This article outlines a few properties of the Normal Inverse Gaussian 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 applications for multi-asset option pricing are considered when the NIG distribution is combined with a copula pricing framework.

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- 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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