Pricing with a smile: an approach using Normal Inverse Gaussian distributions with a SABR-like parameterisation

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 ofsmiles. A parameterisation in terms of SABR inputs is derived. A fewresults 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 NIGdistribution is combined with a copula pricing framework.

Keywords: Normal Inverse Gaussian, NIG, Levy process, SABR,smile, inflation, year-on-year.

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Electronic copy available at: http://ssrn.com/abstract=1968453

Contents

1 Introduction 41.1 Motivation and literature review . . . . . . . . . . . . . . . . 41.2 Paper organization . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Option pricing using the NIG distribution 52.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 A parameterisation of SABR type 73.1 Deriving SABR first four moments when βSABR = 0 . . . . . 73.2 Deriving NIG first four moments . . . . . . . . . . . . . . . . 83.3 Equivalence between original NIG and SABR-like parameter-

isations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.1 Mapping from original NIG to SABR-like parameter-

isation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.2 Mapping from SABR-like to original NIG parameter-

isation . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Switching from σ0 to the ATM volatility in the SABR-like

parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Backbone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Impact of the parameters on the smile 10

5 Inflation options pricing 135.1 Forwards convexity . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Pricing year-on-year options with NIG . . . . . . . . . . . . . 13

6 Empirical results 136.1 RPI Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.1.1 Tables: market data, model smile, parameters of thedistributions . . . . . . . . . . . . . . . . . . . . . . . 14

6.1.2 Graphs: 2Y-7Y-10Y-20Y . . . . . . . . . . . . . . . . 176.2 CHF Libor rate . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2.1 Graphs: Expiries 1Y-3Y-5Y-10Y . . . . . . . . . . . . 19

7 Further applications 217.1 Multi-asset option pricing . . . . . . . . . . . . . . . . . . . . 217.2 Multi-asset option pricing methodology: a quick overview . . 21

A Deriving moments of order 2 and 4 in the SABR model(’gaussian case’) 24A.1 Deriving the variance . . . . . . . . . . . . . . . . . . . . . . . 24A.2 Deriving the kurtosis . . . . . . . . . . . . . . . . . . . . . . . 24

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B Impact of original NIG parameters on the density shape: aheuristic approach 26

C Mathematical foundations of NIG processes 29C.1 From the Poisson process to a pure-jump Levy process . . . . 29

C.1.1 The Poisson process . . . . . . . . . . . . . . . . . . . 29C.1.2 The compound Poisson process . . . . . . . . . . . . . 29C.1.3 Pure jumps Levy processes . . . . . . . . . . . . . . . 30

C.2 Mathematical considerations . . . . . . . . . . . . . . . . . . 30C.3 Levy processes . . . . . . . . . . . . . . . . . . . . . . . . . . 31C.4 NIG processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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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 themarket 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 optionpricing and the intuitiveness about the effect of parameters on the smile. Noapproximation of the implied volatility is needed since the density is knownin 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, canproduce a significant smile for short-dated expiries and performs well in anegative rates environment.

Inflation options smile Investors who receive inflation-linked cash flowstend to have appetite for products which give them a protection in case ofa deflation. As a consequence low-strike floors are by far the most popularyear-on-year options, which results in a pronounced normal volatility skew.This super-normal type of distribution is notoriously difficult to reproduceusing any of the dynamics commonly used in the finance industry.

Institutions tend to manage their entire inflation book using a singleterm-structure model. The first model to be widely used was the applicationof the foreign-currency introduced by [13]. In this approach, both nominaland real short rates are represented by one-factor Vasicek processes while thespot inflation index is lognormal. As the inflation forwards are lognormalthe year-on-year normal smile is almost flat. In the market model introducedby [14], [4] and [17], the forward inflation rate is still lognormal, thus theyear-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 tohave a better fit of the market, their conclusion was that the model was notflexible enough to fit the smiles for all the liquidly traded expiries.

[19] then presented a model where the year-on-year forwards have SABRdynamics, which is more likely to achieve a better fit of the year-on-yearmarket smile than the others. However SABR has a number of drawbacks,and other types of distributions have been proposed as an alternative, suchas [1].

Normal Inverse Gaussian distribution To address these issues we lookfor distributions flexible enough to fit extreme shapes of smiles, and ana-lytical enough to enable fast pricing and risk management. Generalized

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hyperbolic processes typically give good flexibility to capture various shapesof smiles due to their semi-heavy tails. Within this class, a number of pro-cesses appear to be tractable, amongst which the Normal Inverse Gaussiandistribution (NIG). The application of this distribution to finance and tooption pricing has been discussed in depth by [20], [22], [15] or [12], howevermostly 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 andshow that it has a remarkable ability to fit the steep skews seen on inflationand to cope with turbulent market conditions on nominal interest rates.We also provide a SABR-like parameterisation, give a quick overview of ourquadrature technique for pricing vanilla options and show how multi-assetEuropean option pricing may benefit from modeling marginals as (Log-)NIGdistributions.

1.2 Paper organization

This article is organized as follows. We give a definition of the NormalInverse Gaussian distribution in Section 2. A remapping of the parameter-isation into SABR-like user inputs is introduced in Section 3. A heuristicapproach to understand the effect of SABR-like parameters on the smile isgiven in Section 4. Section 5 explains how the NIG distribution can be usedto price inflation options. Numerical results are exposed in Section 6 inwhich are displayed concrete results of calibrated smile on UK RPI inflationoptions and CHF interest rates options. Beyond this application to singleasset option pricing, various types of European payoffs may benefit frombeing modelized using (Log-)NIG distributed marginals. Typical examplesinclude CMS-spread options, basket options and quantos payoffs. In theinflation world, there is also a growing need for pricing the Limited PriceIndex (LPI) in line with the year-on-year option market. Section 7 gives aquick overview of a generic methodology that can be applied to price andrisk-manage these structures.

2 Option pricing using the NIG distribution

2.1 Definition

The Normal Inverse Gaussian distribution is characterised by a set of fourparameters (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 motionstopped 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

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it can be expressed as

X|Z ∼ N (µ+ βZ,Z) ,

with N (µ, σ) the Gaussian distribution with mean µ, standard deviation σand Z defined as follows

Z ∼ IG(δ,√α2 − β2

)where 0 ≤ |β| ≤ α ,

where IG(µ, λ) denotes the inverse Gaussian distribution with parametersµ and λ.

A feature that will be of high importance in our applications is thatmoments are all of finite order and that the density is available in closedform, 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 isgiven by the Levy measure, known in closed-form. Usually, the user willcalibrate the process parameters to the vanilla option prices. Alternatively,calibrating the Levy measure to the historical returns is possible and in somecases 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 toomuch into details. Assuming that the the asset or rate YT is NIG distributedunder the T -forward measure, the undiscounted price of a call struck at Kcan 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-Legendrequadrature is not robust enough to achieve an accurate pricing. Instead,splitting the integration interval into different sub-intervals whose size arecontrolled by statistical characteristics of the distribution leads to more sat-isfactory results. This method generates a smooth and accurate smile, as

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long as vol of vols ν and expiries T are not “too” high - in practice, thethreshold 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 Gaussiandistribution, which corresponds to NIG parameters α and δ that go to infin-ity. In this case, the Bessel function K1 takes a significantly high parameteras input and its usual approximation computed as in [21] may lead to nu-merical instabilities. To handle this case it is worth using an approximationof 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 userthe ability to control the implied smile in an easy way, which is lacking inthe 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 byboth distributions match. In this article we work with the assumptionβSABR = 0. This allows the forward to take negative values, and thuscorresponds to the behaviour of the year-on-year rates and even interestrates in some cases. Since the distributions of the NIG and SABR withβSABR = 0 are each characterized by four parameters, we seek to matchtheir 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 givenby the following SDE:

dFt = σtFβSABRt dW 1

t ,

dσt = νσtdW2t ,

where W 1t and W 2

t are two Brownian motions correlated by a factor ρ,βSABR is a CEV parameter, and ν is the volatility of volatility. We considera 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 ·eν

2T − 1ν2

,

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• 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

)+ 2B · (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 theappendix A.

3.2 Deriving NIG first four moments

If X ∼ NIG(α;β; δ;µ), then

• the forward is equal to µ+ δ · βγ ,

• the variance is given by δ · α2

γ3 ,

• the skewness is directly controlled by βα ∈ [−1, 1] ,

• the total kurtosis is given by 3 + 3(

1 + 4(βα

)2)· 1δγ .

3.3 Equivalence between original NIG and SABR-like pa-rameterisations

With the moments controlled by closed form formulae, it is possible to gofrom 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 givenNIG distribution. We want to find an equivalent set of parameters of SABRtype (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. Inthe following, γ =

√α2 − β2.

1Apart from the skewness, in which case we decide to match ρ with βα

directly.

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• since the mean of the (α, β, δ, µ) NIG distribution is given by E(X) =µ+ δ · βγ , we define F0 as F0 = µ+ δ · βγ ,

• avoiding tedious calculation of third moments, it is natural, since βcontrols the symmetry of the NIG distribution, and |β| ≤ α, and ρsimilarly 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 + 2Ax3 + 3Ax2 +(

4A+ 2B)x+ 5A+ 4B = 3 + 3

1 + 4(βα

)2

δγ,

• finally the overall level σ0 by matching their variances:

σ20 ·eν

2T − 1ν2

= 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-likeset 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 βα 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) isknown .

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 thevariance of the distribution. When the vol of vol remains small, σ0 and theat-the-money (ATM) volatility are close, but this is no longer the case whenthe vol of vol reaches high values. In fact, as ν tends to infinity, a NIGdistribution tends to a Cauchy distribution whose moments are infinite: inparticular, σ0 goes to infinity while ATM volatility has obviously to remain

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constant. Note that in SABR, the ATM volatility also diverges from theinitial local volatility as vol of vol increases. This problem is usually tackledby having the ATM volatility parameter as an input rather than the initiallocal volatility. In that case, using a standard numerical solver it is easy toobtain the initial local volatility from the ATM volatility given that there isa one-to-one relationship between the two, We adopt this approach and fromnow on use the SABR-type parameterisation (F,Σ, ν, ρ), where Σ denotesthe ATM volatility.

3.5 Backbone

Given that the ATM volatility is directly an input, it is possible to havea control over the backbone. Assume for instance that we want to have abackbone corresponding to normal dynamics. We must then have

dV NIGATM

dF=∂V N

ATM

∂F,

where V NIGATM and V N

ATM denote the ATM option prices using respectivelyNIG and normal pricing formulae. This leads to

∂V NIGATM

∂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 isshown in a few graphs in the appendix B. In this Section, we present somegraphs which should help in getting an intuition and show that the impactof the SABR parameters on the smile is as expected. We use the folowingset 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, andthe vol of vol ν by keeping the other parameters constant. The quadratureused appears to be stable and accurate as long as ν2T < 30 is satisfied. Thefollowing graphs show that:

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• 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 ATMvolatility bump generates a perfect parallel shift of the smile.

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Figure 2: Impact of vol of vol on the 5Y smile while keeping ρ and Σconstant: ν ∈ 2%; 20%; 35%; 55%; 150%, ρ = −10%, Σ = 1.1%. ν controlsthe 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.

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5 Inflation options pricing

In this Section, we explain how the NIG distribution can be used to priceyear-on-year options.

5.1 Forwards convexity

We denote by It the inflation index available at time t. The year-on-yearrate with reset date T is defined as

ITIT−1y

− 1 .

Ignoring any payment delay issues, the forward value Yt,T at time t of thisyear-on-year paid at time T is given by

Yt,T = ETt[

ITIT−1y

]− 1 (2)

where ETt denotes the expectation under the T -forward neutral measureconditional to information up to time t. From (2) it is clear that the forwardhas some convexity and is model dependent. Consequently in order to priceforwards and options consistently a term-structure model is necessary.

However in this article our aim is to demonstrate how well the NIGdistribution can fit the market smile. For that reason, we do not attempt tointegrate the NIG distribution to a fully consistent term-structure framework- for this purpose, we would rather refer the reader to [19]. Instead we chooseto consider the convexity-adjusted forwards as an external input, either givenby 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 withconvexity-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 Kon this rate can be priced by first remapping the SABR parameters into NIGparameters as described in Section 3, and then using the pricing formula (1).

Note that whereas the NIG distribution is defined on ] − ∞,+∞[, ayear-on-year cannot reach values lower than −100%. Although this is aninconsistency, 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 UKRPI year-on-year option for various expiries as of Sept, 6th 2011. Thecomputations are done as described in Section 5.

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6.1 RPI Index

This Section gives a list of the calibrated parameters (SABR and originalNIG parameterisations) as well as the market vs model smile for all thequoted 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 oftenused in rates for instance). For each expiry: the strike is shown in thefirst column, the model and market normal implied volatilities are shownin columns 2 and 3 respectively, the calibrated SABR-like parameters incolumn 4 and the calibrated original NIG parameters in column 5.

Maturity 2 yearsStrike 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 25.60% 1.82% 1.82% Rho -16% Beta -4.01% 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%


-2% 2.24% 2.22% Fwd 3.4% Mu 3.3%-1% 2.04% 2.03% ATM vol 1.2% Alpha 2.00% 1.82% 1.81% Rho 10% Beta 0.21% 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%

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15







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6.1.2 Graphs: 2Y-7Y-10Y-20Y

The market implied volatilies are displayed as green triangles. The blue linerepresents the smile implied by the model.

Figure 4: RPI Index: 2Y smile quoted in normal volatilities


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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 capletson CHF Libor rates, the tenor being equal to 3M, as of mid Oct 2011. Wehave chosen the CHF currency to illustrate the ability of NIG model togenerate a smile despite the difficult current market conditions, where someLibor rates are negative.

6.2.1 Graphs: Expiries 1Y-3Y-5Y-10Y

Figure 8: CHF Caplet: 1Y smile quoted in normal volatilities


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7 Further applications

7.1 Multi-asset option pricing

Beyond single asset vanilla option pricing, the financial derivatives industryis also keen in having robust methods for pricing and hedging options whichinvolve more than one asset. When the payoff is European, it is commonpractice to cache each marginal cumulative distribution function (cdf) andjointly use a gaussian copula (or other) combined with a Monte-Carlo frame-work. Dealing with distributions where the density is known in closed-formallows for a fast and efficient way of caching the cdf. In that respect, theNIG distribution offers a competitive advantage over other ways to buildthe cached cumulative distribution function: there is no need to computethe digital prices by bumping the strike, and no strong dependency on theinterpolation method like in strike-based volatility cubes. Also, note thatthe standard SABR approximation typically leads to negative densities inthe wings, and hence to decreasing or even negative cdf.

Regardless of the number of assets involved, the methodology remainsthe same: the cdf of each NIG asset is stored as an array containing carefullychosen points (see Section 7.2) and the cdf values at these points. Computingrisks can be efficiently performed via the SABR parameterisation introducedin Section 3.

A non exhaustive list of payoffs that can be handled by this methodologyincludes CMS-spread options, quanto options, basket options. To some ex-tent, inflation products such as LPIs may also benefit from this methodologywhen 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 differentassets/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 algorithmcommonly 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(zj1, ..., z

jN

),

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– compute xj = (zj)T ·A ,

– set uji = F−1i

(Ψ(xji

))for all i = 1...N , where Ψ denotes the

normal cdf ,

– compute the payoff value pj = P(uj1, ..., u

jN

)for this simulation

j ,

• the final price approximation is given by the average of all pj ’s.

Caching the cumulative distribution functions Using an efficienttechnique 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 twocolumns: the points xi and the corresponding cdf F (xi). The points arechosen so that the cdf values are equally spaced. Thus, if we wish to use Npoints, and assuming that x1, ..., xi have already been computed, xi+1 canbe set such that

F (xi+1)− F (xi) ∼1N.

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 theincrement 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) =1N.

Finally we may want to add more points on the left and right-hand sidesof the cached vector. Using this technique, 500 to 1000 points are generallyenough to achieve good accuracy for PV and risks.

Log-NIG marginal Multi-assets payoffs frequently involve underlyingswhich cannot go negative - such fx rates or equities - and therefore forwhich NIG is not relevant. In that case, it is convenient to assume that theasset is an exponential of a NIG process. The following graph shows thesmile generated by a Log-NIG distribution .

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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%

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A Deriving moments of order 2 and 4 in the SABRmodel (’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[F 2t

]= σ2

0 ·eν

2T − 1ν2

.

A.2 Deriving the kurtosis

Deriving the kurtosis is a bit tedious but still very straightforward whenβSABR = 0. Applying Ito-Dublin’s lemma to F 4

t gives

dF 4t = 6σ2

t F2t dt+ (...)dWt .

To compute E[F 4T

]we then need to derive E

[σ2tF

2t

]. With this in mind,

Ito-Dublin’s lemma applied to (σtFt)2 gives

d(σtFt

)2= σ4

t dt+ ν2σ2t F

2t dt+ 4ρνσ3

t Ftdt+ (...)dWt .

Now if we denote Xt = σ2t F

2t and γt = σ3

t Ft we rewrite last equation as

dXt = σ4t dt+ ν2Xtdt+ 4ρνγtdt+ (...)dWt ,

and thus:

d(Xte

−ν2t)

= σ4t e−ν2tdt+ 4ρνγte−ν

2tdt+ (...)dWt .

We now need to calculate E[σ4t

]and E [γt]. After similar calculations, we

get:

d(γte−3ν2t

)= 3ρνe−3ν2tσ4

t dt+ (...)dWt ,

24

and

E[σ4t

]= σ4

0e6ν2t .

Therefore

E [γt] =ρ

νσ4

0e3ν2t

(e3ν

2t − 1),

and it follows that

E[XT e

−ν2T]

= σ40

∫ T

0e−ν

2te6ν2tdt+ 4ρ2σ4

0

∫ T

0e−ν

2te3ν2t(e3ν

2t − 1)

)dt

= σ40(1 + 4ρ2) · e

5ν2T − 15ν2

− 2ρ2σ40

ν2·(e2ν

2T − 1),

as well as

E [Xt] = A · e6ν2t +B · e3ν2t + C · eν2t ,

where

A =σ4

0

(1 + 4ρ2

)5ν2

,

B = −2ρ2σ40

ν2.

C = −A−B .

Finally the kurtosis is given by

E[F 4T

]E[F 2T

]2 = A(x4 + 2x3 + 3x2 + 4x+ 5

)+ 2B(x+ 2) ,

where

x = eν2T ,

A =1 + 4ρ2

5,

B = −2ρ2 .

25

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 andsteepness of the distribution.

• β controls the symmetry as seen on figure 14: if β < 0 the distributionis left-skewed, if β > 0 it is right-skewed.

• Figure 16 empirically shows that the gaussian distribution is obtainedwhen α, δ →∞ with δ

α → σ2,

Figure 13: α controls the steepness of the density.

26

Figure 14: β controls the symmetry of the density shape (β = 0 makes thedistribution symmetrical).

Figure 15: δ controls the scale of the density.

27

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

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C Mathematical foundations of NIG processes

In this Section, we introduce Levy processes keeping in mind a ’practitioner’spoint 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 NIGprocess, which itself belongs to the more general class of Levy processes.The two main components of a Levy process are the Brownian motion (thediffusion part) and the Poisson process (the jump part). The diffusion parthas become well-known by practitioners when it has been introduced todescribe the Black-Scholes model. Let us now focus on the description ofthe 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 withparameter λ, which means that p(τi ∈ [t, t+ dt]|τi ≥ t) = λdt. Define Tn asTn =

∑ni=1 τi. The process

Nt =∑n≥1

1Tn≤t

is called the Poisson process with parameter λ. The variables τi representthe waiting times between jumps and Nt refers to the number of jumpswhich happen before time t.

As each variable τi is exponentially distributed, waiting times have nomemory: the number of jumps which have arrived before time t does notinterfere with what happens after time t. For example if no jump has arrivedbefore time t then the probability for a jump to arrive between t and t+ dtis still λdt, ] as it was for the period [0, dt]. This property is questionablein practice since periods of turbulence (high volatility) alternate with morequiet periods (low volatility). The non-stationarity of increments in realmarkets motivates the conception of stochastic volatility models, which isactually 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: theunderlying may have jumps, but whose sizes are randomly distributed. Thecompound Poisson process is a generalization where the jump sizes have an

29

arbitrary distribution. However the waiting times between jumps are stillexponential. More precisely, taking N a Poisson process with parameter λand Yi i≥1 be a sequence of independent random variables with law f , theprocess

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 overx’ (an integral) of Poisson processes whose jumps are in a small interval[x, x + dx]. In mathematical terms this is the Levy-Ito’s decompositionwhich is recalled in the next subsection. A useful characteristic of a Levyprocess 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 andtime 1 and which fall into [x, x + dx], for a given path. In practice, theunderlying, if modeled via a Levy process, will have a lot of jumps of verysmall 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 littleinterest).

• 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 jumpsin any finite time interval), otherwise if ν(R \ {0}) = ∞ the processcorresponds to infinite activity.

• however a Levy measure must satisfy this criteria of finiteness

∫R\{0}

min(1, x2

)ν(dx) <∞ ,

30

• pure jumps Levy processes (as Levy processes in general) have sta-tionary, independent increments (which is a limitation of their use tomodel a consistent term-structure of smiles). Stochastic volatility canbe incorporated to remedy this issue (see [7]) .

C.3 Levy processes

The class of Levy processes includes all real-valued processes Xt satisfyingthe following properties

• X0 = 0 ,

• independent increments: ∀s, t, 0 ≤ s ≤ t Xt−Xs is independent of FXs ,

• stationarity of increments: ∀s, t, 0 ≤ s ≤ t Xt−Xs and Xs have the same law ,

• continuity in probability: P (|Xt −Xs| > ε)→ 0 when s→ t ∀ε > 0 .

Levy-Ito’s theorem states that any Levy process can be decomposed as asum of a drifted Brownian motion (continuous part) and a sum of Poissonprocesses 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 asa Brownian motion ut starting at µ, having a constant drift β, stopped at arandom time zt defined as the inverse Gaussian Levy process of parameters(δ, γ). What is particularly useful for finance applicationa is its ability tocapture the smile at maturity T . The Levy measure of this process is givenby

ν(x)dx =δα

π

expβx|x|

K1(α|x|)dx .

Figure 17 shows the Levy measure of a NIG process with parametersα = 5.89, β = −3.15, δ = 0.78%.

31

Figure 17: Levy measure of a NIG processes: ν(x)dx is the average numberof 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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