This paper outlines an approach to option pricing using the Normal Inverse Gaussian (NIG) distribution. The NIG distribution can take on various shapes to fit different volatility smiles. The paper derives a parameterization of the NIG distribution similar to the SABR model, allowing users to control the smile through familiar SABR-like parameters. Empirical results demonstrate the NIG distribution fitting smiles for inflation options on UK RPI and interest rate options on CHF Libor. The paper also discusses how modeling asset marginals as NIG distributions could benefit multi-asset option pricing.

1. Electronic copy available at: http://ssrn.com/abstract=1968453
Pricing with a smile: an approach using Normal
Inverse Gaussian distributions with a SABR-like
parameterisation
Xavier Charvet, Yann Ticot
November 10, 2011
Abstract
This article outlines a few properties of the Normal Inverse Gaus-
sian distribution and demonstrates its ability to fit various shapes of
smiles. A parameterisation in terms of SABR inputs is derived. A few
results related to vanilla options on RPI year-on-year inflation rates,
as well as caplets on CHF Libor rates are exposed. Finally, further ap-
plications for multi-asset option pricing are considered when the NIG
distribution is combined with a copula pricing framework.
Keywords: Normal Inverse Gaussian, NIG, Levy process, SABR,
smile, inflation, year-on-year.
1

2. Electronic copy available at: http://ssrn.com/abstract=1968453
Contents
1 Introduction 4
1.1 Motivation and literature review . . . . . . . . . . . . . . . . 4
1.2 Paper organization . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Option pricing using the NIG distribution 5
2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 A parameterisation of SABR type 7
3.1 Deriving SABR first four moments when βSABR = 0 . . . . . 7
3.2 Deriving NIG first four moments . . . . . . . . . . . . . . . . 8
3.3 Equivalence between original NIG and SABR-like parameter-
isations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3.1 Mapping from original NIG to SABR-like parameter-
isation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3.2 Mapping from SABR-like to original NIG parameter-
isation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Switching from σ0 to the ATM volatility in the SABR-like
parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5 Backbone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Impact of the parameters on the smile 10
5 Inflation options pricing 13
5.1 Forwards convexity . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Pricing year-on-year options with NIG . . . . . . . . . . . . . 13
6 Empirical results 13
6.1 RPI Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.1.1 Tables: market data, model smile, parameters of the
distributions . . . . . . . . . . . . . . . . . . . . . . . 14
6.1.2 Graphs: 2Y-7Y-10Y-20Y . . . . . . . . . . . . . . . . 17
6.2 CHF Libor rate . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2.1 Graphs: Expiries 1Y-3Y-5Y-10Y . . . . . . . . . . . . 19
7 Further applications 21
7.1 Multi-asset option pricing . . . . . . . . . . . . . . . . . . . . 21
7.2 Multi-asset option pricing methodology: a quick overview . . 21
A Deriving moments of order 2 and 4 in the SABR model
(’gaussian case’) 24
A.1 Deriving the variance . . . . . . . . . . . . . . . . . . . . . . . 24
A.2 Deriving the kurtosis . . . . . . . . . . . . . . . . . . . . . . . 24
2

3. B Impact of original NIG parameters on the density shape: a
heuristic approach 26
C Mathematical foundations of NIG processes 29
C.1 From the Poisson process to a pure-jump Levy process . . . . 29
C.1.1 The Poisson process . . . . . . . . . . . . . . . . . . . 29
C.1.2 The compound Poisson process . . . . . . . . . . . . . 29
C.1.3 Pure jumps Levy processes . . . . . . . . . . . . . . . 30
C.2 Mathematical considerations . . . . . . . . . . . . . . . . . . 30
C.3 Levy processes . . . . . . . . . . . . . . . . . . . . . . . . . . 31
C.4 NIG processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3

4. 1 Introduction
1.1 Motivation and literature review
Interest rate vanilla options smile Introduced by Hagan & al. in
[10], the SABR model and its analytical approximation have become the
market standard for pricing interest rates vanilla options. It combines speed,
transparency and most of the time allows a good fit of the market smile.
This document explores the possibility of using the Normal Inverse Gaus-
sian distribution as an alternative without compromising the speed of option
pricing and the intuitiveness about the effect of parameters on the smile. No
approximation of the implied volatility is needed since the density is known
in closed form, which enables an exact pricing via a one-dimensional quadra-
ture. We show that the model is able to cope with the low-strike case, can
produce a significant smile for short-dated expiries and performs well in a
negative rates environment.
Inflation options smile Investors who receive inflation-linked cash flows
tend to have appetite for products which give them a protection in case of
a deflation. As a consequence low-strike floors are by far the most popular
year-on-year options, which results in a pronounced normal volatility skew.
This super-normal type of distribution is notoriously difficult to reproduce
using any of the dynamics commonly used in the finance industry.
Institutions tend to manage their entire inflation book using a single
term-structure model. The first model to be widely used was the application
of the foreign-currency introduced by [13]. In this approach, both nominal
and real short rates are represented by one-factor Vasicek processes while the
spot inflation index is lognormal. As the inflation forwards are lognormal
the year-on-year normal smile is almost flat. In the market model introduced
by [14], [4] and [17], the forward inflation rate is still lognormal, thus the
year-on-year smile is also flat.
[18] extended the lognormal market model by adding a square-root stochas-
tic volatility to the inflation forwards. Although the smile did appear to
have a better fit of the market, their conclusion was that the model was not
flexible enough to fit the smiles for all the liquidly traded expiries.
[19] then presented a model where the year-on-year forwards have SABR
dynamics, which is more likely to achieve a better fit of the year-on-year
market smile than the others. However SABR has a number of drawbacks,
and other types of distributions have been proposed as an alternative, such
as [1].
Normal Inverse Gaussian distribution To address these issues we look
for distributions flexible enough to fit extreme shapes of smiles, and ana-
lytical enough to enable fast pricing and risk management. Generalized
4

5. hyperbolic processes typically give good flexibility to capture various shapes
of smiles due to their semi-heavy tails. Within this class, a number of pro-
cesses appear to be tractable, amongst which the Normal Inverse Gaussian
distribution (NIG). The application of this distribution to finance and to
option pricing has been discussed in depth by [20], [22], [15] or [12], however
mostly in the context of an asset being an exponential of a NIG process.
In this article, we model the asset (rate) directly as a NIG distribution and
show that it has a remarkable ability to fit the steep skews seen on inflation
and to cope with turbulent market conditions on nominal interest rates.
We also provide a SABR-like parameterisation, give a quick overview of our
quadrature technique for pricing vanilla options and show how multi-asset
European option pricing may benefit from modeling marginals as (Log-)NIG
distributions.
1.2 Paper organization
This article is organized as follows. We give a definition of the Normal
Inverse Gaussian distribution in Section 2. A remapping of the parameter-
isation into SABR-like user inputs is introduced in Section 3. A heuristic
approach to understand the effect of SABR-like parameters on the smile is
given in Section 4. Section 5 explains how the NIG distribution can be used
to price inflation options. Numerical results are exposed in Section 6 in
which are displayed concrete results of calibrated smile on UK RPI inflation
options and CHF interest rates options. Beyond this application to single
asset option pricing, various types of European payoffs may benefit from
being modelized using (Log-)NIG distributed marginals. Typical examples
include CMS-spread options, basket options and quantos payoffs. In the
inflation world, there is also a growing need for pricing the Limited Price
Index (LPI) in line with the year-on-year option market. Section 7 gives a
quick overview of a generic methodology that can be applied to price and
risk-manage these structures.
2 Option pricing using the NIG distribution
2.1 Definition
The Normal Inverse Gaussian distribution is characterised by a set of four
parameters (α, β, δ, µ). Consider two uncorrelated Brownian motions start-
ing respectively at α and 0, with constant drifts (β, δ), where δ > 0.
The NIG distribution corresponds to the law of the first Brownian motion
stopped at the time when the second Brownian motion hits a barrier µ > 0.
More formally, a variable X has a Normal Inverse Gaussian distribution if
5

6. it can be expressed as
X|Z ∼ N (µ + βZ, Z) ,
with N(µ, σ) the Gaussian distribution with mean µ, standard deviation σ
and Z defined as follows
Z ∼ IG δ, α2 − β2 where 0 ≤ |β| ≤ α ,
where IG(µ, λ) denotes the inverse Gaussian distribution with parameters
µ and λ.
A feature that will be of high importance in our applications is that
moments are all of finite order and that the density is available in closed
form, given by
p(x; α, β, δ, µ) =
αδK1 α δ2 + (x − µ)2
π δ2 + (x − µ)2
exp (δγ + β(x − µ)) ,
where γ = α2 − β2 and K1 is the modified Bessel function of the second
kind and index 1.
An interesting property of NIG distributions is that they can generate
some smile even for short expiries. This property arises as NIG is a pure-
jump Levy process. The intensity and frequency of jumps that occur is
given by the Levy measure, known in closed-form. Usually, the user will
calibrate the process parameters to the vanilla option prices. Alternatively,
calibrating the Levy measure to the historical returns is possible and in some
cases might be a convenient way of modeling the returns as NIG.
In the Appendix, the reader can find a short introduction on Levy pro-
cesses to which class NIG processes belong.
2.2 Pricing
The pricing of call and put options is done using a quadrature on the density.
This Section gives a quick overview of the methodology without going too
much into details. Assuming that the the asset or rate YT is NIG distributed
under the T-forward measure, the undiscounted price of a call struck at K
can be written as
ET
[YT − K]+
=
∞
−∞
(y − K)+
p(y)dy (1)
where the density p(x) is known in closed-form. A standard Gauss-Legendre
quadrature is not robust enough to achieve an accurate pricing. Instead,
splitting the integration interval into different sub-intervals whose size are
controlled by statistical characteristics of the distribution leads to more sat-
isfactory results. This method generates a smooth and accurate smile, as
6

7. long as vol of vols ν and expiries T are not “too” high - in practice, the
threshold with our quadrature seems to be in the region of ν2 · T = 30. In-
versely, when the vol of vol is very small, NIG degenerates into a Gaussian
distribution, which corresponds to NIG parameters α and δ that go to infin-
ity. In this case, the Bessel function K1 takes a significantly high parameter
as input and its usual approximation computed as in [21] may lead to nu-
merical instabilities. To handle this case it is worth using an approximation
of the Bessel function K1 specific to high input values.
3 A parameterisation of SABR type
In this Section we derive an alternative representation of the NIG distribu-
tion in terms of usual SABR parameters. The objective is to give the user
the ability to control the implied smile in an easy way, which is lacking in
the parameterisation in terms of the original NIG parameters (α, β, δ, µ).
The mapping between the NIG parameters and the SABR-like parameters
(σ0, ρ, ν, F0) is achieved by ensuring that the first moments generated by
both distributions match. In this article we work with the assumption
βSABR = 0. This allows the forward to take negative values, and thus
corresponds to the behaviour of the year-on-year rates and even interest
rates in some cases. Since the distributions of the NIG and SABR with
βSABR = 0 are each characterized by four parameters, we seek to match
their first four moments.
3.1 Deriving SABR first four moments when βSABR = 0
In the SABR model, the underlying is assumed to follow a dynamics given
by the following SDE:
dFt = σtFβSABR
t dW1
t ,
dσt = νσtdW2
t ,
where W1
t and W2
t are two Brownian motions correlated by a factor ρ,
βSABR is a CEV parameter, and ν is the volatility of volatility. We consider
a maturity T. The first moments of FT are then given by:
• the forward is equal to F0 ,
• the variance is given by
E (FT − F0)2
= σ2
0 ·
eν2T − 1
ν2
,
7

8. • the skewness is directly controlled by ρ ∈ [−1, 1] ,
• the total kurtosis is given by
E (FT − F0)4
E (FT − F0)2
2 = ˜A · x4
+ 2x3
+ 3x2
+ 4x + 5 + 2 ˜B · (x + 2) ,
where
x = eν2T
,
˜A =
1 + 4ρ2
5
,
˜B = −2ρ2
.
A detailed derivation of SABR moments of order 2 and 4 is done in the
appendix A.
3.2 Deriving NIG first four moments
If X ∼ NIG(α; β; δ; µ), then
• the forward is equal to µ + δ · β
γ ,
• the variance is given by δ · α2
γ3 ,
• the skewness is directly controlled by β
α ∈ [−1, 1] ,
• the total kurtosis is given by 3 + 3 1 + 4 β
α
2
· 1
δγ .
3.3 Equivalence between original NIG and SABR-like pa-
rameterisations
With the moments controlled by closed form formulae, it is possible to go
from one parameterisation to the other using a numerical solver.
3.3.1 Mapping from original NIG to SABR-like parameterisation
We suppose here that we know the original parameters (α, β, δ, µ) of a given
NIG distribution. We want to find an equivalent set of parameters of SABR
type (F0, σ0, ν, ρ) which generates a SABR distribution whose first 4 mo-
ments match the first 4 moments of the (α, β, δ, µ) NIG distribution 1. In
the following, γ = α2 − β2.
1
Apart from the skewness, in which case we decide to match ρ with β
α
directly.
8

9. • since the mean of the (α, β, δ, µ) NIG distribution is given by E(X) =
µ + δ · β
γ , we define F0 as F0 = µ + δ · β
γ ,
• avoiding tedious calculation of third moments, it is natural, since β
controls the symmetry of the NIG distribution, and |β| ≤ α, and ρ
similarly controls the skew of the SABR smile, to define ρ as ρ = β
α ,
• noting x = eν2T , we find the vol of vol parameter ν by matching the
kurtosis of both distributions:
˜Ax4
+ 2 ˜Ax3
+ 3 ˜Ax2
+ 4 ˜A + 2 ˜B x + 5 ˜A + 4 ˜B = 3 + 3
1 + 4 β
α
2
δγ
,
• finally the overall level σ0 by matching their variances:
σ2
0 ·
eν2T − 1
ν2
= δ ·
α2
γ3
.
3.3.2 Mapping from SABR-like to original NIG parameterisation
Equally, since the transformation above is a one-to-one mapping, it is possi-
ble to go the other way around: in practice, the user will enter a SABR-like
set of parameters to describe a NIG distribution whose original NIG param-
eters are then retrieved using the following procedure:
• δγ is found by matching the Kurtosis. This is do-able since β
α is known
as being equal to ρ ,
• matching variances allows us to retrieve α since δγ is now known ,
• β is then backed out as β = ρα ,
• δ can now be retrieved since both kurtosis and γ = α2 − β2 are
known ,
• finally µ is simply backed out by matching forwards since (α, β, δ) is
known .
3.4 Switching from σ0 to the ATM volatility in the SABR-
like parameterisation
Section 3.1 has shown that the initial local volatility σ0 directly controls the
variance of the distribution. When the vol of vol remains small, σ0 and the
at-the-money (ATM) volatility are close, but this is no longer the case when
the vol of vol reaches high values. In fact, as ν tends to infinity, a NIG
distribution tends to a Cauchy distribution whose moments are infinite: in
particular, σ0 goes to infinity while ATM volatility has obviously to remain
9

10. constant. Note that in SABR, the ATM volatility also diverges from the
initial local volatility as vol of vol increases. This problem is usually tackled
by having the ATM volatility parameter as an input rather than the initial
local volatility. In that case, using a standard numerical solver it is easy to
obtain the initial local volatility from the ATM volatility given that there is
a one-to-one relationship between the two, We adopt this approach and from
now on use the SABR-type parameterisation (F, Σ, ν, ρ), where Σ denotes
the ATM volatility.
3.5 Backbone
Given that the ATM volatility is directly an input, it is possible to have
a control over the backbone. Assume for instance that we want to have a
backbone corresponding to normal dynamics. We must then have
dV NIG
ATM
dF
=
∂V N
ATM
∂F
,
where V NIG
ATM and V N
ATM denote the ATM option prices using respectively
NIG and normal pricing formulae. This leads to
∂V NIG
ATM
∂Σ
∂Σ
∂F
=
∂V N
ATM
∂F
−
∂V NIG
ATM
∂F
,
which gives the change of the ATM volatility Σ with respect to the forward.
4 Impact of the parameters on the smile
The effect of the original NIG parameters on the shape of the density is
shown in a few graphs in the appendix B. In this Section, we present some
graphs which should help in getting an intuition and show that the impact
of the SABR parameters on the smile is as expected. We use the folowing
set of parameters:
• F0 = 2.3% ,
• Σ = 1.1% ,
• ρ = −10% ,
• ν = 55% ,
• T = 5 years ,
and we vary respectively either the ATM volatility Σ, the correlation ρ, and
the vol of vol ν by keeping the other parameters constant. The quadrature
used appears to be stable and accurate as long as ν2T < 30 is satisfied. The
following graphs show that:
10

11. • Σ controls the at-the-money volatility ,
• ρ controls the skew ,
• the vol of vol ν controls the curvature of the smile .
Figure 1: Impact of ATM vol on the 5Y smile while keeping ρ and Σ con-
stant: Σ ∈ 0.5%; 0.8%; 1.1%; 1.4%; 1.8%, ρ = −10%, ν = 55%. An ATM
volatility bump generates a perfect parallel shift of the smile.
11

12. Figure 2: Impact of vol of vol on the 5Y smile while keeping ρ and Σ
constant: ν ∈ 2%; 20%; 35%; 55%; 150%, ρ = −10%, Σ = 1.1%. ν controls
the curvature.
Figure 3: Impact of rho on the 5Y smile while keeping ν and Σ constant:
ρ ∈ −80%; −50%; −20%; 0%; 20%, ν = 55%, Σ = 1.1%. ρ controls the skew.
12

13. 5 Inflation options pricing
In this Section, we explain how the NIG distribution can be used to price
year-on-year options.
5.1 Forwards convexity
We denote by It the inflation index available at time t. The year-on-year
rate with reset date T is defined as
IT
IT−1y
− 1 .
Ignoring any payment delay issues, the forward value Yt,T at time t of this
year-on-year paid at time T is given by
Yt,T = ET
t
IT
IT−1y
− 1 (2)
where ET
t denotes the expectation under the T-forward neutral measure
conditional to information up to time t. From (2) it is clear that the forward
has some convexity and is model dependent. Consequently in order to price
forwards and options consistently a term-structure model is necessary.
However in this article our aim is to demonstrate how well the NIG
distribution can fit the market smile. For that reason, we do not attempt to
integrate the NIG distribution to a fully consistent term-structure framework
- for this purpose, we would rather refer the reader to [19]. Instead we choose
to consider the convexity-adjusted forwards as an external input, either given
by the market, or implied by a model.
5.2 Pricing year-on-year options with NIG
We assume that the year-on-year rate YT,T has a NIG distribution with
convexity-adjusted forward Yt,T under the T-forward measure and SABR-
like parameters Σ, βS = 0, ρ, ν. Then an option of maturity T and strike K
on this rate can be priced by first remapping the SABR parameters into NIG
parameters as described in Section 3, and then using the pricing formula (1).
Note that whereas the NIG distribution is defined on ] − ∞, +∞[, a
year-on-year cannot reach values lower than −100%. Although this is an
inconsistency, in pratice the weight of the part of the distribution below
−100% can be considered as negligible.
6 Empirical results
In this Section we present the result of the calibration of NIG to the UK
RPI year-on-year option for various expiries as of Sept, 6th 2011. The
computations are done as described in Section 5.
13

14. 6.1 RPI Index
This Section gives a list of the calibrated parameters (SABR and original
NIG parameterisations) as well as the market vs model smile for all the
quoted expiries available on the market as of Sept, 6th 2011. Graphs corre-
sponding to some of the expiries are also displayed.
6.1.1 Tables: market data, model smile, parameters of the dis-
tributions
Note that all the volatilities which are considered in this document are nor-
mal implied volatilies (as opposed to the usual lognormal volatilities often
used in rates for instance). For each expiry: the strike is shown in the
first column, the model and market normal implied volatilities are shown
in columns 2 and 3 respectively, the calibrated SABR-like parameters in
column 4 and the calibrated original NIG parameters in column 5.
Maturity 2 years
Strike Model vol Market vol SABR params NIG params
-2% 2.19% 2.14% Fwd 2.5% Mu 2.7%
-1% 2.01% 1.96% ATM vol 1.45% Alpha 25.6
0% 1.82% 1.82% Rho -16% Beta -4.0
1% 1.64% 1.67% Vol vol 58% Delta 1.5%
2% 1.49% 1.54%
3% 1.44% 1.40%
4% 1.51% 1.53%
5% 1.64% 1.67%
6% 1.79% 1.80%
Maturity 5 years
Strike Model vol Market vol SABR params NIG params
-2% 2.24% 2.22% Fwd 3.4% Mu 3.3%
-1% 2.04% 2.03% ATM vol 1.2% Alpha 2.0
0% 1.82% 1.81% Rho 10% Beta 0.2
1% 1.61% 1.61% Vol vol 57% Delta 0.8%
2% 1.39% 1.41%
3% 1.23% 1.20%
4% 1.28% 1.35%
5% 1.48% 1.51%
6% 1.72% 1.67%
14

15. Maturity 7 years
Strike Model vol Market vol SABR params NIG params
-2% 2.03% 2.05% Fwd 3.1% Mu 3.4%
-1% 1.85% 1.85% ATM vol 1.2% Alpha 3.2
0% 1.67% 1.67% Rho -23% Beta -0.7
1% 1.49% 1.48% Vol vol 45% Delta 1.0%
2% 1.31% 1.33%
3% 1.18% 1.14%
4% 1.18% 1.23%
5% 1.29% 1.30%
6% 1.44% 1.39%
Maturity 10 years
Strike Model vol Market vol SABR params NIG params
-2% 1.71% 1.75% Fwd 3.3% Mu 4.4%
-1% 1.59% 1.61% ATM vol 1.0% Alpha 14.1
0% 1.47% 1.47% Rho -57% Beta -8.0
1% 1.34% 1.32% Vol vol 30% Delta 1.7%
2% 1.21% 1.21%
3% 1.09% 1.07%
4% 0.98% 1.01%
5% 0.94% 0.95%
6% 0.95% 0.91%
15

16. Maturity 12 years
Strike Model vol Market vol SABR params NIG params
-2% 1.69% 1.73% Fwd 3.6% Mu 4.9%
-1% 1.56% 1.58% ATM vol 0.9% Alpha 15.0
0% 1.43% 1.42% Rho -75% Beta -11.3
1% 1.29% 1.28% Vol vol 30% Delta 1.1%
2% 1.14% 1.14%
3% 0.98% 0.98%
4% 0.83% 0.86%
5% 0.72% 0.82%
6% 0.71% 0.61%
Maturity 20 years
Strike Model vol Market vol SABR params NIG params
-2% 1.34% 1.40% Fwd 3.8% Mu 5.0%
-1% 1.24% 1.26% ATM vol 0.7% Alpha 13.0
0% 1.14% 1.14% Rho -71% Beta -9.3
1% 1.03% 1.04% Vol vol 23% Delta 1.2%
2% 0.92% 0.90%
3% 0.81% 0.79%
4% 0.69% 0.71%
5% 0.60% 0.62%
6% 0.58% 0.55%
Maturity 30 years
Strike Model vol Market vol SABR params NIG params
-2% 1.19% 1.23% Fwd 3.7% Mu 5.2%
-1% 1.09% 1.12% ATM vol 0.5% Alpha 28.3
0% 0.99% 1.00% Rho -92% Beta -26.1
1% 0.89% 0.85% Vol vol 20% Delta 0.6%
2% 0.78% 0.73%
3% 0.66% 0.60%
4% 0.52% 0.50%
5% 0.38% 0.40%
6% 0.32% 0.30%
16

17. 6.1.2 Graphs: 2Y-7Y-10Y-20Y
The market implied volatilies are displayed as green triangles. The blue line
represents the smile implied by the model.
Figure 4: RPI Index: 2Y smile quoted in normal volatilities
Figure 5: RPI Index: 7Y smile quoted in normal volatilities
17

18. Figure 6: RPI Index: 10Y smile quoted in normal volatilities
Figure 7: RPI Index: 20Y smile quoted in normal volatilities
18

19. 6.2 CHF Libor rate
This Section exhibits a few smiles generated with NIG that are representa-
tive of normal volatilities observed at different strikes and expiries for caplets
on CHF Libor rates, the tenor being equal to 3M, as of mid Oct 2011. We
have chosen the CHF currency to illustrate the ability of NIG model to
generate a smile despite the difficult current market conditions, where some
Libor rates are negative.
6.2.1 Graphs: Expiries 1Y-3Y-5Y-10Y
Figure 8: CHF Caplet: 1Y smile quoted in normal volatilities
Figure 9: CHF Caplet: 3Y smile quoted in normal volatilities
19

20. Figure 10: CHF Caplet: 5Y smile quoted in normal volatilities
Figure 11: CHF Caplet: 10Y smile quoted in normal volatilities
20

21. 7 Further applications
7.1 Multi-asset option pricing
Beyond single asset vanilla option pricing, the financial derivatives industry
is also keen in having robust methods for pricing and hedging options which
involve more than one asset. When the payoff is European, it is common
practice to cache each marginal cumulative distribution function (cdf) and
jointly use a gaussian copula (or other) combined with a Monte-Carlo frame-
work. Dealing with distributions where the density is known in closed-form
allows for a fast and efficient way of caching the cdf. In that respect, the
NIG distribution offers a competitive advantage over other ways to build
the cached cumulative distribution function: there is no need to compute
the digital prices by bumping the strike, and no strong dependency on the
interpolation method like in strike-based volatility cubes. Also, note that
the standard SABR approximation typically leads to negative densities in
the wings, and hence to decreasing or even negative cdf.
Regardless of the number of assets involved, the methodology remains
the same: the cdf of each NIG asset is stored as an array containing carefully
chosen points (see Section 7.2) and the cdf values at these points. Computing
risks can be efficiently performed via the SABR parameterisation introduced
in Section 3.
A non exhaustive list of payoffs that can be handled by this methodology
includes CMS-spread options, quanto options, basket options. To some ex-
tent, inflation products such as LPIs may also benefit from this methodology
when year-on-year rates are assumed to be NIG distributed.
7.2 Multi-asset option pricing methodology: a quick overview
We assume here that the payoff to be priced is European and can be ex-
pressed as P (u1, ..., uN ) at expiry T, where u1, ..., uN represent the different
assets/rates involved in the payoff. Each marginal ui is assumed to be Fi-
distributed under the T-forward measure, where Fi denotes the cdf of ui,
and the dependence structure is given by the correlation matrix R.
Monte-Carlo algorithm In a Monte-Carlo framework, the algorithm
commonly used for Gaussian copula is as follows:
• find a decomposition of the correlation matrix R such that R = A·AT
(Cholesky decomposition for instance) ,
• for each simulation j = 1...K, where K is the total number of simula-
tions:
– draw N independent standard gaussian numbers zj
1, ..., zj
N ,
21

22. – compute xj = (zj)T · A ,
– set uj
i = F−1
i Ψ xj
i for all i = 1...N, where Ψ denotes the
normal cdf ,
– compute the payoff value pj = P uj
1, ..., uj
N for this simulation
j ,
• the final price approximation is given by the average of all pj’s.
Caching the cumulative distribution functions Using an efficient
technique to invert the cdf during the MonteCarlo step is crucial as oth-
erwise it may be time-consuming and lead to unaccurate PV and poor risks.
The cdf of each asset is computed and stored as an array containing two
columns: the points xi and the corresponding cdf F(xi). The points are
chosen so that the cdf values are equally spaced. Thus, if we wish to use N
points, and assuming that x1, ..., xi have already been computed, xi+1 can
be set such that
F (xi+1) − F (xi) ∼
1
N
.
A simple Taylor expansion of the cdf at the first order gives
F (xi+1) ∼ F (xi) + f (xi) · (xi+1 − xi) ,
and the fact that the pdf is known in closed-form allows us to compute the
increment dxi to be used in the cache as
xi+1 ∼ xi +
1
N · f (xi)
.
The very first point x0 is the only one that has to be computed numerically,
so that
F (x0) =
1
N
.
Finally we may want to add more points on the left and right-hand sides
of the cached vector. Using this technique, 500 to 1000 points are generally
enough to achieve good accuracy for PV and risks.
Log-NIG marginal Multi-assets payoffs frequently involve underlyings
which cannot go negative - such fx rates or equities - and therefore for
which NIG is not relevant. In that case, it is convenient to assume that the
asset is an exponential of a NIG process. The following graph shows the
smile generated by a Log-NIG distribution .
22

23. Figure 12: A typical example of the smile that can be produced with a Log-
NIG distribution. Here, we use F = 1, Σ = 20%, ν = 50% and ρ = −20%
23

24. A Deriving moments of order 2 and 4 in the SABR
model (’gaussian case’)
In this Section, we assume that βSABR = 0, and denote
˜Ft = Ft − F0 .
A.1 Deriving the variance
Applying Ito-Dublin’s lemma to Ft and taking expectations gives
E ˜F2
t = σ2
0 ·
eν2T − 1
ν2
.
A.2 Deriving the kurtosis
Deriving the kurtosis is a bit tedious but still very straightforward when
βSABR = 0. Applying Ito-Dublin’s lemma to F4
t gives
d ˜F4
t = 6σ2
t
˜F2
t dt + (...)dWt .
To compute E ˜F4
T we then need to derive E σ2
t F2
t . With this in mind,
Ito-Dublin’s lemma applied to (σt
˜Ft)2 gives
d σt
˜Ft
2
= σ4
t dt + ν2
σ2
t
˜F2
t dt + 4ρνσ3
t
˜Ftdt + (...)dWt .
Now if we denote Xt = σ2
t
˜F2
t and γt = σ3
t
˜Ft we rewrite last equation as
dXt = σ4
t dt + ν2
Xtdt + 4ρνγtdt + (...)dWt ,
and thus:
d Xte−ν2t
= σ4
t e−ν2t
dt + 4ρνγte−ν2t
dt + (...)dWt .
We now need to calculate E σ4
t and E [γt]. After similar calculations, we
get:
d γte−3ν2t
= 3ρνe−3ν2t
σ4
t dt + (...)dWt ,
24

25. and
E σ4
t = σ4
0e6ν2t
.
Therefore
E [γt] =
ρ
ν
σ4
0e3ν2t
e3ν2t
− 1 ,
and it follows that
E XT e−ν2T
= σ4
0
T
0
e−ν2t
e6ν2t
dt + 4ρ2
σ4
0
T
0
e−ν2t
e3ν2t
e3ν2t
− 1 )dt
= σ4
0(1 + 4ρ2
) ·
e5ν2T − 1
5ν2
−
2ρ2σ4
0
ν2
· e2ν2T
− 1 ,
as well as
E [Xt] = A · e6ν2t
+ B · e3ν2t
+ C · eν2t
,
where
A =
σ4
0 1 + 4ρ2
5ν2
,
B = −
2ρ2σ4
0
ν2
.
C = −A − B .
Finally the kurtosis is given by
E ˜F4
T
E ˜F2
T
2 = ˜A x4
+ 2x3
+ 3x2
+ 4x + 5 + 2 ˜B(x + 2) ,
where
x = eν2T
,
˜A =
1 + 4ρ2
5
,
˜B = −2ρ2
.
25

26. B Impact of original NIG parameters on the den-
sity shape: a heuristic approach
• Figures 13 and 15 show that α and δ parameters control the scale and
steepness of the distribution.
• β controls the symmetry as seen on figure 14: if β < 0 the distribution
is left-skewed, if β > 0 it is right-skewed.
• Figure 16 empirically shows that the gaussian distribution is obtained
when α, δ → ∞ with δ
α → σ2,
Figure 13: α controls the steepness of the density.
26

27. Figure 14: β controls the symmetry of the density shape (β = 0 makes the
distribution symmetrical).
Figure 15: δ controls the scale of the density.
27

28. Figure 16: When δ
α → σ2, δ, α → ∞, the distribution becomes gaussian
28

29. C Mathematical foundations of NIG processes
In this Section, we introduce Levy processes keeping in mind a ’practitioner’s
point of view’. The organization of the Section is inspired by [23].
The NIG distribution arises as the distribution at a given time t of a NIG
process, which itself belongs to the more general class of Levy processes.
The two main components of a Levy process are the Brownian motion (the
diffusion part) and the Poisson process (the jump part). The diffusion part
has become well-known by practitioners when it has been introduced to
describe the Black-Scholes model. Let us now focus on the description of
the jump part.
C.1 From the Poisson process to a pure-jump Levy process
C.1.1 The Poisson process
Let {τi}i≥1 be a sequence of independent exponential random variables with
parameter λ, which means that p(τi ∈ [t, t + dt]|τi ≥ t) = λdt. Define Tn as
Tn = n
i=1 τi. The process
Nt =
n≥1
1Tn≤t
is called the Poisson process with parameter λ. The variables τi represent
the waiting times between jumps and Nt refers to the number of jumps
which happen before time t.
As each variable τi is exponentially distributed, waiting times have no
memory: the number of jumps which have arrived before time t does not
interfere with what happens after time t. For example if no jump has arrived
before time t then the probability for a jump to arrive between t and t + dt
is still λdt, ] as it was for the period [0, dt]. This property is questionable
in practice since periods of turbulence (high volatility) alternate with more
quiet periods (low volatility). The non-stationarity of increments in real
markets motivates the conception of stochastic volatility models, which is
actually possible even when the underlying is driven by a Levy process
(instead of a simple Brownian motion), see for instance [7].
C.1.2 The compound Poisson process
Note that the Poisson process is a jump process with only a single pos-
sible jump size: 1. For financial applications, this is too restrictive: the
underlying may have jumps, but whose sizes are randomly distributed. The
compound Poisson process is a generalization where the jump sizes have an
29

30. arbitrary distribution. However the waiting times between jumps are still
exponential. More precisely, taking N a Poisson process with parameter λ
and Yi i≥1 be a sequence of independent random variables with law f, the
process
Xt =
Nt
i=1
Yi
is called a compound Poisson process.
C.1.3 Pure jumps Levy processes
Pure jumps Levy processes are a generalization of compound Poisson pro-
cesses in the sense that they can be represented as a ’continuous sum over
x’ (an integral) of Poisson processes whose jumps are in a small interval
[x, x + dx]. In mathematical terms this is the Levy-Ito’s decomposition
which is recalled in the next subsection. A useful characteristic of a Levy
process is its Levy measure defined as
ν(dx) = E N
[x,x+dx]
1
where N1 represents the number of jumps that occur between time 0 and
time 1 and which fall into [x, x + dx], for a given path. In practice, the
underlying, if modeled via a Levy process, will have a lot of jumps of very
small size and much less jumps of bigger size in any finite time interval.
C.2 Mathematical considerations
• the Levy measure ν is defined on R {0} (jumps of size 0 are of little
interest).
• ν does not have to integrate to a finite number: if ν(R {0}) < ∞
one refers to finite activity (the process has a finite number of jumps
in any finite time interval), otherwise if ν(R {0}) = ∞ the process
corresponds to infinite activity.
• however a Levy measure must satisfy this criteria of finiteness
R{0}
min 1, x2
ν(dx) < ∞ ,
30

31. • pure jumps Levy processes (as Levy processes in general) have sta-
tionary, independent increments (which is a limitation of their use to
model a consistent term-structure of smiles). Stochastic volatility can
be incorporated to remedy this issue (see [7]) .
C.3 Levy processes
The class of Levy processes includes all real-valued processes Xt satisfying
the following properties
• X0 = 0 ,
• independent increments: ∀s, t, 0 ≤ s ≤ t Xt−Xs is independent of FX
s ,
• stationarity of increments: ∀s, t, 0 ≤ s ≤ t Xt−Xs and Xs have the same law ,
• continuity in probability: P(|Xt − Xs| > ) → 0 when s → t ∀ > 0 .
Levy-Ito’s theorem states that any Levy process can be decomposed as a
sum of a drifted Brownian motion (continuous part) and a sum of Poisson
processes as follows
Xt = µt + σWt +
|x|<1
x(Nt(dx) − tν(dx)) +
|x|≥1
xNt(dx) .
C.4 NIG processes
NIG processes are a sub-class of Levy processes. A NIG process is defined as
a Brownian motion ut starting at µ, having a constant drift β, stopped at a
random time zt defined as the inverse Gaussian Levy process of parameters
(δ, γ). What is particularly useful for finance applicationa is its ability to
capture the smile at maturity T. The Levy measure of this process is given
by
ν(x)dx =
δα
π
exp βx
|x|
K1(α|x|)dx .
Figure 17 shows the Levy measure of a NIG process with parameters
α = 5.89, β = −3.15, δ = 0.78%.
31

32. Figure 17: Levy measure of a NIG processes: ν(x)dx is the average number
of jumps which fall into [x, x + dx] per unit time. α = 5.89, β = −3.15,
δ = 0.78%, µ = 3.1%
32

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34