ;π) = {Χ: Ω→ s.t. Χ is _T-measurable and ^μ [ π_T Χ < ∞]} where μ is a probability measure on (Ω,). Let (π^N_t)_0 ≤ t be the (nominal) pricing kernel process. If we consider some claim  Χ∈ℒ_T^1(;π^N), then by  standard no-arbitrage theory, see e.g. <cit.>, the process (V^χ _t  )_0 ≤ t ≤ T, defined by

    V^χ_t  = 1/π_t^N_t  [  π_T^NΧ],

is an arbitrage-free price process.  The notation _t[·] is short-hand for [ · |ℱ_t]. Following <cit.>[Proposition 2.2], we have:

Consider n assets with price processes (S^1_t),…,(S^n_t) satisfying Eq. (<ref>), i.e., such that (π_t^N S_t^i ) is a -martingale for i=1,…,n. Assume the asset with strictly positive price process (S_t^1) is traded. Then, the market is free of arbitrage.


By Eq. (<ref>), the process ξ_t := S^1_t π^N_t /S^1_0 is a strictly positive martingale with ξ_0=1. A measure  may be defined by ξ_t = . /|_ _t on any finite interval, and by the Bayes' rule one obtains

    _t  [ S^i_T/S^1_T  ] = _t  [ξ_T (S^i_T/S^1_T)  ]/ξ_t = _t  [π^N_T S^i_T  ]/π_t S^1_t  = S^i_t/S^1_t,

for 2≤ i ≤ n. Thus,  is a risk-neutral measure associated with the numeraire (S^1_t). The existence of a pricing kernel, here the process (π^N_t)_0≤ t, guarantees absence of arbitrage, also in the case of uncountably many assets. Here we refer to the no-arbitrage notion of “no asymptotic free lunch with vanishing risk” (NAFLVR) developed in <cit.>.

We are agnostic as to how the asset with price process (S^1_t) is chosen; for example it may be a zero-coupon bond. From formula (<ref>) it follows that, for 0≤ t≤ T, the nominal zero-coupon bond price system,

    P^N_tT  = 1/π_t^N_t  [  π_T^N],
 
is free of arbitrage opportunities. Assuming that P^N_tT is differentiable in T, the short rate process (r^N_t)_0 ≤ t ≤ may be obtained by the well-known relation 
r^N_t = -∂_T ln(P_tT^N)| _T=t.
This tells that (π^N_t)  determines simultaneously the inter-temporal risk-adjustment and the discounting rate. 

The goal is to produce models, which facilitate the pricing of inflation-linked derivatives. To this end, we equip the framework with a real-market analogous to the foreign economy in the foreign-exchange analogy. If we assume that (π_t^R)_0≤ t is a pricing kernel for the real market, then the foreign-exchange analogy establishes the relationship

    C_t = π_t^R/π_t^N

where (C_t)_0≤ t denotes the CPI process that acts as an exchange rate from the nominal to the real economy, see, e.g. <cit.>.

As in <cit.>, <cit.>, <cit.> and  <cit.>,  we introduce an extra degree of flexibility and model prices with respect to an auxiliary measure  . This extra degree of freedom allows for simplified calculations or more tractable modelling under the -measure while desirable statistical properties may still be captured under the -measure. In fact it is also possible to build in  terminal distributions or “views” under , in the spirit of <cit.>, and as explicitly obtained in <cit.>. This is a feature expected by practitioners of inflation-linked trading, motivated by the fact that inflation is an area that often receives significant attention from monetary policymakers and is subject to so-called “forward guidance”. With regard to how to induce the measure change for such a purpose, we refer to <cit.>, <cit.> for the multivariate generalisation, and <cit.> for an application in a multi-curve term structure setup. We shall model the Radon-Nikodym process (M_t)_0 ≤ t with M_0 = 1 as a strictly positive,   martingale and fix some time <∞.  Then,
(A) = [M_ A],
for A∈_, defines an equivalent  measure. By setting h^N_t =π_t^N / M_t, with no loss of generality, we can express the fundamental pricing equation (<ref>) under  by the Bayes formula: 

    V^χ _t   = 1/π_t^N_t  [  π_T^NΧ] 
    =1/h_t^N M_t_t  [ M_T h_T^NΧ] = 1/h_t^N_t  [  h_T^NΧ],

for 0 ≤ t≤ T  < ∧ and Χ∈ℒ^1_T ( ;π^N).  That is, (h_t^N) is the nominal pricing kernel under the -measure. Similarly,  the relationship π_t^R = M_t h^R_t introduces the real pricing kernel (h^R_t) under . It follows that, under , (h_t^N) and (h_t^R)   are strictly positive, semimartingales, see <cit.>[III Theorem 3.13],  and that C_t h_t^N =   h_t^R for all t≥ 0.



Modelling convention. Let s_t:=1/C_t for t≥ 0, for modelling convenience. From the relation (<ref>), it then follows h^N_t=s_t h^R_t. We model (h^R_t) and (s_t), where h^R_0=1/s_0=C_0, as strictly positive semimartingales under 𝕄 such that h^R_t and h^N_t have finite expectation for t≥ 0. 
[Real-kernel spread model]
Let the triplet (h_t^R,s_t,M_t)_0≤ t be such that (h_t^R)_0≤ t, (s_t)_0≤ t and (M_t)_0 ≤ t   are strictly-positive, , h^R_0=1/s_0=C_0 and M_0=1. Furthermore assume (h_t^R)_0≤ t and (s_t)_0≤ t  are semimartingales and that  (M_t)_0 ≤ t is a martingale. Denote by  the measure induced by (M_t). Assume that h_t^R and h_t^R  s_t have finite expectation for all t≥ 0 under . We call such a triplet a real-kernel spread model (RSM).

Often, the pricing of inflation-linked instruments is performed under either the nominal risk-neutral measure ^N or the real risk-neutral measure ^R. In the general setting presented so far, one is not necessarily in a position to get consistent prices under these measures. In Section <ref>, we treat this issue in the context of some well-known models, which use from the outset a risk-neutral measure. In Section <ref> we proceed to the pricing of primary inflation-linked securities in the backdrop of a  more specific model class. In Section <ref> we discuss the change to risk-neutral measures in the same model class.  


 §.§ Comparison with other models
  
In this section, we discuss other models and in a few cases show that our specification can be regarded as a generalisation. The comparisons shall help to understand our modelling approach in that they show how our model ingredients would look in known models.

In the case of equity pricing, the benchmark model is the geometric Brownian motion specification of <cit.>. In this sense, the most natural translation of this to inflation modelling is done by <cit.> specifying the inflation index under the nominal risk-neutral measure by

    C_t  = C_t  ( r^N(t)  - r^R(t) )     t + C_tσ_C   W^C_t,

where r^N and r^R are the deterministic nominal and real interest rates and (W^C_t)_0≤ t is a Brownian motion. Black-Scholes-type pricing formulae are derived for ZC caps with payoff function max[C_T/C_0-K,0], and <cit.> derives a pricing formula of a similar type for YoY caplets with payoff function max[C_T_i/C_T_i-1-K,0]. We refer to <cit.>  for the exact formulae. The formulae for the ZC cap and YoY caplet as functions of the volatility parameter σ_C  can be inverted to implied volatilities as it is commonly done for equity options. This will be relevant in Section <ref> when we calibrate some specific rational pricing models. 

<cit.> produce an important generalisation that allows for the pricing of inflation-linked securities with stochastic interest rates. In practice, the popular model specification is to assume that the nominal and the real interest rates have Hull-White dynamics. Under the nominal risk-neutral measure, such a model specification takes the form

    r^N_t = [ θ_N(t) -  a_N r^N _t  ]  t + σ_N W^N _t,       r^R_t = [ θ_R(t) - ρ_RCσ_Cσ_R - a_R r^R_t   ]  t + σ_R W^R_t,   
        C_t  =  C_t    ( r_t^N  - r_t^R  )     t + C_t  σ_C W^C_t,

where (W^N_t), (W^R_t) and (W^C_t) are dependent Brownian motions, and where θ_N(t) and θ_R(t) are functions chosen to fit the term-structure of interest rates, see <cit.>[Chapter 15] and <cit.>. 


 
 The <cit.> and the <cit.> models are RSM triplets where (h^R_t, s_t, M_t)_0≤ t is given by 

    h^R_t=exp(-∫_0^t r^R_s   s) .  ^R/^N|_ _t,      s_t = 1 /h_t^Rexp(-∫_0^t r^N_s  s),      M_t =. ^N/|_ _t.

The nominal pricing kernel (h^N_t)_0≤ t is determined by h^N_t=s_t h^R_t.


The measure change to ^N is given in <cit.>[Footnote 5], the measure change to ^R is similar and, in the Black-Scholes case, the results are standard. 

The choice above is not unique. One could, e.g., set M_t = 1 and change the other processes accordingly, which would  amount to specifying and matching the models under , instead.



 §.§ Primary inflation-linked instruments
  
We now proceed to the pricing of the primary inflation-linked products, suchlike the ZC swap and the YoY swap, which serve as the fundamental hedging instruments against inflation risk and the swap rates as underlying of exotic inflation-linked derivatives. To this end, we propose a specific class of rational pricing kernels:

[Rational pricing kernel system]  
Let  be a measure equivalent to  induced by a Radon-Nikodym process (M_t)_0≤ t.  Let (A_t^R)_0≤ t and (A_t^S
)_0≤ t  be positive martingales under  with A^S_0=A^R_0=1.  Let  ( A_t^R A_t^S) have finite expectation under  for all t≥ 0. Let the real pricing kernel (h^R_t)_0≤ t be given by  h_t^R = R(t)[ 1 + b^R(t) ( A_t^R -1 ) ] where R(t)∈ C^1 is a strictly positive deterministic function with  R(0)=C_0, and where b^R(t)∈ C^1 is a deterministic function that satisfies 0<b^R(t)<1. Furthermore, let s_t = S(t) A_t^S
where S(t)∈ C^1 is a strictly positive deterministic function with S(0)=1/C_0, and set 
h^N_t=s_t  h^R_t.
We call ( h_t^R, s_t, M_t)_0≤ t thus specified a rational pricing kernel system (RPKS).

By Ito's lemma, (h_t^R) and (s_t) are strictly positive semimartingales. An RPKS is in particular an RSM-triplet and therefore, by Section <ref>, it produces a nominal and a real market, both free of arbitrage opportunities. The martingale (A^R_t) generates the randomness in the real market, while the joint law of (A^S_t) and (A^R_t) generates the randomness in the nominal market. All  derivations throughout will be obtained under the assumption of having an RPKS.
[Affine payoffs evaluated in an RPKS]  
Assume an RPKS. The price process (V^χ_t)_0≤ t≤ T of a contract with payoff function Χ = a_1+ a_2 C_T, for a_1, a_2∈, at the fixed date T≥ t≥ 0 is given by 

    V^χ_t  = a_2 (b_0(T) +   b_1(T) A_t^R) + a_1 (b_2(T) A_t^S +  b_3(T)  _t[  A_T^R A_T^S] ) / b_2(t) A_t^S + b_3(t)  A_t^R A_t^S

where,  for 0≤ t ≤ T, 
b_0(t) = R(t)  (1-b^R(t)),    b_1(t) =  R(t) b^R(t),
b_2(t) = R(t)  (1-b^R(t))S(t),  b_3 (t) = R(t) b^R(t)S(t). If a_1=0, i.e. the payoff is linear in C_T,  the price process V^χ_t is a rational function of A^R_t and A^S_t.


It follows by the -pricing equation (<ref>).

The price process (P^N_tT)_0≤ t≤ T of the nominal ZC bond follows from Eq. (<ref>) for a_1=1 and a_2=0. We have,

    P_tT^N =   b_2(T) A_t^S + b_3(T)  _t[  A_T^R A_T^S]  / b_2(t) A_t^S + b_3(t)  A_t^R A_t^S

with b_2(t) and b_3(t) given in Proposition <ref>. 
It then follows that the initial nominal term structure P^N_0t, 0≤ t≤ T, is given by
P^N_0t=R(t)S(t)( 1+  b^R (t)([A^R_tA^S_t] -1)).
In particular, the parameter function S(t) appearing in both, the price processes of the nominal ZC bond and the contract (<ref>), can thus be used for calibrating to the market-observed prices P^N_0t, 0≤ t≤ T, according to
S(t)=P^N_0t/[ R(t)(1+  b^R (t)([A^R_tA^S_t] -1))].
We note that should  t↦[A^R_t A^S_t] not belong to C^1, one can calculate its value in all relevant time points and use a C^1-interpolation, and nevertheless produce the same price for any financial product whose payoff only depends on state variables at those times.

The most basic inflation-linked product is the ZC swap, which gives exposure to the CPI value at the swap maturity T for an annualised fixed payment. Its price process (V_t^ZCS) _0≤ t≤ T can be written in the form

    V^ZCS_t = 1/h^N_t_t [h^N_T  ( C_T/C_0 -K )]  = P_tT^IL  - K  P_tT^N
 
where  
P_tT^IL =_t[h^N_T   C_T/C_0]/h^N_t
is the price of an inflation-linked ZC bond at t≤ T. ZC swaps are highly liquid for several maturities and therefore it is reasonable to consider an actual term-structure of ZC swaps and aim at constructing models able to calibrate to the relevant market data in a parsimonious manner. By Eq. (<ref>), given a nominal term-structure,  a ZC swap term-structure is equivalent to an inflation-linked ZC bond term-structure,  and fitting either is equivalent.  The price of an inflation-linked ZC bond within an RPKS follows directly from Proposition <ref>:

    P_tT^IL = 1/C_0b_0(T) +   b_1(T) A_t^R/ b_2(t) A_t^S + b_3(t)  A_t^R A_t^S
 
with b_0(t) and b_1(t) given in Proposition <ref>.  We see that by matching the degree of freedom R(t) to the initial term structure P^IL_0t of inflation-linked bonds as implied from the market, i.e. R(t) = P^IL_0t C_0, the model replicates the term structure of ZC swaps. For ZC swaps, a de-annualised fair rate is quoted, namely a number  such that for K=(1+)^T the initial value of the swap is zero. Given P^N_0T, the initial term structure P^ IL_0T is implied from the ZC swap market fair rates k^ ZC_0T via 
k^ ZC_0T  =  (P^N_0T/P^ IL _0T ) ^1/T-1.
The price process (P^R_tT)_0≤ t≤ T of a real ZC bond is

    P^R_tT  = 1/h_t^R_t[ h^R_T] =  b_0(T) + b_1(T) A_t^R/ b_0(t) + b_1(t) A_t^R,

with b_0(t) and b_1(t) as in Proposition <ref>.  In accordance with the foreign-exchange analogy, it holds that
P^R_tT  C_t = P^IL_tT.

Next, we consider the Year-on-Year swap (YoY swap) which exchanges yearly percentage increments of CPI against a fixed rate. The YoY swap can be decomposed into swaplets, so we consider first the price V_tT_i ^YoYSL at time  t<T_i-1 of a such over the period [T_i-1, T_i]. By the pricing relation (<ref>) we have  

    V_tT_i ^YoYSL =   1/h_t^N_t  [h_T_i^N (  C_T_i/C_T_i-1  - K  )  ] 
     =P_tT_i^IL S(T_i-1) A_t^S      +    b_2(T_i)  S( T_i-1)     _t^[ A^R_ T_i-1 ,A^S_ T_i-1]  / b_2(t) A_t^S + b_3(t)  A_t^R A_t^S  -   K  P^N_t T_i

with b_2(t) and b_3(t) as in Proposition <ref>.
For YoY swaps the fair rate  is quoted in financial markets such that K=1+. The price of the swap is V_tT_N^YoYS = ∑_i=1^N V^YoYSL_tT_i, from which the fair rate ^YoY can be extracted:

    ^YoY_tT_N = 1 /∑_i=1^N  P^N_t T_i∑_i=1^N    (      P_tT_i^IL  S(T_i-1) A_t^S   +   b_2(T_i)  S( T_i-1)     _t^[ A^R_ T_i-1 ,A^S_ T_i-1]  / b_2(t) A_t^S + b_3(t)  A_t^R A_t^S) -1.

If independence between  (A_t^R) and (A_t^S) is assumed, the YoY swap rate at time t=0 becomes

    k^YoY_0T_N = 1 /∑_i=1^N  P^N_0 T_i∑_i=1^N     ( P^IL_0T_i-1/P^N_0T_i-1P_0T_i^IL  ) -1.

Thus, if the independence assumption is imposed, the swap rate is completely determined by the inflation-linked and nominal term structures and hence can be expressed in a model-independent fashion. The difference between the market-observed swap rate and the above expression is often referred to as the convexity correction for the YoY swap of length T_N.



 §.§ On short-rate representation
  
In the following section, we study the question of the existence of a classical savings accounts in an RPKS, which is related to the work of <cit.>. We shall see that the obtained class of nominal pricing kernels can rarely be represented in terms of a short rate model. We study the reasons behind the lack of such a property. If it is possible to decompose a pricing kernel into

    h_t =  . /|__t^-∫_0^t r_u  u

where (r_t) is a short rate process and  is the corresponding numeraire measure, then the prices obtained in either model would be equal, i.e. the same prices could instead have been obtained using a short rate approach. To investigate whether a decomposition like (<ref>) exists, some technical material is needed. 

We recall that a special semimartingale is a process (X_t) with a unique decomposition X_t = X_0+ B_t+ N_t, where (B_t) is predictable and of finite variation, (N_t) is a local martingale and B_0=N_0=0. The decomposition is called the canonical (additive) decomposition. We recall <cit.>
 
Let (X_t) be a semimartingale with X_0=1, such that (X_t) and (X_t-) are strictly positive. Then,  (X_t) is a special semimartingale if and only if it admits a multiplicative decomposition 

    X_t =  M_t L_t,

 where (M_t) is a strictly positive and  -local martingale, (L_t) is a positive, predictable process with locally finite variation, and M_0=L_0=1. When the decomposition exists, it is unique and is given by	

    M_t= (∫_0^·1/X_s- + Δ B_s N_s  )_t,      L_t = (-∫_0^·1/X_s- + Δ B_s B_s  )_t^-1

where (B_t) and (N_t) are the processes in its canonical additive decomposition.

Comparing Eq. (<ref>) and Eq. (<ref>) in Theorem <ref> we see that a number of conditions need to be satisfied for a short rate representation to be available. The pricing kernel (h_t) has to satisfy the assumptions of Theorem <ref>, (M_t) has to be a true martingale and act as a measure change, (L_t) needs to have the specific form in (<ref>) and finally the resulting ZC bond prices need to be sufficiently differentiable. More precisely, following <cit.>, when (<ref>) exists for a pricing kernel, one calls A_t = L_t^-1 an implied savings account. When in addition A_t = 1 + ∫_0 ^t ϕ_s  s is satisfied, where 
(ϕ_t) is adapted and ∫_0^t ϕ_s s ∈ L^1(), then the forward and short rates exist, that is, (<ref>) holds, and (A_t) is termed a classical savings account.

Our first endeavour is to characterise the real-economy risk-neutral measure ^R. In the case that the short-rate process exists, we denote it by (r^R_t)_0≤ t. If in addition (r^R_t) is absolutely integrable, then the discount factor (D^R_t) exists, and we define

    D_t^R = exp(-∫_0^t r^R_s  s).

We introduce the process (I^R_t )_0 ≤ t, given by

    I^R_t =  ∫_0^t   b^R(s) /    1+ b^R(s) (  A_s-^R -1 )  A_s ^R,

and note that  Δ I^R_t > -1, for all t≥ 0.
Next, we denote by (·) the stochastic exponential and define

    ξ^R_t= (  I^R ) _t,
 
which is strictly positive for all t≥ 0.
 
Assume an RPKS. Then (h_t^R,) has a  savings account if and only if  (ξ_t ^R) in (<ref>) is an -martingale. In this case, 
	ξ_t ^R  =^R/| __t, and the savings account is classical with short rate process  (r_t^R) given by
	
    r_t^R = -1/h_t^R(b_0'(t)  +b_1'(t) A_t^R).


  
	By the relation (<ref>), we obtain expression (<ref>). Now, h_t^R= b_0(t)+b_1(t)A_t^R and Ito's formula shows
	
    h_t^R = -r_t^R h_t^R t  + R(t) b^R(t)  A_t^R,    h_0^R = C_0

	which exposes the unique additive decomposition of (h_t^R). One can now either calculate the dynamics of ξ_t^R =h_t^R /D_t ^R or apply the formula for the multiplicative decomposition.

We now examine the nominal market processes; this endeavour is slightly more elaborate. In the case that the nominal short-rate process (r^N_t)_0≤ t and the associated discount factor (D^N_t) exist, we write

    D_t^N = exp(- ∫_0^t r^N_s  s).

We furthermore define 

    I^S_t  = ∫_0^t 1/s_s- s_s = ∫_0^t 1 /  A^S_s- A_s ^S,

and m_t(T) = _t[A_T^R A_T^S], which is differentiable in T=t in the case that the nominal short rate exists in an RPKS. In the case that ∫_0 ^t m'_s(s)b_3(s)/h_s^N s < ∞, we define the stochastic exponential
ξ_t^N = (  ∫_0 ^· -m'_s(s)b_3(s)/h_s^N s + I^R+ I^S  + [ I^R, I^S] )_t.
We note that Δ ( I^R_ t + I^S_t  + [ I^R, I^S]_ t  ) >-1 ∀  t ≥  0, i.e. ξ_t^N > 0 for all t≥ 0. We define the “pseudo short-rate” 

    _t =-1/h_t^N( b_2'(t) A_t^S + b_3'(t) A_t^R A_t^S).

  
	Assume an RPKS and that (r_t^N) exists. Consider the “pseudo short-rate” (<ref>). Then (h_t^N,) has a classical savings account with short rate (r_t^N), given by 
	
    r_t^N  = r_t - m'_t(t) b_3(t)/h_t^N,

	if and only if ∫_0 ^t m'_s(s)b_3(s)/h_s^N s < ∞  and (ξ_t^N) is an -martingale.


	By (<ref>) we get expression (<ref>).
	Since h_t^N = b_2(t) A_t^S + b_3(t) A_t^R A_t^S  Ito's formula shows that
	
    h_t^N =    
    	 - h_t^N_t t  + b_3(t) A_t-^S A_t^R
    	+ (b_2(t) +b_3(t) A_t-^R )  A_t^S + b_3(t) [A^R,A^S]_t   
    
    	=     ( - h_t^N r_t^N  - m_t'(t)b_3(t)   )  t  + b_3(t) A_t-^S A_t^R
    	+ (b_2(t) +b_3(t) A_t-^R )  A_t^S +  b_3(t) [A^R,A^S]_t  ,

	with h^N_0=1.
	We define ξ_t^N= h_t^N/D_t^N. An application of Ito's quotient rule shows that
	
    ξ_t ^N =  ξ_t^N (  -m'_t(t) b_3(t)/h_t^N t   + I^R_t   + I^S_t +[I^R,I^S]_t  ),

	i.e. (ξ_t ^N) is the stated stochastic exponential.

Lemma <ref> is a weaker result than Lemma <ref>, since (ξ_t^N ) is not necessarily a local martingale. The result is still interesting because (r_t^N) is the most tempting candidate for a short-rate process in a discount factor emerging from a nominal pricing kernel. Lemma <ref> shows that, in general, we cannot expect (h_t^N,) to have a classical savings account with short rate (r_t^N). This rules out though neither the existence of a savings account nor a classical savings account with a different “short-rate”. To apply the theory we need (h_t^N) to be a special semimartingale.

	Assume an RPKS and that (h_t^N) is a special semimartingale. Then the canonical additive decomposition is given by
	h_t^N =  1+ B_t + N_t,
	where

    B_t =    h_t^N_t t +  b_3(t)  ⟨ A ^R, A^S⟩_t,         (B_0=0)  
     N_t =       b_3(t) A_t-^S A_t^R+(b_2(t) +b_3(t) A_t-^R )  A_t^S  +b_3(t) (  [A^R,A^S]_t- ⟨ A^R, A^S⟩_t ),          (N_0=0).

The multiplicative decomposition is h_t^N =  M_t L_t where

    M_t = ( ∫_0 ^·1/h_s-^N + Δ B_s N_s  )_t,      L_t = ( -∫_0 ^·1/h_s-^N + Δ B_s B_s  )_t ^-1,

moreover, a savings account exists if and only if (M_t) is a martingale.


	The dynamics of (h_t^N) was derived in (<ref>). By <cit.>, 
	∫_0^t b_3(s)  [A ^R,A^S]_s
	has locally integrable variation, and therefore it has a predictable compensator 
    ∫_0^t b_3(s) ⟨ A^R,A^S⟩_s,
 see <cit.>. 

When there are no simultanous jumps in (A^R_t) and (A^S_t) the situation is simpler:

	Assume an RPKS, and that Δ A_t^RΔ A_t^S = 0 a.s. Then (h_t^N) is a special semimartingale and the canonical decomposition is given by	h_t^N =  1+ B_t + N_t,
	where

    B_t =    h_t^N_t t+  b_3(t)   [A^R,A^S]_t,        (B_0=0) 
     N_t =       b_3(t) A_t-^S A_t^R+(b_2(t) +b_3(t) A_t-^R )  A_t^S,           (N_0=0)

	and h_t^N =  M_t L_t where

    M_t = (I^R + I^S )_t,      L_t =( -∫_0^·  h_s^N_s s +[I^R,I^S])_t ^-1,

and a savings account exists if and only if (M_t) is a martingale.  If additionally [A^R , A^S]_t  is absolutely continuous,  write [A^R , A^S]_t = ∫_0 ^t a_s  s and define λ_t = a_t b_3(t)/h_t^N for t≥ 0. Then,

    L_t =exp ( ∫_0^· (h_s^N_s - λ_s)  s  )_t

and, if ∫_0^t  exp(-∫_0 ^s (h_u^N _u - λ_u)  u )  h_s^N _s - λ_s s ∈ L^1(), then a classical savings account exists.


	By assumption, we have that [A^R,A^S]_t=⟨ (A^R)^c, (A^S)^c ⟩_t and thus 
	the decomposition above has a predictable bounded variation part. 
	By change of variables, L_t^-1 = 1+∫_0 ^t ∂/∂ sexp(-∫_0 ^s (h_u^N _u - λ_u)  u )  s. That is, by <cit.>, integrability implies existence of a classical savings account.

A simple example where the first condition is satisfied is if (A_t^R) or (A_t^S) is continuous. If both (A_t^R) and (A_t^S) are Ito processes then [A^R , A^S]_t is absolutely continuous.

Finally we present a lemma giving a condition in an RPKS to check whether (h_t^N) is special, this will be particularly simple to check in the following setting.
 
	Assume an RPKS and that  (A_t^R) and (A_t^S)  are locally square-integrable. Then (h_t^N) is a special semimartingale.


	By <cit.>, the process ([A^R,A^S]_t) has locally integrable variation and its compensator (⟨ A^R, A^S⟩_t) exists. Therefore, the decomposition in (<ref>) is warranted.



§ CONSTRUCTION OF THE EXPONENTIAL-RATIONAL CLASS
  
Our next goal is to derive explicit price formulae for financial derivatives based on the ZC and the YoY swap rates, and for the so-called limited price-index (LPI) swap. For its flexibility, tractability and good calibration properties, we choose to work with a sub-class among the rational pricing kernel systems, namely the exponential-rational pricing kernels. We next construct this class. 
[Exponential-rational pricing kernels] 
Assume an RPKS and let  (X_t)_0≤ t be a d-dimensional stochastic process.  Assume that  (A^R_t)_0≤ t and (A^S_t)_0≤ t in Definition <ref> are on the form  A^R_t=exp(w_RX_t) and A^S_t= exp(w_SX_t). We call this class the exponential-rational pricing kernel models. If (X_t) is an additive process, we call this class the  additive exponential-rational pricing kernel models.

Remember that an RPKS requires that (A_t^R) and (A_t^S) are martingales, that (A_t^RA_t^S) has finite expectation for all t≥ 0 and that A_0^S=A_0^R=1, it is implicit that w_R,w_S and (X_t) in Definition <ref> are chosen such that this is satisfied. 
[Additive process]  
 Let  (X_t)_0≤ t be a d-dimensional stochastic process. Following  <cit.>, we say (X_t) is additive if it has a.s.  paths, X_0=0, and
 
 
  * Independent increments: for any n≥ 1, 0≤ t_0 < t_1 < …<t_n-1 <  t_n,  the random variables X_t_0, X_t_1- X_t_0,…, X_t_n-X_t_n-1 are independent,
 
  * Stochastic continuity: for any t≥ 0 and  ϵ >0,  lim_s → t  (  X_t - X_s > ϵ ) =0.
   

In the remainder of the paper, the derivations will be based on exponential-rational pricing kernel models. The additive exponential-rational pricing kernel models will be given particular attention, so we  now recall some facts about additive processes and provide some examples. The independent increments property gives a -Khintchine representation

    [ ^i   z   X_t] =   ^  i ψ_t(z),  
    ψ_t(z) =     i zμ_t  - 1/2 zΣ_t  z  +∫_^d  (  ^i zx-1 - i zxx≤ 1 ) ν_t( x) ,

where the -Khintchine triplet (μ_t,Σ_t,ν_t) is unique and satisfies a number of conditions (see <cit.>). The -Khintchine triplet also  determines the sample path properties of (X_t)_0≤ t by the -Ito decomposition  (see  <cit.>[Chapter 4]). Next, we consider  examples of how additive processes can be obtained. 
[Time change]  
Let (τ_t)_0≤ t be a continuous, increasing process with τ_0=0. For t≥ 0, define pathwise X_t = L_τ_t  for (L_t)_0≤ t a  process. Then (X_t)_0 ≤ t inherits the independent increments of (L_t) and is therefore additive. A particular simple case is obtained by letting τ_t=τ(t) be deterministic.  Then, we may write 
[ exp ( i  z   X_t ) ] =  exp ( τ(t)  ψ^L(z)),
where ψ^L(z) is the characteristic exponent of (L_t) in t=1.


[Stacking independent additive processes]  
Let (X_t^1),…,(X_t^N) be one-dimensional additive processes with characteristic exponents ψ^1_t ,…,ψ^N_t. Then (X_t) = ( X^1_t,…,X^N_t)  is an N-dimensional additive process with characteristic exponent 
ψ_t(z) = ∑_i=1^N ψ^i_t(z_i),
where z= (z_1,…,z_N). Furthermore, for w∈^N (wX_t)_0 ≤ t is a one-dimensional additive process with characteristic exponent z ↦ψ_t(z w).


A fact about  additive processes is that they are convenient to construct martingales. Define

    (X) = { z ∈ℂ^d :  ∫_{x ∈^d   :  x> 1 }exp( zx)   ν_t( x) < ∞,   ∀ t≥ 0 },

then for z ∈(X) it holds that [ ^zX_t]<∞ and the Laplace exponent κ_t(z) = ψ_t (  -i  z ) is well-defined, see <cit.>[Theorem 25.17].
It follows from the independent increments property that for w∈(X), one has

    ^wX_t/[^wX_t]  = ^wX_t  - κ_t(w)
 
is a martingale. We can build exponential martingales by taking an additive process and let the drift absorb the mean in (<ref>). This produces the condition that, if

    wμ_t =  -1/2wΣ_t  w - ∫_^d(  exp(wx)-1 -  wxx≤ 1  )ν_t( x),

then (^wX_t)_0≤ t is a martingale. Eq. (<ref>) is useful for additive-exponential rational models. Recall Definition <ref>, Eq. (<ref>)  shows that  (A_t^R A_t^S) has finite expectation for all t≥ 0 if w_S+w_R∈(X). Similarly, recalling Lemma <ref>, (A_t^R) and (A_t^S) are locally square-integrable if 2 w_S∈(X) and 2 w_R∈(X).

We proceed to calculate a number of expressions needed in both the previous and next sections. First,   for w∈(X) and 0 ≤ t≤ T,

    _t[ exp ( wX_T)] =  exp(wX_t) exp ( κ_tT (  w) ),

where we define  the forward Laplace exponent κ_tT(·) = κ_T (· ) -κ_t(·). For the  YoY swap (<ref>) we also need

    ^_t[ exp ( w_1X_T) , exp( w_2X_T )] 
    									=    ^w_1+w_2X_t (  ^κ_tT (w_1+w_2)    - ^κ_tT(w_1) +κ_tT(w_2)   )  
    
    									=    ^w_1+w_2X_t^κ_tT (w_1+w_2)   ( 1  - ^κ_tT(w_1) +κ_tT(w_2)- κ_tT (w_1+w_2)    ),

where w_1,w_2, (w_1+w_2) ∈(X) is assumed.
 We notice that the sign, and to some extent the magnitude of the covariance, depends on the non-linearity of z↦κ_tT(z).
 For the subsequent derivation of Fourier-inversion formulae, we will also need the multiperiod generalized characteristic function. 
  
Let (X_t)_0≤ t be an additive process.  Assume t ≤ T_0≤ T_1 ≤…≤ T_N, set u = ( u_1, …, u_N) and define
 
    q_t(u) =   _t [  exp(    ∑_i=1^N  u_i w_iX_T_i )  ].

 Set  z_N =  u_N w_N and  z_i-1 = z_i +  u_i-1 w_i-1 for i=2,…,N. Assume that z_i ∈(X) for i=1,…,N.
Then,

    q_t(u)  =exp( z_1(u) X_t ) exp (  ∑_i=1^N  κ_ T_i-1 T_i  ( z_i(u ) )    ),

which is well-defined and finite.
 
 
 The statement follows from iterated expectation and the independent increments property.
 


We next build on the ideas of the Examples <ref> and <ref> with two concrete specifications for the  additive exponential-rational pricing kernel models. Very similar models will be calibrated in Section <ref>.

[Time-changed ]      
Let (L_t)_0≤ t be a   process satisfying the condition ^(L_1)=1. Let  (t_1,a_1) , (t_2,a_2) , … , (t_n,a_n)  be given points such that both coordinates are increasing and let τ(t) be a continuous, non-decreasing  interpolation. Then   t ↦^(L_τ(t)) = τ(t)  interpolates  the given points. This property can be utilised for, e.g., fitting a term-structure of at-the-money implied volatilities. Building on this motivation, we let 

    X_t = (L^R_τ^R(t) + μ^R(t) ,L^S_τ^S(t)+ μ^S(t)).

Let (t μ̃^R,t  (σ^R)^2,  t ν^R) and (t μ̃^S, t (σ^2)^S,t  ν^S) denote the -Khintchine triplets of (L_t^R) and (L_t ^S).
Then (X_t) has the -Khintchine triplet

    μ_t = (   μ^R(t) +τ^R(t) μ̃^R ,   μ^S(t) + τ^S(t) μ̃^S),    Σ_t =  [  (σ^R)^2   τ^R(t)                 0;                 0 (σ^S)^2    τ^S(t) ],  
       ν_t(B)  =   τ^R(t)   ν^R ( B_1 )  +  τ^S(t)   ν^S ( B_2 )

where B_1 = { x ∈: (x,0) ∈ B } and B_2 =  { x∈ : (0,x) ∈ B }. 
By choosing the drifts μ^R(t) and μ^S(t) according to (<ref>) we can turn A^R_t=exp(w_RX_t) and A^S_t= exp(w_SX_t) into martingales. This construction  generalises to higher dimensions in a straightforward way.



[Time-changed Wiener process]   
As a special case of Example <ref>, we consider  time-changing independent Wiener processes. 
We set 

    X_t = (  W^R_τ^R(t) + μ^R(t), W^S_τ^S(t) + μ^S(t) ) .

This corresponds to having the -Khintchine triplet given by 

    μ_t =    ( μ^R(t), μ^S (t) ) ,     ν_t =     0 ,     Σ_t =     [ τ^R(t)      0;      0 τ^S(t) ].

We can choose (A^R_t) and (A^S_t) to be martingales as in Specification <ref>.



 §.§ Option pricing
  
By use of the exponential-rational pricing kernel models, tractable expressions can be derived for inflation-linked derivatives, such as the YoY floor and the ZC floor. Under the stronger assumption of  additive exponential-rational pricing kernel models, we can find a similarly tractable formula for the LPI swap.


  §.§.§ Year-on-Year floors
 
The payoff of the YoY floor can be written in terms of a series of floorlets. A floorlet has payoff function ( K- C_T_i/C_T_i-1)^+ paid at time T, typically  is quoted with K=1+. In practice it is often observed that T> T_i  to ensure that there is a reliable observation of CPI available at maturity.  We want our framework to be able to accommodate this feature. The next theorem is a pricing formula for the YoY floorlet.
  
Assume an exponential-rational pricing kernel model. Let Y_1=c_1 +   w_S X_T_i-1-  w_S X_T_i, Y_2=w_S X_T and Y_3 =c_3 + w_RX_T
where c_1=ln[S(T_i-1)/(K S(T_i))], c_3=ln[b^R(T)/(1-b^R(T))],   
and q_t(z)= _t[^z  (Y_1,Y_2, Y_3)]. Let R>0 and assume that 

    q_t(-R,1,1)+q_t(-R,1,0)<∞.

Let  t≤ T and consider

    V^YoYFl_t =   1  /h_t^N_t  [ h_T^N ( K-  C_T_i/C_T_i-1 )^+  ],

by Eq. (<ref>), the price of the YoY floorlet. Then we have: 

    V^YoYFl_t    =  c_0 K /π  h_t^N∫_^+ϑ_t  ( u )  /(R+i u) (1+R+iu)  u,

where c_0  = R(T) (1-b^R(T))S(T) and ϑ_t ( u ) =q_t(-(R+iu),1,0)+ q_t(-(R+iu),1,1). If T_i ≤ t ≤ T, then  V^YoYFl_t  = (K-  C_T_i/C_T_i-1)^+ P_tT ^N.


By the pricing formula (<ref>) the price at any time t < T_i   is

    V^YoYFl_t  =    1  /h_t^N_t  [ h_T^N ( K- C_T_i/C_T_i-1 )^+  ]
    	   =   c_0 K /h_t^N_t [(1+^Y_3)  ^Y_2   (1- ^Y_1)^+ ].

 We can directly apply Lemma <ref>, found in the appendix, to get (<ref>).  Note that if T_i-1≤ t<T_i  then a part of Y_1 is measurable. The formula for T_i ≤ t ≤ T follows by observing that the payoff function is T_i-measurable and recalling Eq. (<ref>) combined with the relation (<ref>).

A point to note about (<ref>) is the quadratic convergence of the numerator in the integral, which makes the formula particularly tractable. 
In the case that we use an additive exponential-rational pricing kernel model, q_t(z) follows from an   application of Lemma <ref>, and the integrability (<ref>)  is satisfied if -R w_S∈(X).
Note that the difference between a YoY caplet and floorlet is a YoY swaplet, potentially with time-lag. The price is 

    V_t^YoYSL= 1/h_t^N_t  [ h_T^N  (C_T_i/C_T_i-1 -K  )   ] =       c_0 K/h_t^N (  q_t( 1,1,0)+q_t( 1,1,1)  )-KP_tT^N,

where q_t(z) and c_0 are given in Theorem <ref>.  This can be used in conjunction with Theorem <ref> to get the price of a caplet or used directly to price the time-lagged swaplet. 
 

  §.§.§ Zero-Coupon floors
  
Next we focus on the pricing of the ZC floor, which together with the ZC cap and  YoY caps and floors,  are the most liquidly traded inflation-linked derivatives. The structure of this section closely follows   the previous one, since the calculations   are   similar.  The payoff at time T of the ZC floor can  be written in the form ( K-C_T_i/C_0)^+ with T ≥ T_i akin to the YoY floor. Typically, the strike quoted is , where K=(1+)^T.
 
Assume an exponential-rational pricing kernel model. Let Y_1= c_4 +   w_S X_T_i, Y_2=   w_S X_T and   Y_3=c_3 +  w_R X_T, where

    c_4=ln(1   / K S(T_i) )        ,    c_3= ln(b^R(T)/1-b^R(T)),

and q_t(z)= _t [^z  (Y_1,Y_2, Y_3)]. 
Assume R>0 and

    q_t( -R,1,0)+q_t(-R,1,1) < ∞  .

Consider  T_0 ≤ t≤ T_i≤ T and let

    V^ZCF_t  =   1  /h_t^N_t  [ h_T^N (  K -C_T_i/C_T_0 )^+   ]

 be the price at time t of a ZC floor. Then we have 

    V^ZCF_t    =  c_0 K /π  h_t^N∫_^+ϑ_t  ( u )  /(R+i u) (1+R+iu) u,

where c_0=R(T) (1-b^R(T)) S(T) and
ϑ_t  ( u ) =q_t(-(R+iu),1,0)+ q_t(-(R+iu),1,1). If T_i ≤ t ≤ T, then  V^ZCF_t  = ( K- C_T_i/C_T_0)^+ P_tT ^N. 


Exactly as for the YoY case, see Theorem <ref>.


If we assume the additive exponential-rational pricing kernel,  q_t(z) follows directly from (<ref>), and the assumption (<ref>) is satisfied if  -R w_S∈(X). Analogous to the YoY cap we  have the time-lagged ZC swap has price

    V_t^ZCS  = 1/h_t^N_t  [ h_T^N ( C_T_i/C_T_0-K  )   ]= c_0/h_t^N ( q_t(1,1,0)+ q_t(1,1,1)  ) - K P_tT^N,
 
where c_0 and q_t(z) are given in Theorem <ref>. Which can also be used to obtain the price of ZC caps. 


  §.§.§ Limited price index swap
 
The tractability of the model specification we have used so far allows to find semi-closed-form price formulae for the  exotic limited price index (LPI) swap. This, contrary to the previous theorems, does rely on the assumption that the driving stochastic process (X_t)_0≤ t is additive.
The LPI is defined by 

    C^LPI _ T_k  = C^LPI _ T_k-1 mid( 1+K_f , C_T_k/C_T_k-1 , 1+K_c) ,

where k=1,…,N and T_k is a periodic fixed date, typically yearly. The contracts have maturities up to 30 years. Similar to the ZC swap, the LPI swap  has payoff  C^LPI _T_N -K. We will consider the payoff to be settled at the fixed time T≥ T_N. The pricing relation (<ref>) gives the swap price at time t:

    V^LPIS_t  =1/h_t^N_t[ h_T^N  C^LPI _T_N -K] = P_t T^LPI  - K P_tT^N

where P_t T^LPI =_t [ h_T^N C^LPI_T_N  ]/h_t^N is the price process of the  LPI-linked ZC bond. We therefore need to derive the price at time t≤ T of the LPI-linked  ZC bond.
 
Assume an additive exponential-rational pricing kernel model.  Assume, without loss of generality, that T_0 ≤ t < T_1< T_2 <… < T_N ≤ T.
Let, for k=1,…, N, 

    q^1k ( z_1,z_2) =    [  exp  ( (z_1+z_2) w_SX_T_k   -X_T_k-1∨ t   )],   
    
    q^2k ( z_1,z_2) =    [  exp (  z_1 w_SX_T_k   -X_T_k-1∨ t     + z_2  w_R+w_SX_T_k   -X_T_k-1∨ t  )] ,

and R_k>0  be such that 

    ∑_k=1^N  (q^1k ( -R_k,1)+q^2k ( -R_k,1)  ) < ∞.

Then,

    P_tT^LPI  =  1h_t^N  C^LPI_ T_0 (   c_0 V_t^11∏_k=2^N V^1k + c_5 A_t^R  V_t^21∏_k=2^N V^2k )   A_t^S

where 

    c_0 =       R(T) (1-b^R(T)) S(T)  ,      c_5 =    R(T) b^R(T) S(T)  exp( κ_ T_N T  (   w_R+w_S  )).

Furthermore, for j=1,2

    V_t ^j1=    β_c exp (  κ_ t T_1    (   ω^j )   ) +   1/π∫_ ^+ β_c  ϑ^j1(α^1c_t,u) + β_f  ϑ^j1(α^1f_t,u) /(R+iu) (1+R+iu)  u,  
    
    V ^jk=    β_c exp (  κ_  T_k-1 T_k    (  ω^j )   ) +   1/π∫_ ^+ β_c  ϑ^jk(α^kc,u) + β_f  ϑ^jk(α^kf,u) /(R+iu) (1+R+iu)  u,

where ϑ^jk (α, u) =     α^-(R+iu  )  q^jk   ( -(R+iu) ,1  ) ,  and

    α^1c_t =    β_c^-1  S(T_1) /  S( T_0 ) A^S_T_0 / A_t^S  ,      α^1f_t =   β_f^-1  S(T_1) /  S( T_0 ) A^S_T_0 / A_t^S ,   
    α^kc =    β_c^-1 S(T_k-1  )/S(T_k) ,    α^kf =   β_f^-1 S(T_k-1  )/S(T_k) ,    for  k=2,…,N, 
    β_c =    (1+K_c) ,       β_f =    (1+K_f), 
    ω^1 =     w^S ,    ω^2 =     w^S+w^R .

 

First  we  write 

    C^LPI_ T_N =     C^LPI_ T_0∏_k=1^N  Z_T_k,   
    
     Z_T_k =        (1+K_c )  - (1+ K_c - C_T_k/C_T_k-1 )^++  ( 1+ K_f -  C_T_k/C_T_k-1 )^+ .

Note that Z_T_k is _T_k-measurable and independent of _T_k-1. 
Using the tower property and the independent increments property we have: 

    P_tT_N^LPI   =       C^LPI_ T_01 /h_t^N_t    [ h_T ^N∏_k=1^N  Z_T_k ]  
    
    							  =     C^LPI_ T_0c_0 /h_t^N_t    [  A_T_N-1^S∏_k=1^N-1  Z_T_k   ]   [   A_T_N^S/A_T_N-1^S Z_T_N  ]     
       +C^LPI_ T_0c_5 /h_t^N_t   [ A_T_N-1^R   A_T_N-1^S∏_k=1^N-1  Z_T_k  ]    [   A_T_N^R  A_T_N^S/ A_T_N-1^R A_T_N-1^S Z_T_N  ]      
      
    =        C^LPI_ T_0c_0 /h_t^N A^S_t  _t    [   A_T_1^S/A_ t^S Z_T_1  ]   ∏_k=2^N     [   A_T_k^S/A_T_k-1^S Z_T_k  ]    
       + C^LPI_ T_0c_5 /h_t^N   A^R_t A^S_t _t   [    A_T_1^R A_T_1^S/ A_ t^R A_ t^S Z_T_1  ] ∏_k=2^N   ^  [   A_T_k^R  A_T_k^S/ A_   T_k-1^R A_  T_k-1^S Z_T_k  ]   .

All the expectations can be calculated by Lemma <ref>, in the appendix, to obtain

    _t   [       A_T_1^R A_T_1^S/ A_ t^R A_ t^S Z_T_1   ]
        = .    [    A_T_1^R A_T_1^S/ A_ t^R A_ t^S ( (1+K_c )  - (1+ K_c -x C_T_1/C_t )^++  ( 1+ K_f -x C_T_1/C_t )^+  )     ]    |_x= C_t/C_T_0
       =β_c  exp ( κ_t  T_1 (  w_S +w_R)  ) +   1/π∫_ ^+ β_c  ϑ^21(α^1c_t,u) +β_f  ϑ^21(α^1f_t,u) /(R+iu) (1+R+iu)  u.

The remaining remaining  expectations are calculated in the same way. 
 
We note that each V is calculated like a ZC floor or YoY caplet, i.e. the evaluation is no more complicated than for a YoY cap. To price  multiple LPI-linked ZC bonds, the shorter maturity bond prices can be found from the factors needed for the longer maturity ones.
 


 §.§ Gaussian formulae
 
In this section we  derive the results equivalent to  Theorems <ref>, <ref>  and <ref> under the assumption of the model in Specification <ref>. The results will be Black-Scholes-style formulae.


     
Assume the additive exponential-rational pricing kernel where (X_t)_0 ≤ t is the time-changed Wiener process of Specification <ref>. Assume that t ≤ T_i-1 < T_i and denote by  V^YoYFl_t the price of the floorlet, as in Theorem <ref>. Then 

    V^YoYFl_t   =c_0 K /h^N_t      A_t^S (  ^δ_1  ( - d_1 )  -   α^δ_1 + μ_y + 12 ( 1+ 2b_1 )  σ_y^2  ( -( d_1+ σ_y ) )    )      
       +c_6 K /h_t^N  A_t^R  A_t^S (  ^δ_2    (  -d_2)   -   α^δ_2 + μ_y + 12 ( 1+ 2b_2 )  σ_y^2    ( - ( d_2 +σ_y ))    ) ,

where 

    μ_x_1 =w_Sμ_T_i - μ_T_i-1 ,  
       σ_x_1   = w_S ( Σ_T_i -Σ_T_i-1) w_S ,  
       μ_y  =    - w_Sμ_T_i - μ_T_i-1 ,
       σ_x_1 y   =    -w_S  ( Σ_T_i -Σ_T_i-1 ) w_S ,
               μ_x_2  = w_R+w_Sμ_T_i - μ_T_i-1 , 
       σ_x_2   = w_R+w_S ( Σ_T_i -Σ_T_i-1) (w_R+w_S ) ,  
       σ_ y   = w_S ( Σ_T_i -Σ_T_i-1 ) w_S ,   
       σ_x_2 y   =   - w_S ( Σ_T_i -Σ_T_i-1) ( w_R+w_S ) ,

 see Specification (<ref>) for the values of μ_t and Σ_t, 
 and, for j=1,2,
δ_j =  a_j + b_j μ_y + (b_j σ_y)^2/2, d_j =1σ_y  (  lnα + b_jσ_y^2 + μ_y),
a_j =μ_x_j - σ_x_j y/σ_y^2μ_y - (σ_x_j^2-σ^2_x_j yσ_y^2)/2, b_j=σ_x_j yσ_y^2.
Moreover,
 
α=       S(T_i-1)   / K  S(T_i)     ,     c_0=   R(T)  (1-b^R(T))   S(T),   

c_6 =     R(T) b^R(T) S(T)  exp( κ_ t T_i-1  (   w_R+w_S  ) ) exp( κ_ T_i T  (   w_R+w_S  ) ).
     


Using the independent increments property 
 
    V^YoYFl_t  =     1/h^N_t_t  [ R(T)  ( 1+ b^R(T) ( A_T^R -1)) S(T) A_T^S  ( K-  C_T_i/C_T_i-1  )^+   ] 
    
    			   =    c_0 K /h^N_t     A_t^S  [     A_T_i^S/A_T_i-1^S (  1- 1/K  C_T_i/C_T_i-1 )^+  ] +  c_6 K /h_t^N  A_t^R    A_t^S  [    A_T_i^R A_T_i^S/A_T_i-1^R A_T_i-1^S (1-    1/K  C_T_i/C_T_i-1  )^+  ]

Now we apply  Lemma <ref>  to each term to obtain the result.

  The case where T_i-1 < t < T_i is derived similarly, see Proposition <ref>. 
The price formula for the ZC floor is derived analogously.
   
Assume the additive exponential-rational pricing kernel where (X_t)_0 ≤ t is the time-changed Wiener process of Specification <ref>.
Let  T_0 ≤ t < T_i ≤ T, then the price of the ZC floor is

    V^ZCFl_t  =      c_0/h^N_t1/S(T_i)(    α_t  ^μ_y + 12σ_y^2    (d^1_t + σ_y)  -   (  d^1_t )    )  +    c_7/h^N_t1/S(T_i)   A_t^R (     α_t  ^δ + μ_y + 12 (1+2 b) σ_y^2    (   d^2_t +σ_y  ) - ^δ  (  d^2_t)    )

where
μ_x  =   w_Rμ_T_i - μ_t, σ_x   =   w_R ( Σ_T_i  -Σ_t)  w_R,  μ_y  =    w_Sμ_T_i - μ_t, σ_ y   =   w_S ( Σ_T_i  -Σ_t ) w_S, σ_x y   =  w_R ( Σ_T_i -Σ_t ) w_S,
 see Specification (<ref>) for the values of μ_t and Σ_t. Furthermore,
  d^1_t = 1σ_y  (  lnα_t +μ_y),  d^2_t =1σ_y  (  lnα_t + b σ_y^2 + μ_y), a=μ_x - σ_x y/σ_y^2μ_y - 1/2 (σ_x^2-σ^2_xyσ_y^2),  b=σ_xyσ_y^2,  δ = a+ b μ_y + (b σ_y)^2/2,     
α_t =K S(T_i) A_t^S,  c_0=   R(T) (1-b^R(T)) S(T),  c_7 = R(T) b^R(T)   S(T)  ^κ_T_i T ( w_R+w_S ).    


Using the properties of the conditional expectation and the independence of the increments, we may write 
 
    V^ZCF_t   =     1/h^N_t_t  [ R(T)  ( 1+ b^R(T) ( A_T^R-1) ) S(T) A_T^S ( K- C_T_i )^+   ] 
    
    			  =    .    c_0/h^N_t1/S(T_i) [  (  1/x   K    C_t /C_T_i -1   )^+  ] |_x=C_t  +.  c_7/h^N_t A_t^R/S(T_i) [  A_T_i^R/A_t^R (   1/x   K    C_t /C_T_i -1   )^+  ]    |_x= C_t .

Now applying  Lemma <ref>  to each term yields the result.

The formula for the price process of the LPI-linked ZC bond follows in the same way.
[Limited price index bond] 
Assume the additive exponential-rational pricing kernel with (X_t)_0 ≤ t the time-changed Wiener process of Specification <ref>.   Assume without loss of generality that T_0 ≤ t < T_1< T_2 <… < T_N ≤ T. 
Then

    P_tT^LPI  =  1h_t^N  C^LPI_ T_0 (   c_0 V^11_t  ∏_k=2^N V^1k  + c_5 A_t^R V_t^21∏_k=2^N V^2k )  A_t^S

where 
c_0=R(T) (1-b^R(T) ) S(T),  c_5 =R(T) b^R(T) S(T)exp( κ_ T_N T  (   w_R+w_S  ) ), and for k=2,…,N, j=1,2

    V^j1_t=     (1+K_c  )  ^κ_t  T_1 ( ω^j)  +     ^δ^j1 ( - d_t^j1c  )    - α_t^1c^δ^j1 + μ^1_y + 12  (1+2b^j1 )   ( σ^1_y  )^2   (   - d_t^j1c  -σ^1_y    )   
        +     ^δ^j1 ( -  d_t^j1f  )    - α_t ^1f^δ^j1 + μ^1_y + 12 (1+2b^j1 )    ( σ^1_y  )^2  (  -  d_t^j1f  - σ^1_y     )    
    
    V^jk=     (1+K_c  )  ^κ_ T_k-1  T_k  ( ω ^j ) +     ^δ^jk  ( -  d^jkc  )    - α^kc^δ^jk + μ^k_y + 12  (1+2b^jk )  ( σ^k_y)^2   (  -    d^jkc  - σ^k_y    )   
        +     ^δ^jk  ( -   d^jkf )   - α^kf^δ^jk + μ^k_y +12 (1+2b^jk  )  ( σ^k_y)^2    (  -  d^jkf  -σ^k_y  ),

where, for k=1,…,N,

    μ^k_x_1 =   w_Sμ_T_k - μ_T_k-1∨  t,   
       σ^k_x_1   =  w_S ( Σ_T_k -Σ_T_k-1∨  t) w_S,   
       μ^k_y  =  - w_Sμ_T_k - μ_T_k-1∨  t ,
       σ^k_x_1 y   =   -w_S ( Σ_T_k -Σ_T_k-1∨  t ) w_S,  
       ω^1   =  w^S, 
               μ^k_x_2  =   w_R+w_Sμ_T_k - μ_T_k-1∨  t   ,
       σ^k_x_2   =   w_R+w_S  ( Σ_T_k -Σ_T_k-1∨  t  ) (  w_R+w_S )  , 
       σ^k_ y   =  w_S  ( Σ_T_k -Σ_T_k-1∨  t ) w_S  ,   
       σ^k_x_2 y   =  - w_S (Σ_T_k -Σ_T_k-1∨  t )  ( w_R+w_S )  , 
       ω^2   = w^S+ w^R,

   see Specification (<ref>) for the values of μ_t and Σ_t. For j=1,2, k=1,…,N, we have

    δ^jk  = a^jk + b^jkμ^k_y + (b^jkσ^k_y)^2/2,    b^jk =   σ^k_x_jy/(σ_y^k)^2,      a^jk  = μ^k_x_j - σ^k_x_j y/(σ^k _y)^2μ^k_y - 1/2 ((σ^k_x_j ) ^2-( σ^k_x_j y)^2(σ^k_y)^2 ),
 
 for k=2,…,N and  l = c,f

    d_t^j1l  =    1σ^k_y ( lnα^1l_t  +   b^1j  (σ^1_y)^2 + μ^1_y  ) ,     α^1l_t =      S(T_1) / (1+K_l)  S( T_0 ) A^S_T_0 / A_t^S ,  
    
    d^jkl  =    1σ^k_y (lnα^kl + b^jk   (σ^k_y)^2 + μ^k_y  )   , 
        α^kl  =     S(T_i-1)   /  (1+K_l)   S(T_i)     .



We may use the proof of Theorem <ref> up to  the expectations in Eq. (<ref>) which can be calculated by Lemma <ref> in the Appendix.  




 §.§ Nominal products
   
An important nominal linear interest rate derivative is the swap which pays the difference between a fixed rate and a floating rate. Loosely speaking we refer to this rate as the  LIBOR. Suppose we have a sequence of  time points T_0<T_1<…< T_N, and let δ_i=T_i-T_i-1.  A payer's swap pays  δ_i(L(T_i,T_i-1,T_i)-K) at each T_i, where L(T_i,T_i-1,T_i) is the LIBOR spot rate. We assume for ease of exposition that payments on the fixed leg K and floating leg L(T_i,T_i-1,T_i) both occur at time T_i. It follows that the price of the swap at time t≤ T_0 is given by 

    V^Sw_t =∑_i=1^N δ_i(  _t[ h_T_i^N/h_t^NL(T_i,T_i-1,T_i) ] -KP^N_tT_i ) .

[Single-curve setup]
If LIBOR rates are spanned by a single system of nominal bonds for all tenors, we say that we are in the single-curve setup.
  
We refer to <cit.> for an overview of single- and multi-curve interest rate models.  Within the single-curve setup, we have the no-arbitrage relation 

    L(T_i,T_i-1,T_i)= 1/δ_i(1/P^N_T_i-1 T_i-1).

It follows that

    _t[ h_T_i^N/h_t^NL(T_i,T_i-1,T_i)]=1/δ_i(P_t T_i-1^N-P^N_tT_i),

and thus the swap price is given  by

    V^Sw_t =∑_i=1^N [ P^N_t T_i-1 -(1+δ_i K)P^N_tT_i].




  §.§.§ Swaptions
   
A swaption is an option to enter  a swap at  some future time. If we let this point in time be  T_k and denote the maturity of the underlying swap by T_N, the swaption price at t≤ T_k is given by Eq. (<ref>) and we have

    V^Swn_t  = N /h_t^N_t  [ h^N _T_k( V^Sw _T_k) ^+ ],

where N is the notional.

 
Assume an additive exponential-rational pricing kernel model and assume the single curve setup.  Let  V^Swn_t,  as in  (<ref>), be the swaption price at time 0≤ t ≤ T_k. 
Let T_k+1<T_k+2<… <T_N denote the payment dates of the underlying swap. Set 

    c_0 =   ∑_i=k+1^N  (   R(T_i-1) (1-b^R(T_i-1) ) S(T_i-1) - (1+ δ_i K )R(T_i)(1-b^R(T_i))     S(T_i )    ),  
    
    c_1 =    ∑_i=k+1^N       (    R(T_i-1)  b^R(T_i-1 )   S(T_i-1)  ^κ_T_kT_i-1 (  w_R+w_S)  - (1+ δ_i K ) R(T_i)  b^R(T_i )  S(T_i )     ^κ_T_kT_i (  w_R+w_S)   ).

If c_0<0 and c_1<0, then   V^Swn_t=0, and if c_0>0 and c_1>0, then  

    V^Swn_t =1/h_t^N  A_t^S ( c_0 +A_t^R  c_1 ^κ_tT_k (  w_R + w_S  )  )  .
 
If (c_0 ) ≠ (c_1), define
Y_1=      w_S X_T_k, Y_2=      w_R X_T_k and q_t(z)= _t[^ z  (Y_1,Y_2) ].
Let  R<-1 if c_0<0  and  R>0 if c_0>0. Assume that  q_t(-R,1) < ∞.
Then

    V^Swn_t = c_0/π h_t^N∫_^+ϑ_t(u) / (R+iu)(1+R+iu) u

where
ϑ_t(u)  = α^-(R+iu)   q_t(-(R+iu) ,1 )
and α = c_1/c_0.


From the nominal bond pricing formula (<ref>) we have that

    h_T_k^N P_T_k T_i^N = R(T_i ) S(T_i) ( A_T_k^S(1-b^R(T_i))+  b^R(T_i) _T_k[A^R_T_i A_T_i^S] ) .

Then, inserting this into the swap formula (<ref>), we obtain 

    h_T_k^N V^Sw_T_k  =    ∑_i=k+1^N   (R(T_i-1) S(T_i-1)( A_T_k^S(1-b^R(T_i-1))  + b^R(T_i-1) _T_k[A^R_T_i-1 A_T_i-1^S])
         - (1+δ_i K )   R(T_i) S(T_i)  ( A_T_k^S(1-b^R(T_i))  + b^R(T_i) _T_k[A^R_T_i A_T_i^S])) .

We may write 

    _T_k[A^R_T_i A_T_i^S] =A^R_T_k   A^S_T_k^κ_T_kT_i  (  w_R +w_S) .
 
Collecting terms and using the fact that A^S_t>0 for any t ≥ 0, we arrive at

    (h^N_T_k V^Sw_T_k)^+  = A_T_k^S( c_0    + c_1  A_T_k^R)^+ .

 If c_0>0 and c_1<0, then 

    ( c_0    + c_1  A_T_k^R)^+  =    c_0 ( 1    - α  A_T_k^R)^+,

where we recall that  α = c_1/c_0. The result follows from Lemma <ref>.
If c_0<0 and c_1>0, then

    ( c_0    + c_1  A_T_k^R)^+  =   c_0 ( α A_T_k^R  -1   )^+

and the result follows from  Lemma <ref>. The two remaining cases are straightforward.

The independent increments property of (X_t) is only used to obtain Eq. (<ref>).


Under the assumption of  Specification <ref> the counterpart to Proposition <ref> is obtained by applying  Lemma <ref> and <ref> to  Eq. (<ref>) and Eq. (<ref>). 





  §.§.§ Multi-curve interest rate setting
 
We can, at a relatively low cost, allow our model to incorporate multi-curve-features. This is done by modelling (<ref>) as a rational function of state variables not fully spanned by the ones driving the nominal bonds. We model  the  forward LIBOR by

    L(t,T_i-1,T_i):= 1/h_t^N_t[ h_T_i^N  L(T_i,T_i-1,T_i)].

<cit.> propose the following definition, which we shall adopt.
[Rational multi-curve setup]
Let 

    L(t,T_i-1,T_i):= L(0,T_i-1,T_i)+b^L(T_i-1,T_i) ( A_t^L-1 ) /h_t^N,

where (A_t^L)_0≤ t is an -martingale with A_0^L=1, and where b^L(·,·) and  L(0,·,·) are deterministic functions.


We consider
A^L _t = exp( w_LX_t),
where w_L is chosen such that (A^L_t) is a martingale. This is analogous to how (A_t^R) and (A_t^S) are modelled. Adding a multi-curve dimension to the nominal markets has no effect  on any of the formulae derived for the inflation products.  It does though impact the swaption formula.
  
Assume an additive exponential-rational pricing kernel model and the multi-curve setup.  Consider a swaption with maturity T_k written on a swap with payments dates   T_k+1<T_k+2<… <T_N.  The swaption price V^Swn_t at  0 ≤ t≤ T_k  is given by

    V^Swn _t   = 1/h_t^N∫_^n H^M(x) ^+   m_T_k ( x)

where m_T_k is the distribution of X_T_k and

    H^M(x) =      c_0 + c_Lexp ( w_L x ) +  c_Sexp ( w_Sx  ) + c_SRexp ( w_R +w_S x ).

Furthermore,

    c_0=   ∑_i=k+1^N   δ_i    (L(0,T_i-1,T_i)-b^L(T_i-1,T_i)  ),      c_L = ∑_i=k+1^N    δ_i   b^L(T_i-1,T_i)   
       c_S =  - ∑_i=k+1^N    δ_i K       R(T_i) S(T_i)  (1-b^R(T_i) ),      c_SR = -∑_i=k+1^N    δ_i K    R(T_i)  b^R(T_i)   S(T_i)  ^κ_T_k T_i (  w_R +w_S)  .



 Using (<ref>) we may write

    h_T_k^N V^Sw _T_k  =    ∑_i=k+1^N  δ_i(L(0,T_i-1,T_i) + b^L(T_i-1,T_i) (A^L_T_k -1) 
        -  K  R(T_i)  S(T_i) ( A_T_k^S(1-b^R(T_i))  + b^R(T_i) _T_k[A^R_T_i A_T_i^S])).

By collecting the terms, the result follows.

We note that, analogous to the single-curve setup, the independent increments of (X_t) are only used to evaluate _T_k [A^R_T_i A_T_i^S]. In our applications, since  (X_t) is bi-variate, we may apply  the two-dimensional cosine method of <cit.>. 
The most immediate method for handling (<ref>) in higher dimensions is in the style of <cit.>, where 
H(x)^+ ≈  H(x)  G  .
with  G = {  H(x) ≥  0 } being exact.  If G = {ωX_T_k > α}, this leads to a one-dimensional integral, see <cit.> and  <cit.>. If G = {β_1  exp( ω_1x ) + β_2 exp ( ω_2x)  > α}  the inversion formula becomes a two-dimensional <cit.>-type formula.


§ CALIBRATION EXAMPLES
 
In this section, we show the calibration properties of the models on real data.  We consider EUR data from Bloomberg from 1 January 2015. The necessary data consists of OIS zero-yields constructed from EONIA overnight indexed swaps, LIBOR discrete curves based on EURIBOR and a term structure of ZC forward rates, as well as YoY cap and floor prices and EURIBOR swaptions. There is no LPI traded on EUR data. The OIS and EURIBOR curves are constructed directly in the Bloomberg system. We then set the nominal curve equal to the OIS curve, and the initial real (or equivalently the initial inflation-linked) curve is implied from the OIS curve and zero-coupon inflation forward rate using the methodology described in Section <ref>. The prices for YoY caps and floors are available to us for maturities 2, 5, 7, 10, 12, 15, 20 and 30 years. The strikes for the floors range from -1% to 3% and caplets from 1% to 6%. Quotes for YoY swap rates are not available to us, but the overlap in strikes for YoY caps and floors allows us to use put-call (cap-floor) parity to imply YoY swap rates consistent with the option prices.


In Figure <ref>, all the curves are plotted on the left-hand side. The real curve is plotted as zero-coupon rates, and we note that on this day, there is a consistently negative real curve with a widening gap to the nominal as the maturity increases. The 3m EURIBOR and OIS curve are plotted as discrete forward rates with 3m increments to be directly comparable, and we can note a significant spread between the two curves in the short and most liquid end of the maturity spectrum, which warrants the use of a multi-curve model to price nominal products. Finally, we observe that the option-implied YoY swap rates are close to the ZC swap rates. This relation implies only small levels of the convexity correction as seen directly in the right-hand-side of Figure <ref> where the convexity correction, as described in Section <ref>, is plotted for different swap lengths.  
An implied lognormal volatility surface is constructed from the prices of these options (selecting out-of-the-money options where available) using a geometric Brownian motion model for the CPI index as described in Section <ref>. Two of the prices for the two year maturity are identically zero and are thus removed from the dataset. We find the at-the-money implied volatility of the YoY cap using the piecewise constant hermite interpolation. The surface is plotted in the left hand side of Figure <ref>, and one can see a significant volatility smile, but also volatility levels that are quite low, around only 1.5-3%. Finally we consider a EURIBOR term structure of swaptions with maturities ranging from 3m, 6m, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 15Y, 20Y to 30Y. Since the focus of the paper is on the inflation component we limit our modelling to one curve—the 3m tenor curve. We thus calibrate only to swaptions with a one-year underlying swap length, since this swaption, by EUR market convention, contains payments involving only 3m EURIBOR. We refer to <cit.> for a more extensive calibration involving matching the volatility of both the 3m and 6m EURIBOR curves in a rational model resembling this one, but without the inflation component.  Due to the lognormal assumption for swap rates precluding negative interest, rates it is now customary to quote swaption prices in normal or Bachelier implied volatility as opposed to lognormal. This is done on the right panel in Figure <ref>. The data is from 1 January 2015, and we calibrate directly to these volatilities. 

Very similar to Specification <ref>, our model setup is the following:

    X_t=   (X_t^R +μ^R(t) ,X_τ^S(t)^S + μ^S(t), μ^L(t)  ),

where  μ^i(t) for i=L,R,S are deterministic martingalizing functions and thus the model is a two factor model. We assume that (X^R_t,X^S_t) is a two-dimensional  process with independent marginals  and that τ^S(t) is a deterministic time-change. The two independent  processes are defined by their Laplace exponents 

    κ_i(z)=ln ([^z X^i_1]  ),    (i=R,S)

at t=1. Thus we are in the additive exponential-rational pricing kernel setup with

    A_t^R=^w_RX_t ,   A_t^S=^w_SX_t ,   A_t^L=^w_LX_t.

We set  w_S=(0,1,0), w_R =(a_R   ,b a_R ,0) and w_L=a_L w_R  + (0,0,1). This means that the b parameter determines the dependence between the (A^R_t) and (A^S_t) and it furthermore means that the randomness in (A^L_t) is merely a (log)-linear transformation of the randomness in (A^R_t). As in (<ref>), when w_R,w_S,w_L∈(X) we can solve for the martingalizing drifts to obtain

    μ_S(t) =    -τ^S(t)κ_S(1),  μ_R (t) =  -τ^S(t)( κ_S(a_R b)/a_R-bκ_S(1))-tκ_R  (a_R ),
    μ_L(t)=    -τ^S(t)  (κ_S(a_L a_R b)-a_Lκ_S(a_R b)  )  -t  ( κ_R (a_L a_R )-a_Lκ_R (a_R )  ).

We set the deterministic time-change τ^S(t) =∫_0^t a(s)  s, where a(t) is a piecewise constant function

    a(t) = a_k,  t∈ (T_k-1,T_k].

Here {T_0,T_1,…,T_8}={0,2,5,7,10,12,15,20,30}, is the set of maturities quoted in the YoY option market. We calibrate the constants a_1,…,a_8 starting from the smallest to the largest maturity by matching to the YoY cap/floor volatility surface allowing a perfect fit to at least one strike per maturity. The dependence structure between the R and S component is fully determined by the parameter b thus reducing the model to a two-factor setup where the calculated expressions for YoY caplets, YoY swap prices and swaption prices can be applied directly without approximation. 

The nominal and the real curve are fitted by construction, but fitting the term-structure of YoY swap rates is less straightforward, since the swap rate depends on the full parameter set of the model, see Eq. (<ref>). We choose to calibrate the b^R(t) function to this term structure. There is enough flexibility in the b^R(t) function to fit the YoY swap rates without error, but direct calibration results in a quite volatile b^R(t) function which is hardly desirable. Therefore we instead fit an eight-knot Hermite polynomial with a non-smoothness penalty – a similar choice is made in <cit.> – and we find that the loss of accuracy when doing this is insignificant. The flexible shape means that the correlation parameter b and volatility parameter a_R in practice cannot be identified simultaneously with b^R(t) from the YoY swap curve. We solve this issue by simply fixing the b and the a_R parameters before calibration. In practice one needs only to avoid setting these parameters too low because the convexity correction becomes zero, by construction, if b=0 or a_R =0. In both of our calibration examples we fix these values at b=30 and a_R =0.25. 

Since swap rates are determined not only by the b^R(t) function, but the full parameter set of the model, one cannot calibrate b^R (t) independently of a(t) and the parameters determining the (X_t) process. On the other hand, YoY cap and floor prices are primarily affected by the (A_t^S) component and thus not very sensitive to the changes in values of b^R(t) unless the correlation between (X_t^R) and (X_t^S) is very high, which means this dual identification problem is in fact easily solved in practice. The overall calibration algorithm can be reduced to:


  * Set b^R(t)=1, and calibrate a_1,…,a_8 and the parameters of determining the law of (X_t^S) to YoY cap/floor implied volatilities.

  * Calibrate b^R(t) to the curve of YoY swap rates rates using least squares minimization with a penalty for b_R (t)∉(0,1).

  * Repeat Step 1 using instead the updated values of b^R(t). 

  * Calibrate  b_L(·,·) to swaption prices.

The swaption calibration is done by calibrating the b_L(·,·) function sequentially. This means that the a_L parameter, which also determines overall variance cannot be calibrated at the same time and we therefore fix it at a_L=1.3 below. We parametrise the b_L(·,·) function by setting b_L(t,t+3m)=P^N(0,t)L(0,t,t+3m)+b̃_L(t). The function b̃_L(t) is then piecewise constant in relation to the swaption maturities we can observe, i.e.

    b̃_L(t)=b̃_k,  t∈ (T_k,T_k+1]
 
with {T_0,T_1,…,T_11}={3m,6m,1Y,2Y,3Y,5Y,7Y,10Y,15Y,20Y,30Y,31Y}.


 §.§ Gaussian example

We first assume that (X^R_t) and (X^S_t) are independent standard Brownian motions with Laplace exponent at t=1 κ_R(z)=κ_S (z)=1/2 z^2, i.e. in the spirit of Specification <ref>. We would not expect a Gaussian or log-normal model to be well suited to reproduce implied volatility smiles, but we nevertheless believe that a Gaussian setup is illustrative as a benchmark case of study.

As discussed above, we first fix the b=30 and a_R =0.25, and then proceed with the calibration algorithm described above. In Step 1 we choose to calibrate a_1,…,a_8 to at-the-money implied volatility. This is done sequentially starting with calibrating a_1 to the two-year YoY implied volatility and a_2 to the five-year YoY implied volatility, and so forth. We note that the value of a_1 affects not just the two-year maturity but all YoY option maturities (larger than two years) since we are calibrating directly to caps, which have annual payments every year until maturity. Thus the sequential nature of the calibration of these parameters is key. 
The result of this calibration can be seen in Figure <ref> where we plot at-the-money implied volatility from the market. We furthermore plot an example of the model smile in the for a fixed maturity of five years. While the model smile is not completely flat, the Gaussian structure is, by construction, not suited for smile fitting. Finally the YoY swap rates are fitted without error by adjusting the b_R (t) function. YoY swap rates requires only mild adjustment of the b_R (t) function away from its default value of one.[ All parameter values are available upon request. ]

 
When fitting to swaptions we fix a_L=1.3 as explained above. Then we sequentially fit the b_L(t) function directly to swaption normal implied volatility starting from the three-month maturity up to the thirty-year maturity. The model is made to fit at-the-money, swaptions only, and the results are plotted in the lower right quadrant of Figure <ref>. 



 §.§ NIG example

 To produce a model more in line with the volatility smile, we instead assume that (X^R_t,X^S_t) are independent Normal Inverse Gaussian (NIG) processes, see for example <cit.>. We have that the Laplace exponent at t=1 is given by 

    κ_i (z) = -ν_i (√(ν_i^2 - 2 zθ_i - z^2 σ_i^2) -ν_i), i=R,S,

expressed in terms of the parametrisation (ν_i  , θ_i ,σ_i ) as
where ν_i , σ_i  > 0 and θ_i ∈. Since we want to control variance primarily using the time-change τ^S(t), we set σ_i = √(1-θ_i^2/ν_i^2) so that X^R_1 and X^S_1 both have variance of 1. Since we are only calibrating to the YoY cap/floor smile the full distribution of both marginals in  (X_t^R, X_t^S) is not identified by the data. For simplicity, we also set ν_S=ν_R and θ_S=θ_R. As in the Gaussian case we prefix  b=30 and  a_R =0.25. In the NIG case we split Step 1 in the calibration process by first fixing the rate of time at a constant, i.e. a_i=a, and then calibrate a,ν_S,θ_S to the whole YoY cap/floor implied volatility surface using the lsqnonlin algorithm in Matlab. Thereafter, the individual a_1,…,a_8 are calibrated sequentially such that the model fits the at-the-money implied volatilities without error. The rest of the algorithm is followed exactly like in the Gaussian case.

The fit to YoY swap rates is indistinguishable from the graph in Figure <ref> and is not plotted again. We set a_L=1.3 and fit the b_L(t) function to the same dataset of swaptions on one-year underlying swaps. The resulting fit is again indistinguishable from the fit in Figure <ref> and the calibrated b_L(t) function is available upon request.
In Figure <ref>, we plot model vs market volatility smiles for select maturities. We have only used one time-dependent scaling, so the model fits the at-the-money level without error. The remaining option prices are in principle fitted using only two parameters ν_S and θ_S. Thus we would not expect a perfect fit for all maturities. In general, any  process is well known to exhibit a flattening smile as maturities are increased which often results in a slightly too steep smile in the short end and too flat in the long end. These problems could be resolved by introducing further time-inhomogeneity or by applying stochastic time-changes, but with the virtue of model simplicity taken into account, we view the calibrated setup as satisfactory.


§ CONCLUSIONS

This paper focuses primarily on the theoretical development of stochastic, rational term-structure models using pricing kernels suitable for the pricing of nominal and inflation-linked financial instruments. We demonstrate how this model class can be constructed with a view towards calibration to market data. We furthermore show how the models extend the classical short rate approach to inflation modelling. We expect future research to be focused more on the numerics of risk management within the model as well as calibration to a broader set of market instruments such as joint calibration of year-on-year and zero-coupon caps, as well as including time-series information in the calibration problem. 


§ APPENDIX



§ LEMMAS

This section contains a number of lemmas used for the derivation of the formulae for option pricing. 
  
Let q_t(z) = _t [ exp ( zY)] be the moment generating function of Y = [Y_1,Y_2], a random vector with conditional distribution m_t. Assume R>0, 
q(-R,1)<∞ and α >0. Then, 

    _t  [ ^Y_2 (1- α^Y_1   )^+  ]	=  1/π∫_^+α ^-(R+ i u)    q_t(-(R+iu),1)/(R+i u) (1+R+iu) u.
 

We omit the proof of this lemma since it is standard.
  
Let q_t(z) = _t [ exp ( zY)] be the moment generating function of Y = [Y_1,Y_2], a random vector with conditional distribution m_t. Assume R<-1, q_t(-R,1)<∞ and α >0. Then, 

    _t  [ ^Y_2 (α^Y_1  -1   )^+  ]	=  1/π∫_^+α ^-(R+ i u)    q_t(-(R+iu),1)/(R+i u) (1+R+iu) u.
 

 
Let q_t(z) = _t [ exp ( zY)] be the conditional moment generating function of Y = [Y_1,Y_2,Y_3], a random vector with conditional distribution m_t. Assume R>0 and that 
q_t(-R,1,1)+q_t(-R,1,0)<∞.
Then 

    _t  [ (1+ ^Y_3 ) ^Y_2 ( 1- ^Y_1   )^+  ]	=  1/π∫_^+  q_t(-(R+iu),1,0)+ q_t(-(R+iu),1,1) /(R+i u) (1+R+iu) u
 


Write 

    _t  [ (1+ ^Y_3 ) ^Y_2 ( 1- ^Y_1   )^+  ] = _t  [ ^Y_2 ( 1- ^Y_1   )^+  ]  + _t  [  ^Y_3 + Y_2 ( 1- ^Y_1   )^+  ]

 and apply Lemma <ref> to each term. 

 
Let X ∼ N(μ_x, σ_x^2) and Y∼ N(μ_y,σ_y^2)  with [X,Y] = σ_xy and assume that α > 0. Then

    [^X ( α^Y -1)^+ ]  = α^δ + μ_y + 12 ( 1+ 2b ) σ_y^2  N(  d +σ_y ) -   ^δ N (    d  )

 where a = μ_x - σ_xy/σ_y^2μ_y+ (σ_x^2-σ^2_xy/σ_y^2)/2, b=σ_xy/σ_y^2, δ= a + b μ_y + (b σ_y)^2/2 and d = 1σ_y  (  lnα+ bσ_y^2 + μ_y). 


By conditional of normals  [^X    Y] = ^a+bY. Noting that - (x-μ)^22 σ^2  + bx   = b μ +b^2 σ^22 - (x-μ- b σ^2)^22 σ^2, the tower property yields

    [^X ( α^Y  - 1 )^+   ]  =  [^a+bY ( α^Y -1) 1 _{Y ≥ln1α} ]  =  α^δ + μ_y +  12 ( 1+ 2b ) σ_y^2  N(  d +σ_y ) -   ^δ N (    d  ),

as sought.

 
Let X,Y and α be as in  Lemma <ref>. Then

    [^X ( 1 - α^Y )^+ ]  =    ^δ N (-d)   - α^δ + μ_y + 12 ( 1+ 2b ) σ_y^2  N(-d-σ_y )

where a,b, δ and d are as in Lemma <ref>. 


As Lemma <ref>.




