Simple framework for pricing LPI swaps using Normal Inverse Gaussian distributions and a Gaussian copula.

1. 56 Risk April 2013
CUTTING EDGE. INFLATION DERIVATIVES
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.
Framework
Following 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
= + { }{ }( )=
∏ 1
1
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
N
N
T
N
N
= [ ]( ) − −E
1
1
/
(3)
where E
i
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 i
i i
i
E / , /
(4)
with I(t, T) the forward inflation index of maturity T.
Model
Model 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 smile
Inflation 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

2. 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 x
K x
x
NIG ; , , , expα β δ μ
αδ α δ μ
π δ μ
δγ( ) =
+ −( )( )
+ −( )
1
2 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 measure
The cumulative distribution function F of each YoY rate under its
1 Impact of NIG parameters onYoY caplet normal smile and comparison with market quotes, UK RPI as of April 25, 2012
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 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 6.0
Strike (%)
YoY 10y caplet NIG smile
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 6.0
Strike (%)
YoY 10y caplet NIG smile
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
% %
% %

3. 58 Risk April 2013
CUTTING EDGE. INFLATION DERIVATIVES
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
x
xi j i j
i
i 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
x
Y x F y dy
m
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 N
i
N
N
i i
i
N
=
−
=
∑ ∏
⎛
⎝⎜
⎞
⎠⎟ + ( )( )( )⎡
⎣
⎢
⎤
⎦
⎥ =
1
1
1
1E φ 11+( )zN
TN
(13)
■ Measure adjustment with Vasicek nominal rates We use the
approximation (12) and the fact that
exp ,a
Y
Y
i N
N
i
i
i
( )=
+[ ]
+[ ]
E
E
1
1
(14)
By no arbitrage
E EN
i
i
N
i i N
i i
iY
P T
P T
P T T
P T T
Y1
0
0
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 T
P T
P T T
P T T
Y
i N
i
N
i i N
i i
i
( )=
( )
( )
( )
( )
+(
0
0
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
T
u
si
du ds= −( )∫ ∫exp0 0
(17)
2 UK LPI spreads, market vs copula with NIG, as of March 30, 2012
-60
-50
-40
-30
-20
-10
0
10
20
30
0 5 10 15 20 25 30
Maturity (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
-160
-140
-120
-100
-80
-60
-40
-20
0
0 5 10 15 20 25 30
Maturity (years)
0%–3% LPI spread (basis points)
Totem consensus
Copula NIG
Totem consensus
+ range
Totem consensus
– range
Copula NIG no volvol
0
10
20
30
40
50
60
70
80
90
0 5 10 15 20 25 30
Maturity (years)
0%–inf LPI spread (basis points)
Totem consensus
Copula NIG
Totem consensus
+ range
Totem consensus
– range
Copula NIG no volvol

4. risk.net/risk–magazine 59
Therefore, the measure adjustments are related to each other as
follows for k, l > i
a a
l
i k i l
k 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 rates
Plenty 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 γ
i 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 copula
Pricing method
We 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
N
i
i
N
0 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
.
Convergence
The 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 quotes
After 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
3 UK LPI spreads implied by different YoY correlation structures, as
of March 30, 2012
4 UK RPI 30y zero-coupon option lognormal smile, as of March 30,
2012. ATM = 3.6%
-60
-50
-40
-30
-20
-10
0
10
20
0 5 10 15 20 25 30
Maturity (years)
0%–5% LPI spread (basis points)
Flat
corr=14%
Flat
corr=0%
Flat
corr=27%
Flat
corr=39%
-50
-40
-30
-20
-10
0
10
20
0 5 10 15 20 25 30
Maturity (years)
0%–5% LPI spread (basis points) average correlation=14%
Gamma=0.01,
lambda=1.9
Gamma=0.25,
lambda=1.15
Gamma=0.5,
lambda=0.75
Gamma=1,
lambda=0.35
Gamma=1.5,
lambda=0.09
0
2
4
6
8
10
12
14
16
18
0 1 2 3 4 5 6 7
Strike (%)
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%

5. 60 Risk April 2013
CUTTING EDGE. INFLATION DERIVATIVES
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
5 Zero-coupon delta, ATM vega, daily P&L and vega-hedge ratios for LPI spread 0–5 at different points in time
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-0.4
-0.3
-0.3
-0.2
-0.2
-0.1
-0.1
0.0
0 5 10 15 20 25 30 35
Maturity (years)
0–5 LPI spread zero-coupon delta Feb 28/12
Mar 31/12
Apr 30/12
May 31/12
Jun 30/12
Jul 31/12
Aug 31/12
Sep 30/12
Oct 31/12
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25 0–5 LPI spread YoY ATM vol vega
-100
-80
-60
-40
-20
0
20
40
60
80
100
Jul03/12
Jul05/12
Jul07/12
Jul09/12
Jul11/12
Jul13/12
Jul15/12
Jul17/12
Jul19/12
Jul21/12
Jul23/12
Jul25/12
Jul27/12
Jul29/12
Jul31/12
Daily profit and loss 0%–5% LPI
LPI
Hedged
LPI
0
5
10
15
20
25
Jul01/12
Jul03/12
Jul05/12
Jul07/12
Jul09/12
Jul11/12
Jul13/12
Jul15/12
Jul17/12
Jul19/12
Jul21/12
Jul23/12
Jul25/12
Jul27/12
Jul29/12
Jul31/12
Aug02/12
0%–5% LPI vega-hedge ratios – YoY caplets 1y
2y
3y
4y
5y
6y
7y
8y
9y
10y
0 5 10 15 20 25 30 35
Maturity (years)
Feb 28/12
Mar 31/12
Apr 30/12
May 31/12
Jun 30/12
Jul 31/12
Aug 31/12
Sep 30/12
Oct 31/12
%

6. 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 options
Another 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
( ) ( ) = +( )
=
∏/ 0
1
1
(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 hedges
Dynamics
The 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.
Hedges
In 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.
Conclusion
We 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. ■
Yann Ticot is director of inflation derivatives and Xavier Charvet is vice-
president of inflation derivatives, at Bank of America Merrill Lynch. The
authors are indebted to Alexander Lipton for his invaluable help and advice.
Email: yann.ticot@baml.com; charvetx@yahoo.fr
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