Rational Models for Inflation-Linked Derivatives

Henrik Dam 1, Andrea Macrina 2,3, David Skovmand1, David Sloth 4

1 Department of Mathematical Sciences, University of Copenhagen, Denmark.2 Department of Mathematics, University College London, United Kingdom.

3 African Institute of Financial Markets & Risk ManagementUniversity of Cape Town, South Africa.4 Danske Bank, Copenhagen, Denmark.

July 17, 2020

Abstract

We construct models for the pricing and risk management of inflation-linked derivatives. The modelsare rational in the sense that linear payoffs written on the consumer price index have prices that arerational functions of the state variables. The nominal pricing kernel is constructed in a multiplicativemanner 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 conditionsnecessary for the multiplicative nominal pricing kernel to give rise to short rate models for the nominalinterest rate process. The proposed class of pricing kernel models retains the attractive features of anominal multi-curve interest rate model, such as closed-form pricing of nominal swaptions, and it isolatesthe so-called inflation convexity-adjustment term arising from the covariance between the underlyingstochastic drivers. We conclude with examples of how the model can be calibrated to EUR data.1

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

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

1 Introduction

The inflation market has grown in the aftermath of the 2008 financial crisis. Central banks have beenconducting aggressive quantitative easing to keep inflation off the cliff of deflation, and the ensuing fearshave driven hedging needs. As a consequence, the market for trading inflation has soared to the pointwhere standard inflation derivatives are now cleared on the London Clearing House (LCH) in numbersexceeding 100 bn EUR measured by notional outstanding value in early 2017. As this number only countslinear derivatives, the total market size is likely much larger. Among the products cleared one finds theYear-on-Year swap (YoY swap), swapping annual inflation against a fixed strike, and the Zero-Couponswap (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

1The 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 LondonMathematical Finance Seminar held at King’s College London (30 November 2017) for comments and suggestions. Theauthors especially acknowledge high-quality feedback provided by two anonymous reviewers.

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derivatives market is dwarfed in size by the market for inflation-linked bonds. These bonds are typicallygovernment-issued debt where the principal is linked to the consumer price index (CPI) or similar. Thebonds often have an embedded YoY floor protecting the principal from being adjusted downwards bydeflation. Limited Price Index (LPI) products come with both a lower and upper bound on the principaladjustment creating a path-dependent collar on inflation. Despite its exotic nature LPIs have been inhigh demand by pension funds. All products should ideally be priced in a consistent manner using atractable arbitrage-free model. Cap/floor products display volatility skews and non-flat term structuresof volatility, both of which the model also should be able to capture. Besides, the model should yieldclosed-form solutions for the price of the most traded derivatives, here the YoY and the ZC cap/floor.

(Hughston, 1998) develops a general arbitrage-free theory of interest rates and inflation in the casewhere the consumer price index and the real and nominal interest rate systems are jointly driven by amulti-dimensional Brownian motion. This approach is based on a foreign exchange analogy in which theCPI is treated as a foreign exchange rate, and the “real” interest rate system is treated as if it werethe foreign interest rate system associated with the foreign currency. The often-cited work by (Jarrowand Yildirim, 2003) makes use of such a setup. They consider a three-factor model (i.e., driven by threeBrownian motions) in which the CPI is modelled as a geometric Brownian motion, with deterministictime-dependent volatility and the two interest rate systems are treated as extended Vasicek-type (or Hull-White) models. Similar to (Jarrow and Yildirim, 2003), (Dodgson and Kainth, 2006) use a short-rateapproach where the nominal and the inflation rates are both modelled by Hull-White processes whilediscarding the idea of a real economy. A GBM-based model for the CPI provides the baseline frameworkfor how one might understand implied volatility in such a market, but any GBM model for the CPI doesnot, by construction, reproduce volatility smiles.

Further development of inflation models has paralleled that of interest rates models. For exampleinflation counterparts to the nominal LIBOR Market Model, see for example (Brigo and Mercurio, 2007),have been studied in (Belgrade et al., 2004), (Mercurio, 2005), and (Mercurio and Moreni, 2006). Whilethese models can reproduce smiles—augmented with stochastic volatility or jumps—they rely on numeri-cally intensive algorithms or approximations for the pricing of ZC cap/floors, in particular. One may saysimilarly of the models by (Kenyon, 2008), (Gretarsson et al., 2012), and (Mercurio and Moreni, 2009) whoin a similar manner use forward inflation, or in the case of (Hinnerich, 2008) the forward inflation swaprate, as the model primitive. (Waldenberger, 2017) builds an inflation counterpart to the nominal modelof (Grbac et al., 2015) and (Keller-Ressel et al., 2013). One also finds (Ribeiro, 2013) in the local volatilitycontext, (Kruse, 2011) extending the GBM methodology with (Heston, 1993) stochastic volatility, and(Singor et al., 2013) adding stochastic volatility to the (Jarrow and Yildirim, 2003) model. Our work isinspired 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 compre-hensive empirical study of (Filipovic et al., 2017) who demonstrate that linear-rational models performas well or better than similar affine term structure models. Furthermore, the rational model frameworkhas been extended to model multiple nominal curves and credit risk in (Crepey et al., 2016) and (Macrinaand Mahomed, 2018); it is this approach we follow. This framework allows for analytical expressions forswaptions, which is not the case for affine term structure models. In this paper, we demonstrate howrational models for inflation are constructed, which retain the tractability of the nominal counterpart andcan 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 Dberlein and Schweizer (2001),we study the conditions for a short rate model representation to be obtained. In Section 3 we deriveoption pricing formulae under different assumptions in the driving process, and in Section 4 we end withan example that shows how the model can be simultaneously calibrated to inflation derivatives and amultiple-curve nominal interest rate market.

2

2 Rational term structures

We adopt the pricing kernel approach, which was pioneered by (Constantinides, 1992), (Flesaker andHughston, 1996a), (Flesaker and Hughston, 1996b) and (Rogers, 1997)—for a good summary see (Huntand Kennedy, 2004) and, for a more recent account, (Grbac and Runggaldier, 2015). (Macrina andMahomed, 2018) 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-curvepricing formula for consistent valuation and hedging of financial instruments across curves. Applicationsinclude the pricing of inflation-linked and hybrid fixed-income securities. A property of the pricing kernelapproach is the ease with which the pricing and hedging of multiple currencies can be handled. This is theproperty one benefits from when considering inflation-linked pricing, and nominal and real economies areintroduced in analogy to domestic and foreign economies. Compared to the classical approach, in orderto allow for negative short rates, we relax the paradigm and consider general semimartingale dynamicsfor the pricing kernels. The approach taken next is one where the existence of a pricing kernel model ispostulated and its dynamics are modelled. It is via the pricing kernel that no-arbitrage price processesof tradable assets are generated by imposing that the asset price process, when multiplied by the pricingkernel process, be a martingale with respect to the probability measure the pricing kernel dynamics areproduced. This no-arbitrage notion is one presented in textbooks suchlike, e.g., Hunt and Kennedy (2004)and Bjork (2009).

2.1 General model

We model a financial market by a filtered probability space (Ω,F ,P, (Ft)0≤t), where P denotes the realprobability measure and (Ft)0≤t the market filtration satisfying the usual conditions. A finite time horizonis considered, i.e., a time line 0 ≤ t ≤ T < T <∞, throughout.

Definition 2.1 (Pricing kernel). We call a stochastic process (πt)0≤t with π0 = 1 a pricing kernel if it isa strictly positive, cadlag, semimartingale such that πt has finite expectation for all t ≥ 0.

Let L1T (µ;π) = χ : Ω → R s.t. χ is FT -measurable and Eµ[|πTχ| < ∞] where µ is a probabilitymeasure on (Ω,F). Let (πNt )0≤t be the (nominal) pricing kernel process. If we consider some claimχ ∈ L1T (P;πN), then by standard no-arbitrage theory, see e.g. (Hunt and Kennedy, 2004), the process(V χt )0≤t≤T , defined by

V χt =

1

πNtEPt

[πNTχ

], (2.1)

is an arbitrage-free price process. The notation Et[·] is short-hand for E[ · |Ft]. Following (Nguyen andSeifried, 2015)[Proposition 2.2], we have:

Proposition 2.2. Consider n assets with price processes (S1t ), . . . , (Snt ) satisfying Eq. (2.1), i.e., such

that (πNt Sit) is a P-martingale for i = 1, . . . , n. Assume the asset with strictly positive price process (S1

t )is traded. Then, the market is free of arbitrage.

Proof. By Eq. (2.1), the process ξt := S1t π

Nt /S

10 is a strictly positive martingale with ξ0 = 1. A measure

Q may be defined by ξt = dQdP

∣∣∣Ft

on any finite interval, and by the Bayes’ rule one obtains

EQt

[SiTS1T

]=

EPt

[ξT (SiT /S

1T )]

ξt=

EPt

[πNT S

iT

]πtS1

t

=SitS1t

,

for 2 ≤ i ≤ n. Thus, Q is a risk-neutral measure associated with the numeraire (S1t ). The existence of a

pricing kernel, here the process (πNt )0≤t, guarantees absence of arbitrage, also in the case of uncountably

3

many assets. Here we refer to the no-arbitrage notion of “no asymptotic free lunch with vanishing risk”(NAFLVR) developed in (Cuchiero et al., 2016).

We are agnostic as to how the asset with price process (S1t ) is chosen; for example it may be a zero-

coupon bond. From formula (2.1) it follows that, for 0 ≤ t ≤ T , the nominal zero-coupon bond pricesystem,

PNtT =

1

πNtEPt

[πNT], (2.2)

is free of arbitrage opportunities. Assuming that PNtT is differentiable in T , the short rate process (rNt )0≤t≤T

may be obtained by the well-known relation rNt = −∂T ln(PNtT

)|T=t. This tells that (πNt ) 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-exchangeanalogy. If we assume that (πRt )0≤t is a pricing kernel for the real market, then the foreign-exchangeanalogy establishes the relationship

Ct = πRt /πNt (2.3)

where (Ct)0≤t denotes the CPI process that acts as an exchange rate from the nominal to the real economy,see, e.g. (Bjork, 2009, Proposition 17.11).

As in (Flesaker and Hughston, 1996a), (Flesaker and Hughston, 1996b), (Rutkowski, 1997) and (Rogers,1997), we introduce an extra degree of flexibility and model prices with respect to an auxiliary measureM. This extra degree of freedom allows for simplified calculations or more tractable modelling under theM-measure while desirable statistical properties may still be captured under the P-measure. In fact it isalso possible to build in terminal distributions or “views” under P, in the spirit of (Black and Litterman,1992), and as explicitly obtained in (Macrina, 2014). 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 frommonetary policymakers and is subject to so-called “forward guidance”. With regard to how to induce themeasure change for such a purpose, we refer to (Hoyle et al., 2011), (Macrina, 2014) for the multivariategeneralisation, and (Crepey et al., 2016) for an application in a multi-curve term structure setup. We shallmodel the Radon-Nikodym process (Mt)0≤t with M0 = 1 as a strictly positive, cadlag martingale and fixsome time T < ∞. Then, M(A) = EP[MT 1A], for A ∈ FT , defines an equivalent measure. By settinghNt = πNt /Mt, with no loss of generality, we can express the fundamental pricing equation (2.1) under Mby the Bayes formula:

V χt =

1

πNtEPt

[πNTχ

]=

1

hNt MtEPt

[MTh

NTχ] =

1

hNtEMt

[hNTχ

], (2.4)

for 0 ≤ t ≤ T < T ∧ T and χ ∈ L1T (P;πN). That is, (hNt ) is the nominal pricing kernel under the M-measure. Similarly, the relationship πRt = Mth

Rt introduces the real pricing kernel (hRt ) under M. It follows

that, under M, (hNt ) and (hRt ) are strictly positive, semimartingales, see (Jacod and Shiryaev, 2003)[IIITheorem 3.13], and that Cth

Nt = hRt for all t ≥ 0.

Modelling convention. Let st := 1/Ct for t ≥ 0, for modelling convenience. From the relation (2.3), itthen follows hNt = sth

Rt . We model (hRt ) and (st), where hR0 = 1/s0 = C0, as strictly positive semimartin-

gales under M such that hRt and hNt have finite expectation for t ≥ 0.

Definition 2.3 (Real-kernel spread model). Let the triplet (hRt , st,Mt)0≤t be such that (hRt )0≤t, (st)0≤tand (Mt)0≤t are strictly-positive, cadlag, hR0 = 1/s0 = C0 and M0 = 1. Furthermore assume (hRt )0≤t and(st)0≤t are semimartingales and that (Mt)0≤t is a martingale. Denote by M the measure induced by (Mt).

4

Assume that hRt and hRt st have finite expectation for all t ≥ 0 under M. We call such a triplet a real-kernelspread model (RSM).

Often, the pricing of inflation-linked instruments is performed under either the nominal risk-neutralmeasure QN or the real risk-neutral measure QR. In the general setting presented so far, one is notnecessarily in a position to get consistent prices under these measures. In Section 2.2, we treat this issuein the context of some well-known models, which use from the outset a risk-neutral measure. In Section2.3 we proceed to the pricing of primary inflation-linked securities in the backdrop of a more specific modelclass. In Section 2.4 we discuss the change to risk-neutral measures in the same model class.

2.2 Comparison with other models

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

In the case of equity pricing, the benchmark model is the geometric Brownian motion specification of(Black and Scholes, 1973). In this sense, the most natural translation of this to inflation modelling is doneby (Korn and Kruse, 2004) specifying the inflation index under the nominal risk-neutral measure by

dCt = Ct(rN(t)− rR(t))dt+ CtσCdWC

t ,

where rN and rR are the deterministic nominal and real interest rates and (WCt )0≤t is a Brownian motion.

Black-Scholes-type pricing formulae are derived for ZC caps with payoff function max[CT /C0 − K, 0],and (Rubinstein, 1991) derives a pricing formula of a similar type for YoY caplets with payoff functionmax[CTi/CTi−1 −K, 0]. We refer to (Kruse, 2011) for the exact formulae. The formulae for the ZC capand YoY caplet as functions of the volatility parameter σC can be inverted to implied volatilities as itis commonly done for equity options. This will be relevant in Section 4 when we calibrate some specificrational pricing models.

(Jarrow and Yildirim, 2003) 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 assumethat the nominal and the real interest rates have Hull-White dynamics. Under the nominal risk-neutralmeasure, such a model specification takes the form

drNt =[θN(t)− aNrNt

]dt+ σNdWN

t , drRt =[θR(t)− ρRCσCσR − aRrRt

]dt+ σRdWR

t ,

dCt = Ct(rNt − rRt

)dt+ CtσCdWC

t ,

where (WNt ), (WR

t ) and (WCt ) are dependent Brownian motions, and where θN(t) and θR(t) are functions

chosen to fit the term-structure of interest rates, see (Brigo and Mercurio, 2007)[Chapter 15] and (Hulland White, 1990).

Proposition 2.4. The (Korn and Kruse, 2004) and the (Jarrow and Yildirim, 2003) models are RSMtriplets where (hRt , st,Mt)0≤t is given by

hRt = exp

(−∫ t

0rRs ds

)dQR

dQN

∣∣∣∣Ft

, st =1

hRtexp

(−∫ t

0rNs ds

), Mt =

dQN

dP

∣∣∣∣Ft

.

The nominal pricing kernel (hNt )0≤t is determined by hNt = st hRt .

Proof. The measure change to QN is given in (Jarrow and Yildirim, 2003)[Footnote 5], the measure changeto QR is similar and, in the Black-Scholes case, the results are standard.

5

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

2.3 Primary inflation-linked instruments

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

Definition 2.5 (Rational pricing kernel system). Let M be a measure equivalent to P induced by a Radon-Nikodym process (Mt)0≤t. Let (AR

t )0≤t and (ASt )0≤t be positive martingales under M with AS

0 = AR0 = 1.

Let (ARt A

St ) have finite expectation under M for all t ≥ 0. Let the real pricing kernel (hRt )0≤t be given by

hRt = R(t)[1 + bR(t)(AR

t − 1)]

where R(t) ∈ C1 is a strictly positive deterministic function with R(0) = C0,and where bR(t) ∈ C1 is a deterministic function that satisfies 0 < bR(t) < 1. Furthermore, let st = S(t)AS

t

where S(t) ∈ C1 is a strictly positive deterministic function with S(0) = 1/C0, and set hNt = st hRt . We

call (hRt , st,Mt)0≤t thus specified a rational pricing kernel system (RPKS).

By Ito’s lemma, (hRt ) and (st) are strictly positive semimartingales. An RPKS is in particular anRSM-triplet and therefore, by Section 2.1, it produces a nominal and a real market, both free of arbitrageopportunities. The martingale (AR

t ) generates the randomness in the real market, while the joint lawof (AS

t ) and (ARt ) generates the randomness in the nominal market. All derivations throughout will be

obtained under the assumption of having an RPKS.

Proposition 2.6 (Affine payoffs evaluated in an RPKS). Assume an RPKS. The price process (V χt )0≤t≤T

of a contract with payoff function χ = a1 + a2CT , for a1, a2 ∈ R, at the fixed date T ≥ t ≥ 0 is given by

V χt =

a2(b0(T ) + b1(T )ARt ) + a1(b2(T )AS

t + b3(T )EMt [AR

TAST ])

b2(t)ASt + b3(t)AR

t ASt

(2.5)

where, for 0 ≤ t ≤ T , b0(t) = R(t)(1 − bR(t)), b1(t) = R(t)bR(t), b2(t) = R(t)(1 − bR(t))S(t), b3(t) =R(t)bR(t)S(t). If a1 = 0, i.e. the payoff is linear in CT , the price process V χ

t is a rational function of ARt

and ASt .

Proof. It follows by the M-pricing equation (2.4).

The price process (PNtT )0≤t≤T of the nominal ZC bond follows from Eq. (2.5) for a1 = 1 and a2 = 0.

We have,

PNtT =

b2(T )ASt + b3(T )EM

t [ARTA

ST ]

b2(t)ASt + b3(t)AR

t ASt

(2.6)

with b2(t) and b3(t) given in Proposition 2.6. It then follows that the initial nominal term structurePN0t , 0 ≤ t ≤ T , is given by PN

0t = R(t)S(t)(1 + bR(t)(EM[ARt A

St ] − 1)). In particular, the parameter

function S(t) appearing in both, the price processes of the nominal ZC bond and the contract (2.5),can thus be used for calibrating to the market-observed prices PN

0t, 0 ≤ t ≤ T , according to S(t) =PN

0t/[R(t)(1 + bR(t)(EM[ARt A

St ] − 1))]. We note that should t 7→ EM [AR

t ASt

]not belong to C1, one can

calculate its value in all relevant time points and use a C1-interpolation, and nevertheless produce thesame 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 theswap maturity T for an annualised fixed payment. Its price process (V ZCS

t )0≤t≤T can be written in the

6

form

V ZCSt =

1

hNtEMt

[hNT

(CTC0−K

)]= P IL

tT −KPNtT (2.7)

where P ILtT = EM

t [hNTCT /C0]/hNt 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 swapsand aim at constructing models able to calibrate to the relevant market data in a parsimonious manner.By Eq. (2.7), given a nominal term-structure, a ZC swap term-structure is equivalent to an inflation-linkedZC bond term-structure, and fitting either is equivalent. The price of an inflation-linked ZC bond withinan RPKS follows directly from Proposition 2.6:

P ILtT =

1

C0

b0(T ) + b1(T )ARt

b2(t)ASt + b3(t)AR

t ASt

with b0(t) and b1(t) given in Proposition 2.6. We see that by matching the degree of freedom R(t) tothe initial term structure P IL

0t of inflation-linked bonds as implied from the market, i.e. R(t) = P IL0tC0,

the model replicates the term structure of ZC swaps. For ZC swaps, a de-annualised fair rate is quoted,namely a number k such that for K = (1 + k)T the initial value of the swap is zero. Given PN

0T , the initial

term structure PIL0T is implied from the ZC swap market fair rates k ZC

0T via k ZC0T =

(PN

0T

/P IL

0T

)1/T − 1. Theprice process (PR

tT )0≤t≤T of a real ZC bond is

PRtT =

1

hRtEMt

[hRT]

=b0(T ) + b1(T )AR

t

b0(t) + b1(t)ARt

, (2.8)

with b0(t) and b1(t) as in Proposition 2.6. In accordance with the foreign-exchange analogy, it holds thatPRtT Ct = 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 priceV YoYSLtTi

at time t < Ti−1 of a such over the period [Ti−1, Ti]. By the pricing relation (2.4) we have

V YoYSLtTi =

1

hNtEMt

[hNTi

(CTiCTi−1

−K)]

= P ILtTiS(Ti−1)A

St +

b2(Ti)S(Ti−1) CovMt

[ARTi−1

, ASTi−1

]b2(t)AS

t + b3(t)ARt A

St

−KPNtTi

(2.9)

with b2(t) and b3(t) as in Proposition 2.6. For YoY swaps the fair rate k is quoted in financial marketssuch that K = 1 + k. The price of the swap is V YoYS

tTN=∑N

i=1 VYoYSLtTi

, from which the fair rate kYoY canbe extracted:

kYoYtTN=

1∑Ni=1 P

NtTi

N∑i=1

P ILtTiS(Ti−1)A

St +

b2(Ti)S(Ti−1) CovMt

[ARTi−1

, ASTi−1

]b2(t)AS

t + b3(t)ARt A

St

− 1. (2.10)

If independence between (ARt ) and (AS

t ) is assumed, the YoY swap rate at time t = 0 becomes

kYoY0TN=

1∑Ni=1 P

N0Ti

N∑i=1

(P IL

0Ti−1

PN0Ti−1

PIL0Ti

)− 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

7

difference between the market-observed swap rate and the above expression is often referred to as theconvexity correction for the YoY swap of length TN .

2.4 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 Dberlein and Schweizer (2001). We shall see that the obtained class ofnominal pricing kernels can rarely be represented in terms of a short rate model. We study the reasonsbehind the lack of such a property. If it is possible to decompose a pricing kernel into

ht =dQdM

∣∣∣∣Ft

e−∫ t0 rudu (2.11)

where (rt) is a short rate process and Q is the corresponding numeraire measure, then the prices obtainedin either model would be equal, i.e. the same prices could instead have been obtained using a short rateapproach. To investigate whether a decomposition like (2.11) exists, some technical material is needed.

We recall that a special semimartingale is a process (Xt) with a unique decompositionXt = X0+Bt+Nt,where (Bt) is predictable and of finite variation, (Nt) is a local martingale and B0 = N0 = 0. Thedecomposition is called the canonical (additive) decomposition. We recall (Jacod and Shiryaev, 2003, IITheorem 8.21)

Theorem 2.7. Let (Xt) be a semimartingale with X0 = 1, such that (Xt) and (Xt−) are strictly positive.Then, (Xt) is a special semimartingale if and only if it admits a multiplicative decomposition

Xt = MtLt, (2.12)

where (Mt) is a strictly positive and cadlag M-local martingale, (Lt) is a positive, predictable process withlocally finite variation, and M0 = L0 = 1. When the decomposition exists, it is unique and is given by

Mt = E(∫ ·

0

1

Xs− + ∆BsdNs

)t

, Lt = E(−∫ ·0

1

Xs− + ∆BsdBs

)−1t

where (Bt) and (Nt) are the processes in its canonical additive decomposition.

Comparing Eq. (2.11) and Eq. (2.12) in Theorem 2.7 we see that a number of conditions need tobe satisfied for a short rate representation to be available. The pricing kernel (ht) has to satisfy theassumptions of Theorem 2.7, (Mt) has to be a true martingale and act as a measure change, (Lt) needs tohave the specific form in (2.11) and finally the resulting ZC bond prices need to be sufficiently differentiable.More precisely, following (Dberlein and Schweizer, 2001, Theorem 5 and Proposition 12), when (2.12) existsfor a pricing kernel, one calls At = L−1t an implied savings account. When in addition At = 1 +

∫ t0 φsds

is satisfied, where (φt) is adapted and∫ t0 |φs|ds ∈ L

1(M), then the forward and short rates exist, that is,(2.11) holds, and (At) is termed a classical savings account.

Our first endeavour is to characterise the real-economy risk-neutral measure QR. In the case that theshort-rate process exists, we denote it by (rRt )0≤t. If in addition (rRt ) is absolutely integrable, then thediscount factor (DR

t ) exists, and we define

DRt = exp

(−∫ t

0rRs ds

).

8

We introduce the process (IRt )0≤t, given by

IRt =

∫ t

0

bR(s)

1 + bR(s)(ARs− − 1)

dARs ,

and note that ∆IRt > −1, for all t ≥ 0. Next, we denote by E(·) the stochastic exponential and define

ξRt = E(IR)t, (2.13)

which is strictly positive for all t ≥ 0.

Lemma 2.8. Assume an RPKS. Then (hRt ,M) has a savings account if and only if (ξRt ) in (2.13) is anM-martingale. In this case, ξRt = dQR/dM|Ft, and the savings account is classical with short rate process(rRt ) given by

rRt = − 1

hRt

(b′0(t) + b′1(t)A

Rt

). (2.14)

Proof. By the relation (2.8), we obtain expression (2.14). Now, hRt = b0(t) + b1(t)ARt and Ito’s formula

showsdhRt = −rRt hRt dt+R(t)bR(t)dAR

t , hR0 = C0 (2.15)

which exposes the unique additive decomposition of (hRt ). One can now either calculate the dynamics ofξRt = hRt /D

Rt or apply the formula for the multiplicative decomposition.

We now examine the nominal market processes; this endeavour is slightly more elaborate. In the casethat the nominal short-rate process (rNt )0≤t and the associated discount factor (DN

t ) exist, we write

DNt = exp

(−∫ t

0rNs ds

).

We furthermore define

ISt =

∫ t

0

1

ss−dss =

∫ t

0

1

ASs−

dASs ,

and mt(T ) = EMt [AR

TAST ], which is differentiable in T = t in the case that the nominal short rate ex-

ists in an RPKS. In the case that∫ t0 |m′s(s)b3(s)

hNs|ds < ∞, we define the stochastic exponential ξNt =

E(∫ ·

0 −m′s(s)b3(s)

hNsds+ IR + IS + [IR, IS]

)t. We note that ∆(IRt + ISt + [IR, IS]t) > −1 ∀t ≥ 0, i.e. ξNt > 0

for all t ≥ 0. We define the “pseudo short-rate”

rt = − 1

hNt

(b′2(t)A

St + b′3(t)A

Rt A

St

). (2.16)

Lemma 2.9. Assume an RPKS and that (rNt ) exists. Consider the “pseudo short-rate” (2.16). Then(hNt ,M) has a classical savings account with short rate (rNt ), given by

rNt = rt −m′t(t)b3(t)

hNt, (2.17)

if and only if∫ t0 |m′s(s)b3(s)

hNs|ds <∞ and (ξNt ) is an M-martingale.

9

Proof. By (2.6) we get expression (2.17). Since hNt = b2(t)ASt + b3(t)A

Rt A

St Ito’s formula shows that

dhNt = − hNt rtdt+ b3(t)ASt−dAR

t + (b2(t) + b3(t)ARt−)dAS

t + b3(t)[AR, AS]t (2.18)

=(−hNt rNt −m′t(t)b3(t)

)dt+ b3(t)A

St−dAR

t + (b2(t) + b3(t)ARt−)dAS

t + b3(t)[AR, AS]t,

with hN0 = 1. We define ξNt =hNtDN

t. An application of Ito’s quotient rule shows that

dξNt = ξNt

(−m

′t(t)b3(t)

hNtdt+ dIRt + dISt + d[IR, IS]t

),

i.e. (ξNt ) is the stated stochastic exponential.

Lemma 2.9 is a weaker result than Lemma 2.8, since (ξNt ) is not necessarily a local martingale. Theresult is still interesting because (rNt ) is the most tempting candidate for a short-rate process in a discountfactor emerging from a nominal pricing kernel. Lemma 2.9 shows that, in general, we cannot expect (hNt ,M)to have a classical savings account with short rate (rNt ). This rules out though neither the existence of asavings account nor a classical savings account with a different “short-rate”. To apply the theory we need(hNt ) to be a special semimartingale.

Lemma 2.10. Assume an RPKS and that (hNt ) is a special semimartingale. Then the canonical additivedecomposition is given by hNt = 1 +Bt +Nt, where

dBt = hNt rtdt+ b3(t)d〈AR, AS〉t, (B0 = 0) (2.19)

dNt = b3(t)ASt−dAR

t + (b2(t) + b3(t)ARt−)dAS

t + b3(t) d([AR, AS]t − 〈AR, AS〉t

), (N0 = 0).

The multiplicative decomposition is hNt = MtLt where

Mt = E(∫ ·

0

1

hNs− + ∆BsdNs

)t

, Lt = E(−∫ ·0

1

hNs− + ∆BsdBs

)−1t

,

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

Proof. The dynamics of (hNt ) was derived in (2.18). By (Jacod and Shiryaev, 2003, I Theorem 4.23),∫ t0 b3(s) d[AR, AS]s has locally integrable variation, and therefore it has a predictable compensator∫ t

0b3(s) d〈AR, AS〉s,

see (Jacod and Shiryaev, 2003, I Theorem 3.18).

When there are no simultanous jumps in (ARt ) and (AS

t ) the situation is simpler:

Corollary 2.11. Assume an RPKS, and that ∆ARt ∆AS

t = 0 a.s. Then (hNt ) is a special semimartingaleand the canonical decomposition is given by hNt = 1 +Bt +Nt, where

dBt = hNt rtdt+ b3(t)d[AR, AS]t, (B0 = 0)

dNt = b3(t)ASt−dAR

t + (b2(t) + b3(t)ARt−)dAS

t , (N0 = 0)

10

and hNt = MtLt where

Mt = E(IR + IS

)t, Lt = E

(−∫ ·0hNs rsds+ [IR, IS]

)−1t

,

and a savings account exists if and only if (Mt) is a martingale. If additionally [AR, AS]t is absolutelycontinuous, write [AR, AS]t =

∫ t0 asds and define λt = atb3(t)/h

Nt for t ≥ 0. Then,

Lt = exp

(∫ ·0

(hNs rs − λs)ds)t

and, if∫ t0 exp(−

∫ s0 (hNu ru − λu)du)|hNs rs − λs|ds ∈ L1(M), then a classical savings account exists.

Proof. By assumption, we have that [AR, AS]t = 〈(AR)c, (AS)c〉t and thus the decomposition above has apredictable bounded variation part. By change of variables, L−1t = 1 +

∫ t0∂∂s exp(−

∫ s0 (hNu ru − λu)du)ds.

That is, by (Dberlein and Schweizer, 2001, Proposition 12), integrability implies existence of a classicalsavings account.

A simple example where the first condition is satisfied is if (ARt ) or (AS

t ) is continuous. If both (ARt )

and (ASt ) are Ito processes then [AR, AS]t is absolutely continuous.

Finally we present a lemma giving a condition in an RPKS to check whether (hNt ) is special, this willbe particularly simple to check in the following setting.

Lemma 2.12. Assume an RPKS and that (ARt ) and (AS

t ) are locally square-integrable. Then (hNt ) is aspecial semimartingale.

Proof. By (Jacod and Shiryaev, 2003, I Proposition 4.50), the process ([AR, AS]t) has locally integrablevariation and its compensator (〈AR, AS〉t) exists. Therefore, the decomposition in (2.19) is warranted.

3 Construction of the exponential-rational class

Our next goal is to derive explicit price formulae for financial derivatives based on the ZC and the YoYswap rates, and for the so-called limited price-index (LPI) swap. For its flexibility, tractability and goodcalibration 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.

Definition 3.13 (Exponential-rational pricing kernels). Assume an RPKS and let (Xt)0≤t be a d-dimensionalstochastic process. Assume that (AR

t )0≤t and (ASt )0≤t in Definition 2.5 are on the form AR

t = exp(〈wR, Xt〉)and AS

t = exp(〈wS, Xt〉). We call this class the exponential-rational pricing kernel models. If (Xt) is anadditive process, we call this class the additive exponential-rational pricing kernel models.

Remember that an RPKS requires that (ARt ) and (AS

t ) are martingales, that (ARt ASt ) has finite expec-

tation for all t ≥ 0 and that AS0 = AR

0 = 1, it is implicit that wR, wS and (Xt) in Definition 3.13 are chosensuch that this is satisfied.

Definition 3.14 (Additive process). Let (Xt)0≤t be a d-dimensional stochastic process. Following (Sato,1999, Definition 1.6), we say (Xt) is additive if it has a.s. cadlag paths, X0 = 0, and

1. Independent increments: for any n ≥ 1, 0 ≤ t0 < t1 < · · · < tn−1 < tn, the random variablesXt0 , Xt1 −Xt0 , . . . , Xtn −Xtn−1 are independent,

11

2. Stochastic continuity: for any t ≥ 0 and ε > 0, lims→tM(|Xt −Xs| > ε) = 0.

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

EM[ei〈z,Xt〉

]= eiψt(z),

ψt(z) = i〈z, µt〉 − 1/2〈z,Σtz〉+

∫Rd

(ei〈z,x〉 − 1− i〈z, x〉1‖x‖ ≤ 1

)νt(dx),

where the Levy-Khintchine triplet (µt,Σt,νt) is unique and satisfies a number of conditions (see (Sato,1999, Chapter 2)). The Levy-Khintchine triplet also determines the sample path properties of (Xt)0≤t bythe Levy-Ito decomposition (see (Sato, 1999)[Chapter 4]). Next, we consider examples of how additiveprocesses can be obtained.

Example 3.15 (Time change). Let (τt)0≤t be a continuous, increasing process with τ0 = 0. For t ≥ 0,define pathwise Xt = Lτt for (Lt)0≤t a Levy process. Then (Xt)0≤t inherits the independent increments of(Lt) and is therefore additive. A particular simple case is obtained by letting τt = τ(t) be deterministic.Then, we may write EM [exp(izXt)] = exp(τ(t)ψL(z)), where ψL(z) is the characteristic exponent of (Lt)in t = 1.

Example 3.16 (Stacking independent additive processes). Let (X1t ), . . . , (XN

t ) be one-dimensional addi-tive processes with characteristic exponents ψ1

t , . . . , ψNt . Then (Xt) = (X1

t , . . . , XNt ) is an N-dimensional

additive process with characteristic exponent ψt(z) =∑N

i=1 ψit(zi), where z = (z1, . . . , zN ). Furthermore,

for w ∈ RN (〈w,Xt〉)0≤t is a one-dimensional additive process with characteristic exponent z 7→ ψt(zw).

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

E(X) =

z ∈ Cd :

∫x∈Rd : ‖x‖>1

exp(〈Re z, x〉) νt(dx) <∞, ∀t ≥ 0

, (3.20)

then for z ∈ E(X) it holds that EM[|e〈z,Xt〉|] < ∞ and the Laplace exponent κt(z) = ψt(−iz) is well-defined, see Sato (1999)[Theorem 25.17]. It follows from the independent increments property that forw ∈ E(X), one has

e〈w,Xt〉

EM[e〈w,Xt〉

] = e〈w,Xt〉−κt(w) (3.21)

is a martingale. We can build exponential martingales by taking an additive process and let the driftabsorb the mean in (3.21). This produces the condition that, if

〈w, µt〉 = −1

2〈w,Σtw〉 −

∫Rd

(exp(〈w, x〉)− 1− 〈w, x〉1‖x‖ ≤ 1) νt(dx), (3.22)

then (e〈w,Xt〉)0≤t is a martingale. Eq. (3.20) is useful for additive-exponential rational models. RecallDefinition 2.5, Eq. (3.20) shows that (ARt A

St ) has finite expectation for all t ≥ 0 if wS + wR ∈ E(X).

Similarly, recalling Lemma 2.12, (ARt ) and (ASt ) are locally square-integrable if 2wS ∈ E(X) and 2wR ∈E(X).

We proceed to calculate a number of expressions needed in both the previous and next sections. First,

12

for w ∈ E(X) and 0 ≤ t ≤ T ,

EMt [exp(〈w,XT 〉)] = exp(〈w,Xt〉) exp(κtT (w)),

where we define the forward Laplace exponent κtT (·) = κT (·)−κt(·). For the YoY swap (2.9) we also need

CovMt [exp(〈w1, XT 〉), exp(〈w2, XT 〉)] = e〈w1+w2,Xt〉

(eκtT (w1+w2) − eκtT (w1)+κtT (w2)

)= e〈w1+w2,Xt〉eκtT (w1+w2)

(1− eκtT (w1)+κtT (w2)−κtT (w1+w2)

),

where w1, w2, (w1 + w2) ∈ E(X) is assumed. We notice that the sign, and to some extent the magnitudeof the covariance, depends on the non-linearity of z 7→ κtT (z). For the subsequent derivation of Fourier-inversion formulae, we will also need the multiperiod generalized characteristic function.

Lemma 3.17. Let (Xt)0≤t be an additive process. Assume t ≤ T0 ≤ T1 ≤ · · · ≤ TN , set u = (u1, . . . , uN )and define

qt(u) = EMt

[exp

(N∑i=1

ui〈wi, XTi〉

)].

Set zN = uNwN and zi−1 = zi + ui−1wi−1 for i = 2, . . . , N . Assume that zi ∈ E(X) for i = 1, . . . , N .Then,

qt(u) = exp(〈z1(u), Xt〉) exp

(N∑i=1

κTi−1Ti(zi(u))

), (3.23)

which is well-defined and finite.

Proof. The statement follows from iterated expectation and the independent increments property.

We next build on the ideas of the Examples 3.15 and 3.16 with two concrete specifications for theadditive exponential-rational pricing kernel models. Very similar models will be calibrated in Section 4.

Specification 3.18 (Time-changed Levy). Let (Lt)0≤t be a Levy process satisfying the condition VarM(L1) =1. Let (t1, a1), (t2, a2), . . . , (tn, an) be given points such that both coordinates are increasing and let τ(t)be a continuous, non-decreasing interpolation. Then t 7→ VarM(Lτ(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. Buildingon this motivation, we let

Xt =(LRτR(t) + µR(t), LS

τS(t) + µS(t)).

Let (tµR, t(σR)2, tνR) and (tµS, t(σ2)S, tνS) denote the Levy-Khintchine triplets of (LRt ) and (LS

t ). Then(Xt) has the Levy-Khintchine triplet

µt =(µR(t) + τR(t)µR, µS(t) + τS(t)µS

), Σt =

((σR)2τR(t) 0

0 (σS)2τS(t)

),

νt(B) = τR(t)νR(B1) + τS(t)νS(B2)

where B1 = x ∈ R : (x, 0) ∈ B and B2 = x ∈ R : (0, x) ∈ B. By choosing the drifts µR(t) andµS(t) according to (3.22) we can turn AR

t = exp(〈wR, Xt〉) and ASt = exp(〈wS, Xt〉) into martingales. This

construction generalises to higher dimensions in a straightforward way.

13

Specification 3.19 (Time-changed Wiener process). As a special case of Example 3.18, we considertime-changing independent Wiener processes. We set

Xt =(WRτR(t) + µR(t),W S

τS(t) + µS(t)).

This corresponds to having the Levy-Khintchine triplet given by

µt =(µR(t), µS(t)

), νt = 0, Σt =

(τR(t) 0

0 τS(t)

).

We can choose (ARt ) and (AS

t ) to be martingales as in Specification 3.18.

3.1 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 additiveexponential-rational pricing kernel models, we can find a similarly tractable formula for the LPI swap.

3.1.1 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 − CTi/CTi−1)+ paid at time T , typically k is quoted with K = 1 + k. In practice it is often observedthat T > Ti to ensure that there is a reliable observation of CPI available at maturity. We want ourframework to be able to accommodate this feature. The next theorem is a pricing formula for the YoYfloorlet.

Theorem 3.20. Assume an exponential-rational pricing kernel model. Let Y1 = c1 + 〈wS, XTi−1〉 −〈wS, XTi〉, Y2 = 〈wS, XT 〉 and Y3 = c3 + 〈wR, XT 〉 where c1 = ln[S(Ti−1)/(KS(Ti))], c3 = ln[bR(T )/(1 −bR(T ))], and qt(z) = EM

t [e〈z,(Y1,Y2,Y3)〉]. Let R > 0 and assume that

qt(−R, 1, 1) + qt(−R, 1, 0) <∞. (3.24)

Let t ≤ T and consider

V YoYFlt =

1

hNtEMt

[hNT

(K − CTi

CTi−1

)+],

by Eq. (2.4), the price of the YoY floorlet. Then we have:

V YoYFlt =

c0K

πhNt

∫R+

Reϑt(u)

(R+ iu)(1 +R+ iu)du, (3.25)

where c0 = R(T )(1− bR(T ))S(T ) and ϑt(u) = qt(−(R+ iu), 1, 0) + qt(−(R+ iu), 1, 1). If Ti ≤ t ≤ T , thenV YoYFlt = (K − CTi/CTi−1)+PN

tT .

Proof. By the pricing formula (2.4) the price at any time t < Ti is

V YoYFlt =

1

hNtEMt

[hNT

(K − CTi

CTi−1

)+]

=c0K

hNtEMt

[(1 + eY3) eY2 (1− eY1)+

].

We can directly apply Lemma A.33, found in the appendix, to get (3.25). Note that if Ti−1 ≤ t < Ti thena part of Y1 is measurable. The formula for Ti ≤ t ≤ T follows by observing that the payoff function is

14

Ti-measurable and recalling Eq. (2.2) combined with the relation (2.4).

A point to note about (3.25) is the quadratic convergence of the numerator in the integral, whichmakes the formula particularly tractable. In the case that we use an additive exponential-rational pricingkernel model, qt(z) follows from an application of Lemma 3.17, and the integrability (3.24) is satisfied if−RwS ∈ E(X). Note that the difference between a YoY caplet and floorlet is a YoY swaplet, potentiallywith time-lag. The price is

V YoYSLt =

1

hNtEMt

[hNT

(CTiCTi−1

−K)]

=c0K

hNt(qt(1, 1, 0) + qt(1, 1, 1))−KPN

tT , (3.26)

where qt(z) and c0 are given in Theorem 3.20. This can be used in conjunction with Theorem 3.20 to getthe price of a caplet or used directly to price the time-lagged swaplet.

3.1.2 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 theprevious one, since the calculations are similar. The payoff at time T of the ZC floor can be written inthe form (K − CTi/C0)

+ with T ≥ Ti akin to the YoY floor. Typically, the strike quoted is k, whereK = (1 + k)T .

Theorem 3.21. Assume an exponential-rational pricing kernel model. Let Y1 = c4 + 〈wS, XTi〉, Y2 =〈wS, XT 〉 and Y3 = c3 + 〈wR, XT 〉, where

c4 = ln

(1

KS(Ti)

), c3 = ln

(bR(T )

1− bR(T )

),

and qt(z) = EMt [e〈z,(Y1,Y2,Y3)〉]. Assume R > 0 and

qt(−R, 1, 0) + qt(−R, 1, 1) <∞. (3.27)

Consider T0 ≤ t ≤ Ti ≤ T and let

V ZCFt =

1

hNtEMt

[hNT

(K − CTi

CT0

)+]

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

V ZCFt =

c0K

πhNt

∫R+

Reϑt(u)

(R+ iu)(1 +R+ iu)du,

where c0 = R(T )(1− bR(T )

)S(T ) and ϑt(u) = qt (−(R+ iu), 1, 0) + qt (−(R+ iu), 1, 1). If Ti ≤ t ≤ T ,

then V ZCFt = (K − CTi/CT0)+ PN

tT .

Proof. Exactly as for the YoY case, see Theorem 3.20.

If we assume the additive exponential-rational pricing kernel, qt(z) follows directly from (3.23), andthe assumption (3.27) is satisfied if −RwS ∈ E(X). Analogous to the YoY cap we have the time-lagged

15

ZC swap has price

V ZCSt =

1

hNtEMt

[hNT

(CTiCT0−K

)]=

c0

hNt(qt(1, 1, 0) + qt(1, 1, 1))−KPN

tT , (3.28)

where c0 and qt(z) are given in Theorem 3.21. Which can also be used to obtain the price of ZC caps.

3.1.3 Limited price index swap

The tractability of the model specification we have used so far allows to find semi-closed-form price formulaefor the exotic limited price index (LPI) swap. This, contrary to the previous theorems, does rely on theassumption that the driving stochastic process (Xt)0≤t is additive. The LPI is defined by

CLPITk

= CLPITk−1

mid

(1 +Kf ,

CTkCTk−1

, 1 +Kc

),

where k = 1, . . . , N and Tk is a periodic fixed date, typically yearly. The contracts have maturities up to30 years. Similar to the ZC swap, the LPI swap has payoff CLPI

TN−K. We will consider the payoff to be

settled at the fixed time T ≥ TN . The pricing relation (2.4) gives the swap price at time t:

V LPISt =

1

hNtEMt

[hNTC

LPITN−K

]= PLPI

tT −KPNtT

where PLPItT = EM

t [hNTCLPITN

]/hNt is the price process of the LPI-linked ZC bond. We therefore need to derivethe price at time t ≤ T of the LPI-linked ZC bond.

Theorem 3.22. Assume an additive exponential-rational pricing kernel model. Assume, without loss ofgenerality, that T0 ≤ t < T1 < T2 < · · · < TN ≤ T . Let, for k = 1, . . . , N ,

q1k(z1, z2) = EM [exp((z1 + z2)〈wS, XTk −XTk−1∨t〉

)],

q2k(z1, z2) = EM [exp(z1〈wS, XTk −XTk−1∨t〉+ z2〈wR + wS, XTk −XTk−1∨t〉

)],

and Rk > 0 be such thatN∑k=1

(q1k(−Rk, 1) + q2k(−Rk, 1)

)<∞.

Then,

PLPItT = 1

hNtCLPIT0

(c0V

11t

N∏k=2

V 1k + c5ARt V

21t

N∏k=2

V 2k

)ASt

where

c0 = R(T )(1− bR(T )

)S(T ), c5 = R(T )bR(T )S(T ) exp (κTNT (wR + wS)) .

Furthermore, for j = 1, 2

V j1t = βc exp

(κtT1(ωj)

)+

1

π

∫R+

Reβc ϑ

j1(α1ct , u) + βf ϑ

j1(α1ft , u)

(R+ iu)(1 +R+ iu)du,

V jk = βc exp(κTk−1Tk(ωj)

)+

1

π

∫R+

Reβc ϑ

jk(αkc, u) + βf ϑjk(αkf , u)

(R+ iu)(1 +R+ iu)du,

16

where ϑjk(α, u) = α−(R+iu)qjk(−(R+ iu), 1) , and

α1ct = β−1c

S(T1)

S(T0)

AST0

ASt

, α1ft = β−1f

S(T1)

S(T0)

AST0

ASt

,

αkc = β−1cS(Tk−1)

S(Tk), αkf = β−1f

S(Tk−1)

S(Tk), for k = 2, . . . , N,

βc = (1 +Kc), βf = (1 +Kf ),

ω1 = wS, ω2 = wS + wR.

Proof. First we write

CLPITN

= CLPIT0

N∏k=1

ZTk ,

ZTk = (1 +Kc)−(

1 +Kc −CTkCTk−1

)+

+

(1 +Kf −

CTkCTk−1

)+

.

Note that ZTk is FTk -measurable and independent of FTk−1. Using the tower property and the independent

increments property we have:

PLPItTN

= CLPIT0

1

hNtEMt

[hNT

N∏k=1

ZTk

]

= CLPIT0

c0

hNtEMt

[ASTN−1

N−1∏k=1

ZTk

]EM

[ASTN

ASTN−1

ZTN

]

+ CLPIT0

c5

hNtEMt

[ARTN−1

ASTN−1

N−1∏k=1

ZTk

]EM

[ARTNASTN

ARTN−1

ASTN−1

ZTN

]

=CLPIT0

c0

hNtAStEM

t

[AST1

ASt

ZT1

]N∏k=2

EM

[ASTk

ASTk−1

ZTk

]

+ CLPIT0

c5

hNtARt A

StEM

t

[ART1AST1

ARt A

St

ZT1

]N∏k=2

EM

[ARTkASTk

ARTk−1

ASTk−1

ZTk

].

(3.29)

All the expectations can be calculated by Lemma A.32, in the appendix, to obtain

EMt

[ART1AST1

ARt A

St

ZT1

]= EM

[ART1AST1

ARt A

St

((1 +Kc)−

(1 +Kc − x

CT1Ct

)+

+

(1 +Kf − x

CT1Ct

)+)]∣∣∣∣∣

x=Ct/CT0

= βc exp (κtT1(wS + wR)) +1

π

∫R+

Reβc ϑ

21(α1ct , u) + βf ϑ

21(α1ft , u)

(R+ iu)(1 +R+ iu)du.

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 morecomplicated than for a YoY cap. To price multiple LPI-linked ZC bonds, the shorter maturity bond pricescan be found from the factors needed for the longer maturity ones.

17

3.2 Gaussian formulae

In this section we derive the results equivalent to Theorems 3.20, 3.21 and 3.22 under the assumption ofthe model in Specification 3.19. The results will be Black-Scholes-style formulae.

Proposition 3.23. Assume the additive exponential-rational pricing kernel where (Xt)0≤t is the time-changed Wiener process of Specification 3.19. Assume that t ≤ Ti−1 < Ti and denote by V YoYFl

t the priceof the floorlet, as in Theorem 3.20. Then

V YoYFlt =

c0K

hNtASt

(eδ1Φ(−d1)− αeδ1+µy+

12(1+2b1)σ2

yΦ(−(d1 + σy))

)+c6K

hNtARt A

St

(eδ2Φ(−d2)− αeδ2+µy+

12(1+2b2)σ2

yΦ(−(d2 + σy))

),

where

µx1 = 〈wS, µTi − µTi−1〉,σx1 = 〈wS, (ΣTi − ΣTi−1)wS〉,µy = − 〈wS, µTi − µTi−1〉,σx1y = − 〈wS, (ΣTi − ΣTi−1)wS〉,

µx2 = 〈wR + wS, µTi − µTi−1〉,σx2 = 〈wR + wS, (ΣTi − ΣTi−1)(wR + wS)〉,σy = 〈wS, (ΣTi − ΣTi−1)wS〉,σx2y = − 〈wS, (ΣTi − ΣTi−1)(wR + wS)〉,

see Specification (3.19) for the values of µt and Σt, and, for j = 1, 2, δj = aj + bjµy + (bjσy)2/2, dj =

1σy

(lnα+ bjσ2y + µy), aj = µxj −

σxjy

σ2yµy − (σ2xj −

σ2xjy

σ2y

)/2, bj =σxjy

σ2y

. Moreover,

α =S(Ti−1)

KS(Ti), c0 = R(T )

(1− bR(T )

)S(T ),

c6 = R(T )bR(T )S(T ) exp(κtTi−1(wR + wS)

)exp (κTiT (wR + wS)) .

Proof. Using the independent increments property

V YoYFlt =

1

hNtEMt

[R(T )

(1 + bR(T )(AR

T − 1))S(T )AS

T

(K − CTi

CTi−1

)+]

=c0K

hNtAStEM

[ASTi

ASTi−1

(1− 1

K

CTiCTi−1

)+]

+c6K

hNtARt A

StEM

[ARTiASTi

ARTi−1

ASTi−1

(1− 1

K

CTiCTi−1

)+]

Now we apply Lemma A.35 to each term to obtain the result.

The case where Ti−1 < t < Ti is derived similarly, see Proposition 3.25. The price formula for the ZCfloor is derived analogously.

Proposition 3.24. Assume the additive exponential-rational pricing kernel where (Xt)0≤t is the time-changed Wiener process of Specification 3.19. Let T0 ≤ t < Ti ≤ T , then the price of the ZC flooris

V ZCFlt =

c0

hNt

1

S(Ti)

(αt eµy+

12σ

2yΦ(d1t + σy)− Φ(d1t )

)+c7

hNt

1

S(Ti)ARt

(αt eδ+µy+

12(1+2b)σ2

yΦ(d2t + σy)− eδΦ(d2t )

)where µx = 〈wR, µTi − µt〉, σx = 〈wR, (ΣTi −Σt)wR〉, µy = 〈wS, µTi − µt〉, σy = 〈wS, (ΣTi −Σt)wS〉, σxy =〈wR, (ΣTi −Σt)wS〉, see Specification (3.19) for the values of µt and Σt. Furthermore, d1t = 1

σy(lnαt +µy),

18

d2t = 1σy

(lnαt + bσ2y + µy), a = µx − σxyσ2yµy − 1

2(σ2x −σ2xy

σ2y

), b =σxyσ2y

, δ = a+ bµy +(bσy)2

2 , αt = KS(Ti)ASt ,

c0 = R(T )(1− bR(T )

)S(T ), c7 = R(T )bR(T )S(T )eκTiT (wR+wS).

Proof. Using the properties of the conditional expectation and the independence of the increments, wemay write

V ZCFt =

1

hNtEMt

[R(T )

(1 + bR(T )(AR

T − 1))S(T )AS

T (K − CTi)+]

=c0

hNt

1

S(Ti)EM

[(1

xKCtCTi− 1

)+]∣∣∣∣∣x=Ct

+c7

hNt

ARt

S(Ti)EM

[ARTi

ARt

(1

xKCtCTi− 1

)+]∣∣∣∣∣x=Ct

.

Now applying Lemma A.34 to each term yields the result.

The formula for the price process of the LPI-linked ZC bond follows in the same way.

Proposition 3.25 (Limited price index bond). Assume the additive exponential-rational pricing kernelwith (Xt)0≤t the time-changed Wiener process of Specification 3.19. Assume without loss of generality thatT0 ≤ t < T1 < T2 < · · · < TN ≤ T . Then

PLPItT = 1

hNtCLPIT0

(c0V

11t

N∏k=2

V 1k + c5ARt V

21t

N∏k=2

V 2k

)ASt

where c0 = R(T )(1 − bR(T ))S(T ), c5 = R(T )bR(T )S(T ) exp (κTNT (wR + wS)) , and for k = 2, . . . , N ,j = 1, 2

V j1t = (1 +Kc) eκtT1 (ω

j) + eδj1

Φ(−dj1ct

)− α1c

t eδj1+µ1y+

12(1+2bj1)(σ1

y)2

Φ(−dj1ct − σ1y

)+ eδ

j1Φ(−dj1ft

)− α1f

t eδj1+µ1y+

12(1+2bj1)(σ1

y)2

Φ(−dj1ft − σ1y

)V jk = (1 +Kc) eκTk−1Tk

(ωj) + eδjk

Φ(−djkc

)− αkceδ

jk+µky+12(1+2bjk)(σk

y )2

Φ(−djkc − σky

)+ eδ

jkΦ(−djkf

)− αkfeδ

jk+µky+12(1+2bjk)(σk

y )2

Φ(−djkf − σky

),

where, for k = 1, . . . , N ,

µkx1 = 〈wS, µTk − µTk−1∨t〉,σkx1 = 〈wS, (ΣTk − ΣTk−1∨t)wS〉,µky = −〈wS, µTk − µTk−1∨t〉,σkx1y = −〈wS, (ΣTk − ΣTk−1∨t)wS〉,ω1 = wS,

µkx2 = 〈wR + wS, µTk − µTk−1∨t〉,σkx2 = 〈wR + wS, (ΣTk − ΣTk−1∨t)(wR + wS)〉,σky = 〈wS, (ΣTk − ΣTk−1∨t)wS〉,σkx2y = −〈wS, (ΣTk − ΣTk−1∨t)(wR + wS)〉,ω2 = wS + wR,

see Specification (3.19) for the values of µt and Σt. For j = 1, 2, k = 1, . . . , N , we have

δjk = ajk + bjkµky +(bjkσky )2

2, bjk =

σkxjy

(σky )2, ajk = µkxj −

σkxjy

(σky )2µky −

1

2

((σkxj )

2 −(σk

xjy)2

(σky )

2

),

19

for k = 2, . . . , N and l = c, f

dj1lt = 1σky

(lnα1l

t + b1j(σ1y)2 + µ1y

), α1l

t =S(T1)

(1 +Kl)S(T0)

AST0

ASt

,

djkl = 1σky

(lnαkl + bjk(σky )2 + µky

), αkl =

S(Ti−1)

(1 +Kl)S(Ti).

Proof. We may use the proof of Theorem 3.22 up to the expectations in Eq. (3.29) which can be calculatedby Lemma A.35 in the Appendix.

3.3 Nominal products

An important nominal linear interest rate derivative is the swap which pays the difference between a fixedrate and a floating rate. Loosely speaking we refer to this rate as the LIBOR. Suppose we have a sequenceof time points T0 < T1 < · · · < TN , and let δi = Ti − Ti−1. A payer’s swap pays δi(L(Ti, Ti−1, Ti)−K) ateach Ti, where L(Ti, Ti−1, Ti) is the LIBOR spot rate. We assume for ease of exposition that payments onthe fixed leg K and floating leg L(Ti, Ti−1, Ti) both occur at time Ti. It follows that the price of the swapat time t ≤ T0 is given by

V Swt =

N∑i=1

δi

(EMt

[hNTihNt

L(Ti, Ti−1, Ti)

]−KPN

tTi

). (3.30)

Definition 3.26 (Single-curve setup). If LIBOR rates are spanned by a single system of nominal bondsfor all tenors, we say that we are in the single-curve setup.

We refer to (Grbac and Runggaldier, 2015) for an overview of single- and multi-curve interest ratemodels. Within the single-curve setup, we have the no-arbitrage relation

L(Ti, Ti−1, Ti) =1

δi

(1

PNTi−1Ti

− 1

).

It follows that

EMt

[hNTihNt

L(Ti, Ti−1, Ti)

]=

1

δi

(PNtTi−1

− PNtTi

),

and thus the swap price is given by

V Swt =

N∑i=1

[PNtTi−1

− (1 + δiK)PNtTi

]. (3.31)

3.3.1 Swaptions

A swaption is an option to enter a swap at some future time. If we let this point in time be Tk and denotethe maturity of the underlying swap by TN , the swaption price at t ≤ Tk is given by Eq. (2.4) and we have

V Swnt =

N

hNtEMt

[hNTk

(V SwTk

)+], (3.32)

where N is the notional.

20

Proposition 3.27. Assume an additive exponential-rational pricing kernel model and assume the singlecurve setup. Let V Swn

t , as in (3.32), be the swaption price at time 0 ≤ t ≤ Tk. Let Tk+1 < Tk+2 < · · · < TNdenote the payment dates of the underlying swap. Set

c0 =N∑

i=k+1

(R(Ti−1)

(1− bR(Ti−1)

)S(Ti−1)− (1 + δiK)R(Ti)

(1− bR(Ti)

)S(Ti)

),

c1 =

N∑i=k+1

(R(Ti−1)b

R(Ti−1)S(Ti−1)eκTkTi−1

(wR+wS) − (1 + δiK)R(Ti)bR(Ti)S(Ti)e

κTkTi(wR+wS)

).

If c0 < 0 and c1 < 0, then V Swnt = 0, and if c0 > 0 and c1 > 0, then

V Swnt =

1

hNtASt

(c0 +AR

t c1eκtTk (wR+wS)

).

If sign(c0) 6= sign(c1), define Y1 = 〈wS, XTk〉, Y2 = 〈wR, XTk〉 and qt(z) = EMt [e〈z,(Y1,Y2)〉]. Let R < −1 if

c0 < 0 and R > 0 if c0 > 0. Assume that qt(−R, 1) <∞. Then

V Swnt =

|c0|πhNt

∫R+

Reϑt(u)

(R+ iu)(1 +R+ iu)du

where ϑt(u) = α−(R+iu)qt(−(R+ iu), 1) and α = |c1/c0|.

Proof. From the nominal bond pricing formula (2.6) we have that

hNTkPNTkTi

= R(Ti)S(Ti)(ASTk

(1− bR(Ti)

)+ bR(Ti)EM

Tk

[ARTiA

STi

]).

Then, inserting this into the swap formula (3.31), we obtain

hNTkVSwTk

=N∑

i=k+1

(R(Ti−1)S(Ti−1)

(ASTk

(1− bR(Ti−1)

)+ bR(Ti−1)EM

Tk

[ARTi−1

ASTi−1

])− (1 + δiK)R(Ti)S(Ti)

(ASTk

(1− bR(Ti)

)+ bR(Ti)EM

Tk

[ARTiA

STi

])).

We may writeEMTk

[ARTiA

STi

]= AR

TkASTk

eκTkTi(wR+wS). (3.33)

Collecting terms and using the fact that ASt > 0 for any t ≥ 0, we arrive at(

hNTkVSwTk

)+= AS

Tk

(c0 + c1A

RTk

)+.

If c0 > 0 and c1 < 0, then (c0 + c1A

RTk

)+= |c0|

(1− αAR

Tk

)+, (3.34)

where we recall that α = |c1/c0|. The result follows from Lemma A.31. If c0 < 0 and c1 > 0, then(c0 + c1A

RTk

)+= |c0|

(αAR

Tk− 1)+

(3.35)

and the result follows from Lemma A.32. The two remaining cases are straightforward.

21

The independent increments property of (Xt) is only used to obtain Eq. (3.33).

Remark 3.28. Under the assumption of Specification 3.19 the counterpart to Proposition 3.27 is obtainedby applying Lemma A.34 and A.35 to Eq. (3.34) and Eq. (3.35).

3.3.2 Multi-curve interest rate setting

We can, at a relatively low cost, allow our model to incorporate multi-curve-features. This is done bymodelling (3.30) as a rational function of state variables not fully spanned by the ones driving the nominalbonds. We model the forward LIBOR by

L(t, Ti−1, Ti) :=1

hNtEMt

[hNTiL(Ti, Ti−1, Ti)

].

(Crepey et al., 2016) propose the following definition, which we shall adopt.

Definition 3.29 (Rational multi-curve setup). Let

L(t, Ti−1, Ti) :=L(0, Ti−1, Ti) + bL(Ti−1, Ti)(A

Lt − 1)

hNt, (3.36)

where (ALt )0≤t is an M-martingale with AL

0 = 1, and where bL(·, ·) and L(0, ·, ·) are deterministic functions.

We consider ALt = exp(〈wL, Xt〉), where wL is chosen such that (ALt ) is a martingale. This is analogous

to how (ARt ) and (AS

t ) are modelled. Adding a multi-curve dimension to the nominal markets has no effecton any of the formulae derived for the inflation products. It does though impact the swaption formula.

Proposition 3.30. Assume an additive exponential-rational pricing kernel model and the multi-curvesetup. Consider a swaption with maturity Tk written on a swap with payments dates Tk+1 < Tk+2 < · · · <TN . The swaption price V Swn

t at 0 ≤ t ≤ Tk is given by

V Swnt =

1

hNt

∫Rn

HM(x)+mTk(dx) (3.37)

where mTk is the distribution of XTk and

HM(x) = c0 + cL exp (〈wL, x〉) + cS exp (〈wS, x〉) + cSR exp (〈wR + wS, x〉) .

Furthermore,

c0 =N∑

i=k+1

δi(L(0, Ti−1, Ti)− bL(Ti−1, Ti)

), cL =

N∑i=k+1

δibL(Ti−1, Ti)

cS = −N∑

i=k+1

δiKR(Ti)S(Ti)(1− bR(Ti)

), cSR = −

N∑i=k+1

δiKR(Ti)bR(Ti)S(Ti)e

κTkTi(wR+wS).

Proof. Using (3.36) we may write

hNTkVSwTk

=

N∑i=k+1

δi

(L(0, Ti−1, Ti) + bL(Ti−1, Ti)(A

LTk− 1)

−KR(Ti)S(Ti)(ASTk

(1− bR(Ti)

)+ bR(Ti)EM

Tk

[ARTiA

STi

])).

22

By collecting the terms, the result follows.

We note that, analogous to the single-curve setup, the independent increments of (Xt) are only usedto evaluate EM

Tk[AR

TiASTi

]. In our applications, since (Xt) is bi-variate, we may apply the two-dimensionalcosine method of (Ruijter and Oosterlee, 2012). The most immediate method for handling (3.37) inhigher dimensions is in the style of (Singleton and Umantsev, 2002), where H(x)+ ≈ H(x)1G. withG = H(x) ≥ 0 being exact. If G = 〈ω,XTk〉 > α, this leads to a one-dimensional integral, see (Kim,2014) and (Cuchiero et al., 2019). If G = β1 exp(〈ω1, x〉) + β2 exp(〈ω2, x〉) > α the inversion formulabecomes a two-dimensional (Hurd and Zhou, 2010)-type formula.

4 Calibration examples

In this section, we show the calibration properties of the models on real data. We consider EUR data fromBloomberg from 1 January 2015. The necessary data consists of OIS zero-yields constructed from EONIAovernight indexed swaps, LIBOR discrete curves based on EURIBOR and a term structure of ZC forwardrates, as well as YoY cap and floor prices and EURIBOR swaptions. There is no LPI traded on EURdata. The OIS and EURIBOR curves are constructed directly in the Bloomberg system. We then set thenominal curve equal to the OIS curve, and the initial real (or equivalently the initial inflation-linked) curveis implied from the OIS curve and zero-coupon inflation forward rate using the methodology described inSection 2.3. The prices for YoY caps and floors are available to us for maturities 2, 5, 7, 10, 12, 15, 20and 30 years. The strikes for the floors range from -1% to 3% and caplets from 1% to 6%. Quotes forYoY swap rates are not available to us, but the overlap in strikes for YoY caps and floors allows us to useput-call (cap-floor) parity to imply YoY swap rates consistent with the option prices.

0 5 10 15 20 25 30 35-0.01

-0.005

0

0.005

0.01

0.015

0.02

OIS/Nominal Zero YieldsReal Zero YieldsZC Inflation Swap ratesYoY Inflation Swap rates(Option Implied)3M Euribor

0 5 10 15 20 25 30-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Figure 1: Left: The initial curves. Right: Convexity corrections for YoY swap rates. 1 January 2015.

In Figure 1, all the curves are plotted on the left-hand side. The real curve is plotted as zero-couponrates, and we note that on this day, there is a consistently negative real curve with a widening gap tothe nominal as the maturity increases. The 3m EURIBOR and OIS curve are plotted as discrete forwardrates with 3m increments to be directly comparable, and we can note a significant spread between the twocurves in the short and most liquid end of the maturity spectrum, which warrants the use of a multi-curvemodel to price nominal products. Finally, we observe that the option-implied YoY swap rates are closeto the ZC swap rates. This relation implies only small levels of the convexity correction as seen directlyin the right-hand-side of Figure 1 where the convexity correction, as described in Section 2.3, is plottedfor different swap lengths. An implied lognormal volatility surface is constructed from the prices of theseoptions (selecting out-of-the-money options where available) using a geometric Brownian motion model for

23

the CPI index as described in Section 2.2. Two of the prices for the two year maturity are identically zeroand are thus removed from the dataset. We find the at-the-money implied volatility of the YoY cap usingthe piecewise constant hermite interpolation. The surface is plotted in the left hand side of Figure 2, andone 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 limitour modelling to one curve—the 3m tenor curve. We thus calibrate only to swaptions with a one-yearunderlying swap length, since this swaption, by EUR market convention, contains payments involving only3m EURIBOR. We refer to (Crepey et al., 2016) for a more extensive calibration involving matching thevolatility of both the 3m and 6m EURIBOR curves in a rational model resembling this one, but withoutthe 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 tolognormal. This is done on the right panel in Figure 2. The data is from 1 January 2015, and we calibratedirectly to these volatilities.

Figure 2: On the left: Lognormal implied volatility surface. On the right: Implied normal (Bachelier) volatility in basispoints for swaptions on 1Y swaps. The data is from 1 January 2015.

Very similar to Specification 3.18, our model setup is the following:

Xt =(XRt + µR(t), XS

τS(t) + µS(t), µL(t)),

where µi(t) for i = L,R, S are deterministic martingalizing functions and thus the model is a two factormodel. We assume that (XR

t , XSt ) is a two-dimensional Levy process with independent marginals and that

τS(t) is a deterministic time-change. The two independent Levy processes are defined by their Laplaceexponents

κi(z) = ln(EM[ezX

i1

]), (i = R,S)

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

ARt = e〈wR,Xt〉, AS

t = e〈wS,Xt〉, ALt = e〈wL,Xt〉.

We set wS = (0, 1, 0), wR = (aR, baR, 0) and wL = aLwR + (0, 0, 1). This means that the b parameterdetermines the dependence between the (AR

t ) and (ASt ) and it furthermore means that the randomness in

(ALt ) is merely a (log)-linear transformation of the randomness in (AR

t ). As in (3.21), when wR, wS, wL ∈

24

E(X) we can solve for the martingalizing drifts to obtain

µS(t) =− τS(t)κS(1), µR(t) = −τS(t)

(κS(aRb)

aR− bκS(1)

)− tκR(aR),

µL(t) =− τS(t) (κS(aLaRb)− aLκS(aRb))− t (κR(aLaR)− aLκR(aR)) .

We set the deterministic time-change τS(t) =∫ t0 a(s) ds, where a(t) is a piecewise constant function

a(t) = ak, t ∈ (Tk−1, Tk].

Here T0, T1, . . . , T8 = 0, 2, 5, 7, 10, 12, 15, 20, 30, is the set of maturities quoted in the YoY option mar-ket. We calibrate the constants a1, . . . , a8 starting from the smallest to the largest maturity by matchingto the YoY cap/floor volatility surface allowing a perfect fit to at least one strike per maturity. The de-pendence structure between the R and S component is fully determined by the parameter b thus reducingthe model to a two-factor setup where the calculated expressions for YoY caplets, YoY swap prices andswaption prices can be applied directly without approximation.

The nominal and the real curve are fitted by construction, but fitting the term-structure of YoY swaprates is less straightforward, since the swap rate depends on the full parameter set of the model, seeEq. (2.10). We choose to calibrate the bR(t) function to this term structure. There is enough flexibility inthe bR(t) function to fit the YoY swap rates without error, but direct calibration results in a quite volatilebR(t) function which is hardly desirable. Therefore we instead fit an eight-knot Hermite polynomial with anon-smoothness penalty – a similar choice is made in (Gretarsson et al., 2012) – and we find that the lossof accuracy when doing this is insignificant. The flexible shape means that the correlation parameter band volatility parameter aR in practice cannot be identified simultaneously with bR(t) from the YoY swapcurve. We solve this issue by simply fixing the b and the aR parameters before calibration. In practiceone needs only to avoid setting these parameters too low because the convexity correction becomes zero,by construction, if b = 0 or aR = 0. In both of our calibration examples we fix these values at b = 30 andaR = 0.25.

Since swap rates are determined not only by the bR(t) function, but the full parameter set of the model,one cannot calibrate bR(t) independently of a(t) and the parameters determining the (Xt) process. Onthe other hand, YoY cap and floor prices are primarily affected by the (AS

t ) component and thus not verysensitive to the changes in values of bR(t) unless the correlation between (XR

t ) and (XSt ) is very high,

which means this dual identification problem is in fact easily solved in practice. The overall calibrationalgorithm can be reduced to:

1. Set bR(t) = 1, and calibrate a1, . . . , a8 and the parameters of determining the law of (XSt ) to YoY

cap/floor implied volatilities.

2. Calibrate bR(t) to the curve of YoY swap rates rates using least squares minimization with a penaltyfor bR(t) /∈ (0, 1).

3. Repeat Step 1 using instead the updated values of bR(t).

4. Calibrate bL(·, ·) to swaption prices.

The swaption calibration is done by calibrating the bL(·, ·) function sequentially. This means that the aLparameter, which also determines overall variance cannot be calibrated at the same time and we thereforefix it at aL = 1.3 below. We parametrise the bL(·, ·) function by setting bL(t, t+ 3m) = PN(0, t)L(0, t, t+3m) + bL(t). The function bL(t) is then piecewise constant in relation to the swaption maturities we canobserve, i.e.

bL(t) = bk, t ∈ (Tk, Tk+1]

25

with T0, T1, . . . , T11 = 3m, 6m, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 15Y, 20Y, 30Y, 31Y .

Gaussian example

We first assume that (XRt ) and (XS

t ) are independent standard Brownian motions with Laplace exponentat t = 1 κR(z) = κS(z) = 1

2z2, i.e. in the spirit of Specification 3.19. We would not expect a Gaussian or

log-normal model to be well suited to reproduce implied volatility smiles, but we nevertheless believe thata Gaussian setup is illustrative as a benchmark case of study.

As discussed above, we first fix the b = 30 and aR = 0.25, and then proceed with the calibrationalgorithm described above. In Step 1 we choose to calibrate a1, . . . , a8 to at-the-money implied volatility.This is done sequentially starting with calibrating a1 to the two-year YoY implied volatility and a2 to thefive-year YoY implied volatility, and so forth. We note that the value of a1 affects not just the two-yearmaturity 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 ofthese parameters is key. The result of this calibration can be seen in Figure 3 where we plot at-the-moneyimplied volatility from the market. We furthermore plot an example of the model smile in the for afixed maturity of five years. While the model smile is not completely flat, the Gaussian structure is, byconstruction, not suited for smile fitting. Finally the YoY swap rates are fitted without error by adjustingthe bR(t) function. YoY swap rates requires only mild adjustment of the bR(t) function away from itsdefault value of one.2

Figure 3: Upper Left: At-the-money lognormal implied volatility of YoY caps model vs market using data from 1 January2015. Upper Righ: YoY cap/floor implied volatility. Lower Left: YoY option implied swap rates, market vs. (Gaussian)model. Lower Right: At-the-money normal swaption implied volatility. 1st of January 2015.

When fitting to swaptions we fix aL = 1.3 as explained above. Then we sequentially fit the bL(t)function directly to swaption normal implied volatility starting from the three-month maturity up to the

2All parameter values are available upon request.

26

thirty-year maturity. The model is made to fit at-the-money, swaptions only, and the results are plottedin the lower right quadrant of Figure 3.

NIG example

To produce a model more in line with the volatility smile, we instead assume that (XRt , X

St ) are independent

Normal Inverse Gaussian (NIG) processes, see for example (Barndorff-Nielsen, 1998). We have that theLaplace exponent at t = 1 is given by

κi(z) = −νi(√

ν2i − 2zθi − z2σ2i − νi), i = R,S,

expressed in terms of the parametrisation (νi, θi, σi) as where νi, σi > 0 and θi ∈ R. Since we want to

control variance primarily using the time-change τS(t), we set σi =√

1− θ2i /ν2i so that XR1 and XS

1 both

have variance of 1. Since we are only calibrating to the YoY cap/floor smile the full distribution of bothmarginals in (XR

t , XSt ) is not identified by the data. For simplicity, we also set νS = νR and θS = θR. As in

the Gaussian case we prefix b = 30 and aR = 0.25. In the NIG case we split Step 1 in the calibration processby first fixing the rate of time at a constant, i.e. ai = a, and then calibrate a, νS, θS to the whole YoYcap/floor implied volatility surface using the lsqnonlin algorithm in Matlab. Thereafter, the individuala1, . . . , a8 are calibrated sequentially such that the model fits the at-the-money implied volatilities withouterror. The rest of the algorithm is followed exactly like in the Gaussian case.

-0.02 0 0.02 0.04 0.06

0.015

0.02

0.025

0.03

0.035

NIG ModelMarket implied vol

-0.02 0 0.02 0.04 0.06

0.015

0.02

0.025

0.03

0.035

-0.02 0 0.02 0.04 0.06

0.015

0.02

0.025

0.03

0.035

-0.02 0 0.02 0.04 0.06

0.015

0.02

0.025

0.03

0.035

-0.02 0 0.02 0.04 0.06

0.015

0.02

0.025

0.03

0.035

0 10 20 300.014

0.016

0.018

0.02

0.022

Figure 4: Lognormal implied volatility for YoY cap/floors model vs market using data from 1 January 2015.

27

The fit to YoY swap rates is indistinguishable from the graph in Figure 3 and is not plotted again.We set aL = 1.3 and fit the bL(t) function to the same dataset of swaptions on one-year underlying swaps.The resulting fit is again indistinguishable from the fit in Figure 3 and the calibrated bL(t) function isavailable upon request. In Figure 4, we plot model vs market volatility smiles for select maturities. Wehave only used one time-dependent scaling, so the model fits the at-the-money level without error. Theremaining option prices are in principle fitted using only two parameters νS and θS. Thus we would notexpect a perfect fit for all maturities. In general, any Levy process is well known to exhibit a flatteningsmile as maturities are increased which often results in a slightly too steep smile in the short end and tooflat in the long end. These problems could be resolved by introducing further time-inhomogeneity or byapplying stochastic time-changes, but with the virtue of model simplicity taken into account, we view thecalibrated setup as satisfactory.

5 Conclusions

This paper focuses primarily on the theoretical development of stochastic, rational term-structure modelsusing pricing kernels suitable for the pricing of nominal and inflation-linked financial instruments. Wedemonstrate 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. Weexpect future research to be focused more on the numerics of risk management within the model as well ascalibration to a broader set of market instruments such as joint calibration of year-on-year and zero-couponcaps, as well as including time-series information in the calibration problem.

Appendix

A Lemmas

This section contains a number of lemmas used for the derivation of the formulae for option pricing.

Lemma A.31. Let qt(z) = Et[exp(〈z, Y 〉)] be the moment generating function of Y = [Y1, Y2], a randomvector with conditional distribution mt. Assume R > 0, q(−R, 1) <∞ and α > 0. Then,

Et[eY2(1− αeY1)+

]=

1

π

∫R+

Reα−(R+iu)qt(−(R+ iu), 1)

(R+ iu)(1 +R+ iu)du.

We omit the proof of this lemma since it is standard.

Lemma A.32. Let qt(z) = Et[exp(〈z, Y 〉)] be the moment generating function of Y = [Y1, Y2], a randomvector with conditional distribution mt. Assume R < −1, qt(−R, 1) <∞ and α > 0. Then,

Et[eY2(αeY1 − 1)+

]=

1

π

∫R+

Reα−(R+iu)qt(−(R+ iu), 1)

(R+ iu)(1 +R+ iu)du.

Lemma A.33. Let qt(z) = Et[exp(〈z, Y 〉)] be the conditional moment generating function of Y = [Y1, Y2, Y3],a random vector with conditional distribution mt. Assume R > 0 and that qt(−R, 1, 1) + qt(−R, 1, 0) <∞.Then

Et[(1 + eY3)eY2(1− eY1)+

]=

1

π

∫R+

Reqt(−(R+ iu), 1, 0) + qt(−(R+ iu), 1, 1)

(R+ iu)(1 +R+ iu)du

28

Proof. Write

Et[(1 + eY3)eY2(1− eY1)+

]= Et

[eY2(1− eY1)+

]+ Et

[eY3+Y2(1− eY1)+

]and apply Lemma A.31 to each term.

Lemma A.34. Let X ∼ N(µx, σ2x) and Y ∼ N(µy, σ

2y) with Cov[X,Y ] = σxy and assume that α > 0.

Then

E[eX(αeY − 1)+

]= αeδ+µy+

12(1+2b)σ2

yN(d+ σy)− eδN(d)

where a = µx− σxyσ2yµy + (σ2x− σ2xy/σ2y)/2, b = σxy/σ

2y, δ = a+ bµy + (bσy)

2/2 and d = 1σy

(lnα+ bσ2y +µy).

Proof. By conditional of normals E[eX | Y ] = ea+bY . Noting that − (x−µ)22σ2 + bx = bµ+ b2σ2

2 −(x−µ−bσ2)2

2σ2 ,the tower property yields

E[eX(αeY − 1)+

]= E

[ea+bY (αeY − 1)1

Y≥ln 1α

]= αeδ+µy+

12(1+2b)σ2

yN(d+ σy)− eδN(d),

as sought.

Lemma A.35. Let X,Y and α be as in Lemma A.34. Then

E[eX(1− αeY )+

]= eδN(−d)− αeδ+µy+

12(1+2b)σ2

yN (−d− σy)

where a, b, δ and d are as in Lemma A.34.

Proof. As Lemma A.34.

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