LpIs with a smile

56 RiskApril 2013

CUTTINGEDGE.INFLATIONDERIVATIVES

Index (LPI) swaps are zero-coupon inflation swaps on capped and floored

year-on-year (YoY) rates. They are particularly liquid on the UK market, where some pension plans are linked to this index. The floor is typically at 0% and the caps can be 3%, 5% or +∞. They are quoted as a spread over the inflation-linked zero-coupon rates.

Due to the strong path-dependency of the payoff, LPIs have his-torically been valued by Monte Carlo simulation using term-struc-ture models. However, inflation models tend to imply LPI spreads very different to what is quoted in the market. The reason is that those models often struggle to cope properly with the parameters to which the LPI payoff has high sensitivity, namely the steep mar-ginal YoY smile and the correlations between YoY rates.

In fact, the most standard inflation models, such as Jarrow-Yildirim (2003) and the market models of Mercurio (2005) and Belgrade et al (2004), imply an almost flat normal YoY smile. The analytical approximations for LPI swaps from Brody and Crosby (2008) and Zhang and Mercurio (2011) also rely on log-normal forward CPI index dynamics. Incorporating stochastic volatility seems to be a potential solution. Mercurio and Moreni (2006 and 2009) got closer to reproducing the YoY smile by using stochastic alpha-beta-rho (SABR)-like dynamics. The approach suggested by Oosterlee et al (2011) involves using Hes-ton dynamics on the spot inflation index. However, the param-eters required to reproduce the market smile often end up being extreme – this is typically the case in sterling – which can cause numerical problems: it is not uncommon to see the volatility of variance higher than 300% and the correlation between the sto-chastic volatility and the underlying lower than −90%. More recently, Trovato, Ribeiro and Gretarsson (2012) have proposed an inflation model based on quadratic gaussian dynamics, which achieves a decent fit to the YoY smile. Alternatively, Zhang and Mercurio (2010) got a satisfactory match of LPI market prices by breaking down the LPI payoff into two com-ponents and simply pricing the first by plugging the YoY options market volatilities into Black formulas.

In this article, we aim to introduce a generic and self-consist-ent framework to price LPIs, taking into account the informa-tion about YoY options prices and correlations without resorting to payoff approximations, and to estimate the impact on spreads. Rather than go through the route of a term-structure model, we

use a term-distribution able to reproduce the YoY options mar-ket smile. The marginal distributions of the YoY rates are implied from the option prices and the rates are then jointly simulated via a Gaussian copula. Finally the LPI swap is priced with Monte Carlo. We show that the resulting prices obtained are close to the market.

FrameworkFollowing Zhang and Mercurio (2011), we consider an annual time structure T0 = 0 , T1 , ... , TN, where Ti = i years. Let I(t) denote the inflation index at time t. The YoY rate represents the return of the inflation index over a year

Yi = I (Ti) /I (Ti−1) − 1 (1)

The LPI index of maturity TN capped at C and floored at F is defined as

LPI Y C FN i

i

N= + { }{ }( )

=∏ 11

max min , ,

(2)

where Yi is the YoY rate between dates Ti−1, Ti. The LPI spread cor-responds to the difference between the annualised fixed rate that cancels the value of the LPI leg and the zero-coupon rate. It can be expressed as follows

s LPI zN

NN

TN

N= [ ]( ) − −E

11

/

(3)

where Ei denotes the expectation under the Ti-forward measure QTi, and zi is the zero-coupon rate of maturity Ti, given by

1 0 0 0+( ) = ( ) ( )⎡⎣ ⎤⎦ = ( ) ( )z I T I I T Ii

T ii i

i E / , /

(4)

with I(t, T) the forward inflation index of maturity T.

ModelModel used for YoY options■ Properties of normal inverse Gaussian processes The copula approach presented in this paper requires as input a model for generating the YoY forwards and YoY option prices. Here we con-sider the case of a term-distribution model where the forwards are

LPIs with a smileInflation models tend to be poor at capturing the high sensitivity of Limited Price Index (LPI) swap payoffs to year-on-year smiles and correlations, and consequently miss market quotes. Yann Ticot and Xavier Charvet propose a simple framework for pricing LPI swaps using the Gaussian Copula that gives a handle on these features – and better fits the data

Limited price

risk.net/risk–magazine 57

generated using the Jarrow-Yildirim model and the YoY options are priced with a normal inverse Gaussian (NIG) distribution. The choice of NIG is motivated by the fact that this distribution provides considerable flexibility in terms of shapes of smile, yet allows for fast and efficient numerical methods for option pricing. A process X is NIG if it has the following properties:

X Z Z Z~ ,N μ β=( ) (5)with N(μ,σ) the Gaussian distribution with mean μ, standard deviation σ and Z defined as follows

Z ~ ,IG δ α β2 2−( )

(6)

where 0 ≤ |β| ≤ α and IG(μ, λ) denotes the inverse Gaussian distribution with parameters μ and λ.

All moments are finite under this distribution, the first four being available in closed form, and both the density and the characteristic function have simple expressions. The density pNIG is given by

p xK x

xNIG ; , , , expα β δ μ

αδ α δ μ

π δ μδγ( ) =

+ −( )( )+ −( )

12 2

2 2++ −( )( )β μx

(7)

while the characteristic function φNIG can be expressed as

φ α β δ μ δ α β α βNIG k ik; , , , exp( ) = − − +( ) − −⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞2 2 2 2

⎠⎠⎟

(8)

where γ = √α2 - β2, i is the complex unit and K1 is the modified Bessel function of the second kind and index 1. These properties allow for an efficient computation of the option prices by inte-gration, either against the density or by Fourier methods. For an extensive discussion on these pricing techniques, we refer the reader to Andersen and Piterbarg (2010) and Lipton (2001). Some recent improvements of the accuracy of Fourier pricing based on cosine series expansions can be found in Fang and Oosterlee (2008), with some further developments also sug-gested in Bang (2012).

■ Remapping into SABR parameters The parameterisation α, β, δ, μ does not provide an intuitive control of the smile. However, it is possible to map the NIG parameters onto SABR-like parame-ters: at-the-money (ATM) volatility σATM, correlation ρ and vola-tility of volatility ν.

To do so, we suggest combining matching of the ATM option prices and of some moments, having set the constant elasticity of variance (CEV) exponent to zero in SABR to get some moments in closed-form. This gives good control over the smile and pro-vides enough flexibility to match the market smile (see figure 1).

Distribution under the natural forward measureThe cumulative distribution function F of each YoY rate under its

1ImpactofNIGparametersonYoYcapletnormalsmileandcomparisonwithmarketquotes,UKRPIasofApril25,2012

0.00

0.50

1.00

1.50

2.00

2.50

-2.0 0.0 2.0 4.0 6.0Strike (%)

YoY 10y caplet NIG smile

ATM vol = 0.78%

ATM vol = 0.98%

ATM vol = 1.18%

0.00

0.50

1.00

1.50

2.00

2.50

-2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0Strike (%)


Rho =-87%

Rho = -47%

Rho=-7%

0.00

0.50

1.00

1.50

2.00

2.50

-2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0Strike (%)


Volvol =23%

Volvol =39%

Volvol =54%

0.00

0.50

1.00

1.50

2.00

2.50

-2.0 0.0 2.0 4.0 6.0

Strike (%)

YoY 10y cap smile

Broker quotes

NIG

% %

% %

58 RiskApril 2013


forward measure is obtained by differentiating the YoY option price with respect to the strike. So, the rate Yi can be simulated under measure QTi as

F Xi i− ( )( )1 Φ

(9)

where Φ is the normal cumulative and Xi is a Gaussian variable. For efficiency, the inverse cumulatives should be pre-computed before the Monte Carlo simulation. For each rate Yi the cumula-tive is cached as Fi (xi,j) based on a grid of M steps xi,j, j = 1...M chosen so that the cumulatives are evenly spaced:

x x F F

xxi j i j

ii j, , ,/+ = + Δ

∂

∂( )1

(10)

where ∆F denotes the constant grid spacing. It is also important

to ensure the forward is repriced by the stored cumulatives, since by no-arbitrage, the following must hold

Ei i m ix

xY x F y dym

M[ ] = + − ( )( )∫ 1

(11)

where xm, xM are respectively the minimum and maximum boundaries for the distribution used. To ensure that (11) is veri-fied, it is possible to insert additional grid points in the wings. Finally, at simulation stage, the inverse cumulatives are interpo-lated linearly, as described in Andersen and Piterbarg (2010).

Distribution of YoY rates under the payment measure■ Drift approximation To price an LPI swap of maturity TN, we need the marginal distributions under the pricing measure QTN rather than under the forward measures of the various rates QTi. Considering a given rate Yi, we make the approximation that the drift ai,N on the CPI ratio 1 + Yi due to the change of measure from QTi to QTN is deterministic – as in the Jarrow & Yildirim model. Under this approximation, the rate Yi can be simulated in the pricing measure as

Y a N F Xi i i i= ( ) + ( )( )( ) −−exp , 1 11 φ

(12)

where Xi is a Gaussian variable under QTN. These measure adjust-ments are calibrated by enforcing a number of no-arbitrage rela-tionships.

■ Index-linked zero-coupon payment In the absence of a cap or floor, the LPI payoff collapses to a standard index-linked zero coupon payment, which means the following must hold

exp ,a F Xi Ni

NN

i ii

N

=

−

=∑ ∏

⎛

⎝⎜⎞

⎠⎟+ ( )( )( )⎡

⎣⎢

⎤

⎦⎥ =

1

1

11E φ 11+( )zN

TN

(13)

■ Measure adjustment with Vasicek nominal rates We use the approximation (12) and the fact that

exp ,a

YYi N

Ni

ii

( ) = +[ ]+[ ]

EE

11

(14)

By no arbitrage

E EN

ii

N

i i N

i iiY

P TP T

P T TP T T

Y100

1+[ ] = ( )( )

( )( )

+( ),,

,,

⎡⎡

⎣⎢

⎤

⎦⎥

(15)

where P(t, T) is the nominal zero-coupon bond of maturity T seen at time t, and therefore combining this with (14) the measure adjustment is given by

exp

,,

,,

,a

P TP T

P T TP T T

Y

i N

i

N

i i N

i ii

( ) =( )( )

( )( )

+(00

1E ))⎡

⎣⎢

⎤

⎦⎥

+[ ]Ei iY1

(16)

Since we use the Jarrow-Yildirim model to compute the for-wards, the nominal short rate is a one-factor Vasicek process with mean reversion μt. Consequently, the drift due to the change of measure from QTi to QTN is linked to TN via a linear dependency to φN − φi, where

φ μi

Tu

si du ds= −( )∫ ∫exp0 0

(17)

2UKLPIspreads,marketvscopulawithNIG,asofMarch30,2012

-60

-50

-40

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-20

-10

0

10

20

30

0 5 10 15 20 25 30Maturity (years)

0%–5% LPI spread (basis points)

Totem consensus

Copula NIG

Totem consensus+ range

Totem consensus– range

Broker bid

Broker offer

Copula NIG no volvol

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0

0 5 10 15 20 25 30Maturity (years)


Totem consensus

Copula NIG




0

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Copula NIG





Therefore, the measure adjustments are related to each other as follows for k, l > i

a a

li k i lk i

i, ,=

−

−

φ φ

φ φ (18)

As long as the mean reversion is small and the maturities are not long dated, φi should remain relatively linear as a function of time, and therefore the measure adjustments only have a mild dependency on the mean reversion.

■ Calibration algorithm The measure adjustments are defined uniquely by relationships (13) and (18) and can be obtained itera-tively. Consider for instance that all ai,j are known for i < k and j = 1, ...,N. Then ak,j, j = 1, ..., N can be determined as follows

■ ak,k = 0,■ ak,k+1 is obtained from (13) using a Monte Carlo simulation on a payment made at time Tk+1,■ ak,l for l > k + 1 can then be implied using (18).

■ Calibration of the Jarrow-Yildirim model The calibration of the Jarrow-Yildirim model must ensure that the market YoY swap rates are repriced correctly. Typically, this is achieved by first cali-brating both the nominal and the real economy to one-year caplets and using a low mean reversion (in our case the mean reversion is close to 0%). The various correlations and spot index volatilities have then to be calibrated in order to reprice the YoY swaps market quotes.

Correlations between YoY ratesPlenty of parameterisations are available in the literature to spec-ify the correlation matrix (ρi,j) between the Gaussian variables X1, ...,XN. In this article, we adopt a basic parameterisation because it is good enough to get a decent match of the market prices and to get an estimation of the impact of correlation on LPI spreads. The parametric form is given as

ρ λ γi j i jT T, exp= − −( )

(19)

The combination of parameters λ and γ gives some control over the level and steepness of the correlation surface, with γ = 0 cor-responding to flatness across tenors.

Pricing using the Gaussian copulaPricing methodWe consider an LPI payment made at time TN, with floor F and cap C. The present value is given by

P T Y C FN

Ni

i

N0 1

1, max min , ,( ) + { }{ }( )⎡

⎣⎢

⎤

⎦⎥

=∏E

(20)

which can be evaluated by simulating the various YoY rates using a standard Gaussian copula Monte Carlo approach. In other words, for each simulation, we do the following:

■ generate an N-dimensional Gaussian vector Z■ obtain the correlated Gaussian variables by setting X = CZ, where C is the Cholesky decomposition of (ρi,j)■ simulate the YoY rates Y1, ..., YN by applying the inverse cumu-lative and the measure adjustments as described in (12)

More generally, this method can be used for pricing any Euro-

pean payoff g(Y1, ..., YN) paid at date TN.

ConvergenceThe convergence is fast, both as a function of the number of Monte Carlo simulations and of the number of cumulative distribution function (CDF) points. For various combinations of cap and floor levels (see figure 1) of 30-year maturity, 10,000 Monte Carlo simu-lations and 100 CDF points are enough to get within 1 basis point of the converged value for the LPI spread; using 50,000 Monte Carlo simulations and 1,000 CDF points brings the spreads within 0.2 basis point.

Calibration to LPI spreads market quotesAfter fitting the Jarrow-Yildirim model to the YoY swap market quotes, the NIG parameters are selected by best fit at each expiry to the YoY options market quotes for strikes ranging from 0% to

3UKLPIspreadsimpliedbydifferentYoYcorrelationstructures,asofMarch30,2012

4UKRPI30yzero-couponoptionlognormalsmile,asofMarch30,2012.ATM=3.6%

-60

-50

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-20

-10

0

10

20

0 5 10 15 20 25 30Maturity (years)


Flat corr=14%

Flat corr=0%

Flat corr=27%

Flat corr=39%

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0

10

20

0 5 10 15 20 25 30Maturity (years)

0%–5% LPI spread (basis points) average correlation=14%Gamma=0.01, lambda=1.9

Gamma=0.25, lambda=1.15


Gamma=1, lambda=0.35


0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5 6 7Strike (%)

30y zero-coupon implied lognormal volatilities

Market

NIG flat corr = 0%

NIG flat corr = 14%

NIG flat corr = 10% – cutoff = –6%

NIG flat corr = 0% – cutoff = -10%

60 RiskApril 2013


6%. Then the correlation parameters are calibrated to the LPI market quotes: primarily the 0%−5% LPI spreads, which are the most liquid, but we also take into account the 0%−3% and 0% − +∞% spreads. Interestingly, the market prices imply close-to-zero correlations between YoY rates: λ = 5, γ = 0.1. Note that the simplicity of the implied correlation structure also justifies the use of a parametric form to specify the correlations.

In figure 2, we compare 0%−5%, 0%−3% and 0%−∞ LPI spreads given by the copula with the available market informa-

tion. To give a better idea of the bid-offer on LPI spreads, we dis-play the Markit Totem consensus prices within the “max-min” range, and also add some points obtained from broker quotes. Additionally we include the results obtained with zero volatility of volatility, ie with a normal model on YoY rates calibrated ATM, it peforms poorly. The numbers were obtained using 100,000 Monte Carlo simulations with antithetic variables and 1,000 points for storing the inverse cumulatives.

For the 0%−5% and 0%−∞ LPI spreads, the copula is within 2

5Zero-coupondelta,ATMvega,dailyP&Landvega-hedgeratiosforLPIspread0–5atdifferentpointsintime

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basis points of the Totem consensus price. The 0%−3% spreads lie a bit further: the difference is up to 5 basis points. In order to get an exact match to the market quotes, a non-Gaussian copula could be investigated, to generate some correlation skew.

Impact of correlation■ Flat correlations We set γ to 0.01 and compare the LPI spreads with different flat correlations between the factors: 14%, 27%, 39% (setting λ to 1.9, 1.25 and 0.9). Figure 3 shows that the impact on the LPI spread 0%−5% is significant. Note that the impact of correlation on the LPI spread is two-fold. On the one hand, there is a positive impact due to the increased correlation of the capped and floored YoY rates, but on the other hand the for-wards go down in the payment measure because of the effect of correlation on the drifts. While the two effects cancel each other in the absence of cap and floor, the latter seems to prevail for positive caps and floors.

■ Term-structure of correlation We now try different combina-tions of λ and γ, which all give the same average correlation of 14%, but which imply different steepnesses of term-structures: γ = 0.01, λ = 1.9, γ = 0.25, λ = 1.15, γ = 0.5, λ = 0.75, γ = 1, λ = 0.35 and γ = 1.5, λ = 0.09. Figure 3 shows the impact on the various LPI spread 0%−5%. The LPI appears to be more sensitive to the correlations between YoY rates with close resets than to those with far-apart resets. This is expected since in the LPI payoff for-mula there are more cross-terms related to rates with close resets than for rates with far-apart resets. As a consequence, the effect of the change in short tenors correlations prevails overall.

Zero-coupon optionsAnother test is to price zero-coupon (ZC) options using the parameters calibrated to the LPI spreads market quotes, since the return of the inflation index over a period [T0, TN] can be expressed as a function of the YoY rates

I T I T YN i

i

N( ) ( ) = +( )

=∏/ 011

(21)

Figure 4 shows that the skew implied by the model is steeper than what is seen in the market. Note that the fact that NIG distribution has a fat left tail seems to explain the high implied ZC volatilities at low strikes. The same frame shows that extrapolating the NIG cumulative using a normal distribution below a certain cutoff could potentially lead to a more satisfactory fit of the smile. While it is

disappointing not to achieve a better fit of the market ZC smile, it is a well-known fact that the YoY options, LPI and ZC options mar-kets are not that well connected. Going forwards, to get a better overall match between the market and the copula, it would be inter-esting to investigate how non-Gaussian copulas perform.

Dynamics and hedgesDynamicsThe deltas with respect to zero-coupon swap rates and the vega sen-sitivities with respect to the ATM YoY volatility over eight months are plotted in figure 5. They correspond to the change in LPI spread 0%−5% for a parallel bump of 1bp respectively on the ZC rates and on the ATM YoY volatilities. The results show there is generally satisfactory stability across time; with only the vega of LPI 0%−5% from May to June showing some change in shape.

HedgesIn this section, to test the robustness of risks, we simulate various LPIs and their hedge portfolios with market data as of July 2012 and August 2012, with the additional constraint that we force the ATM YoY volatilities to have random daily variations. Figure 5 also plots the daily profit and loss of LPI 0%−5% versus the daily profit and loss of the delta and vega hedged LPI, as well as the hedge ratios in terms of the YoY caplets.

ConclusionWe have introduced a method that is able to reprice LPIs and YoY options consistently with the market and allows for a fast and robust valuation and risks. We have been able to demonstrate that the market currently quotes LPI spreads assuming very low cor-relations between YoY rates, and that it implies a steeper zero-coupon smile than seen in the market.

First order expansions such as (13) are currently popular, but one must bear in mind that they will work only as long as the LPI market assumes zero correlations between YoY rates. In our view, this copula approach can be seen as an extension of the frame-work of first-order expansions to non-zero correlations between YoY rates. Finally, our results show that the LPI spreads are quite sensitive to correlations. ■

YannTicotisdirectorofinflationderivativesandXavierCharvetisvice-presidentofinflationderivatives,atBankofAmericaMerrillLynch.TheauthorsareindebtedtoAlexanderLiptonforhisinvaluablehelpandadvice.Email:[email protected];[email protected]


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References

LpIs with a smile

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