JOINT PHD THESIS

Universite Libre de BruxellesFaculty of Sciences, Department of Mathematics

Vrije Universiteit BrusselFaculty of Economic, Political and Social Sciences and Solvay

Business School

Essays on Pricing Derivatives bytaking into account volatility and

interest rates risks

Gregory Rayee

A thesis submitted in partial fulfillment of the requirements for the degreeof

Docteur en Sciences(Universite Libre de Bruxelles)

and

Doctor in de Toegepaste Economische Wetenschappen: Handelsingenieur(Vrije Universiteit Brussel)

Supervisors:Prof. dr. Griselda Deelstra (Universite Libre de Bruxelles)

Prof. dr. Steven Vanduffel (Vrije Universiteit Brussel)

Academic year 2011-2012

Jury:

Prof. dr. Griselda Deelstra, Universite Libre de Bruxelles

Prof. dr. Pierre Devolder, Universite Catholique de Louvain

Prof. dr. Siegfried Hormann, Universite Libre de Bruxelles

Prof. dr. Wim Schoutens, Katholieke Universiteit Leuven

Prof. dr. Paul Van Goethem, Vrije Universiteit Brussel

Prof. dr. Steven Vanduffel, Vrije Universiteit Brussel

Acknowledgments:

Pursuing my PhD thesis has been a major undertaking. The last fiveyears have been a long and hard process, but today I feel very lucky to havelived such an interesting experience. Also I realize that I have been fortunateto receive help, advice and support from so many people. I wish to thank allthese people involved.

There are four people who stand out in terms of contributions to thisthesis. I am profoundly thankful to my supervisor Griselda Deelstra for giv-ing me the opportunity to realize this project and for all the support andvaluable advice. It was a real pleasure to realize this thesis under your su-pervision. I would also like to show my gratitude to my co-supervisor StevenVanduffel for the cotutelle opportunity and for all the research ideas, inparticular with respect to the Chapter 4 of this thesis. I would also like tothank Frederic Bossens and Nikos Skantzos for all the research ideas andpractical advice they gave me during the first year of this thesis. Thanks forgiving me the opportunity to discover the world of Quantitative Analystsand for all the support, help and work shared for the realization of the Chap-ter 1. This thesis would not have been possible without all of them.

I would also like to thank my colleagues. It was a pleasure to work withyou during the completion of this thesis. I would particularly like to thankMarc Levy and Xavier De Scheemaekere for all the valuable discussions wehad.

Finally, I would like to thank Jess, all my friends, in particular Bernard,my parents and my brother Terry for their patient, love, encouragement,support and for all the good times we have lived together.

Gregory RayeeMay 2012

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Contents

Summary 5

French Summary 8

Introduction 11

1 Vanna-Volga Methods Applied to FX Derivatives: From The-ory to Market Practice 261.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2 Description of first-generation exotics . . . . . . . . . . . . . . 291.3 Handling Market Data . . . . . . . . . . . . . . . . . . . . . . 30

1.3.1 Delta conventions . . . . . . . . . . . . . . . . . . . . . 311.3.2 At-The-Money Conventions . . . . . . . . . . . . . . . 321.3.3 Smile-related quotes and the brokers Strangle . . . . . 32

1.4 The Vanna-Volga Method . . . . . . . . . . . . . . . . . . . . 371.4.1 The general framework . . . . . . . . . . . . . . . . . . 381.4.2 Vanna-Volga as a smile-interpolation method . . . . . . 40

1.5 Market-adapted variations of Vanna-Volga . . . . . . . . . . . 431.5.1 Survival probability . . . . . . . . . . . . . . . . . . . . 441.5.2 First exit time . . . . . . . . . . . . . . . . . . . . . . . 451.5.3 Qualitative differences between surv and fet . . . . . . 461.5.4 Arbitrage tests . . . . . . . . . . . . . . . . . . . . . . 481.5.5 Sensitivity to market data . . . . . . . . . . . . . . . . 49

1.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 511.6.1 Definition of the model error . . . . . . . . . . . . . . . 521.6.2 Shortcomings of common stochastic models in pricing

exotic options . . . . . . . . . . . . . . . . . . . . . . . 531.6.3 Vanna-Volga calibration . . . . . . . . . . . . . . . . . 55

1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A : Definitions of notation used . . . . . . . . . . . . . . . . . 60

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B : Premium-included Delta . . . . . . . . . . . . . . . . . . . 60

2 Local Volatility Pricing Models for Long-dated FX Deriva-tives 632.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 The three-factor pricing model with local volatility . . . . . . 682.3 The local volatility function . . . . . . . . . . . . . . . . . . . 69

2.3.1 Forward PDE . . . . . . . . . . . . . . . . . . . . . . . 692.3.2 The local volatility derivation . . . . . . . . . . . . . . 71

2.4 Calibrating the Local Volatility . . . . . . . . . . . . . . . . . 742.4.1 Numerical approaches . . . . . . . . . . . . . . . . . . 752.4.2 Comparison between local volatility with and without

stochastic interest rates . . . . . . . . . . . . . . . . . 772.4.3 Calibrating the local volatility by mimicking sto-chastic

volatility models . . . . . . . . . . . . . . . . . . . . . 792.5 Hybrid volatility model . . . . . . . . . . . . . . . . . . . . . 84

2.5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . 882.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Pricing Variable Annuity Guarantees in a Local Volatilityframework 913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 The local volatility model with stochastic interest rates . . . . 943.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.3.1 The local volatility function . . . . . . . . . . . . . . . 953.3.2 The Monte-Carlo approach . . . . . . . . . . . . . . . 973.3.3 Comparison between local volatility with and without

stochastic interest rates . . . . . . . . . . . . . . . . . 1003.3.4 Calibrating the local volatility by mimicking stochastic

volatility models . . . . . . . . . . . . . . . . . . . . . 1003.4 Variable Annuity Guarantees . . . . . . . . . . . . . . . . . . 102

3.4.1 Guaranteed Annuity Options . . . . . . . . . . . . . . 1023.4.2 Guaranteed Minimum Income Benefit (Rider) . . . . . 1053.4.3 Barrier GAOs . . . . . . . . . . . . . . . . . . . . . . 106

3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 1083.5.1 Calibration to the Vanilla options Market . . . . . . . 1093.5.2 A numerical comparison between local volatility with

and without stochastic interest rates . . . . . . . . . . 1113.5.3 GAO results . . . . . . . . . . . . . . . . . . . . . . . 1123.5.4 GMIB Rider . . . . . . . . . . . . . . . . . . . . . . . 1163.5.5 Barrier GAOs . . . . . . . . . . . . . . . . . . . . . . 118

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3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A : Explicit formula for the GAO price in the BSHW andSZHW models . . . . . . . . . . . . . . . . . . . . . . . 124

B : Down-and-in GAO results . . . . . . . . . . . . . . . . . . 127C : Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4 Using bounds for a faster pricing of Asian style options 1354.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.2 Bounds as control variates in Levy markets . . . . . . . . . . 137

4.2.1 Geometric Lower Bound GLB . . . . . . . . . . . . . . 1424.3 Market settings: subordinated Brownian motion . . . . . . . 144

4.3.1 The Variance Gamma model . . . . . . . . . . . . . . . 1454.3.2 The Normal Inverse Gaussian model . . . . . . . . . . 146

4.4 ALB and GLB as an expression of power calls . . . . . . . . . 1484.4.1 Calculation of power call options in the Variance Gamma

model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.4.2 Calculation of power call options in the Normal Inverse

Gaussian model . . . . . . . . . . . . . . . . . . . . . . 1514.5 Lower bound derived using the stochastic clock . . . . . . . . 1524.6 Numerical illustrations . . . . . . . . . . . . . . . . . . . . . . 155

4.6.1 Efficiency Measure . . . . . . . . . . . . . . . . . . . . 1554.6.2 Numerical results for Asian options . . . . . . . . . . . 156

4.7 Applications to other products . . . . . . . . . . . . . . . . . 1604.7.1 Unit Linked Insurance . . . . . . . . . . . . . . . . . . 1604.7.2 Ratchet equity-indexed annuities (EIAs) . . . . . . . . 162

4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Conclusions 165

4

Summary

Over the past decade, the stock and derivative markets have often made theJournal Headlines. They are very attractive, thanks to the returns they canprovide, but they are also very dangerous since they are able to cause conse-quent losses of money and are responsible of many bankruptcies, especiallyin the banking sector. Among the most popular derivative products, we findthe European call and put options, created initially as hedging tools for fi-nancial risks, but quickly converted into products of speculation for investorsinterested by high profits. A call option (resp. put) is a financial contractbetween a buyer and a seller that gives to the buyer the right to buy (resp.sell) an underlying asset at a fixed price (the strike price) and a fixed time inthe future (the maturity date). This derivative product depends on a randomvariable, namely the price of the underlying asset at the options expirationdate, and consequently, the pricing problem of such product is complicated.To calculate this price it is essential to use dynamics for the underlying finan-cial asset which are as close as possible to the real market dynamics. Manyresearchers have attempted to solve this option pricing problem. Discrete-time methods have been developed such as the binomial method introducedby Cox et al. (1979) which consists in modelling the dynamics of the under-lying via a binomial tree. Today, the most popular methods are based oncontinuous-time dynamics and consist in modelling the dynamics of the assetwith stochastic differential equations. The most famous model is probablythe Black and Scholes (1973) model in which the underlying asset is mod-elled by a geometric Brownian motion. Unfortunately this model is based onseveral assumptions which are not consistent with the reality and can lead toserious pricing and hedging errors. Therefore, its use by banks and other fi-nancial institutions is becoming rare. One of the most important problems ofthe Black and Scholes model is the assumption of a constant volatility duringthe lifetime of the option. Twenty years later, two famous types of modelsprovide a solution to this problem. The first type are stochastic volatilitymodels such as the Heston model (see Heston (1993)) and the second typeare the local volatility models introduced by Dupire (1994) and Derman and

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Kani (1994). Currently, these are probably the most popular models in themarket and are all consistent with the implied volatility of the Europeanoptions market.

There are also many types of so-called exotic options, among which wehave the popular barrier type options that can be activated or deactivated ifthe underlying asset reaches a certain fixed barrier level during the lifetimeof the option. Generally, local and stochastic volatility models behave differ-ently in the pricing of exotic options. They often do not give the same results,nor agree with the market. In Chapter 1, we present a new approach to eval-uate this type of exotic options based on a method known as the Vanna-Volgamethod. This new approach allows for a fast and easy calibration which isdirectly done on the barrier options market. It allows to price these optionswith a tool in accordance with the barrier options market. We also compareour results with those coming from the Dupire and Heston models. Further-more, we study the sensitivity of the Vanna-Volga method with respect tothe market data. We give a new theoretical justification for the Vanna-Volgamethod. More precisely, we show that the Vanna-Volga options price canbe seen as a first-order Taylor expansion of the Black-Scholes option pricearound the at-the-money volatility.

In Chapter 2, we study a model able to capture the market implied volatil-ity effects and which also takes into account the market variability of theinterest rates. This relaxes the assumption of constant interest rates presentin the Black-Scholes model and solves a second main problem encounteredin the latter, which can have large consequences in the valuation and hedg-ing strategies especially for long maturity products. More precisely, we workin the foreign exchange (FX) market, with a local volatility model for thedynamics of the foreign exchange spot rates in which the domestic and for-eign interest rates are also assumed stochastic. We derive the expression ofthe local volatility and various results particularly useful for the calibrationof the model. Finally, we derive useful results for the calibration of hybridvolatility models where the volatility of the FX spot rate is a mix of a localvolatility and a stochastic volatility and we develop a calibration method forthis model.

In Chapter 3, we apply the local volatility model with stochastic inter-est rates developed in the previous chapter to the pricing of life insurancederivatives. Since the maturity of such options is the retirement age, theycan be considered as long maturity products. For the calibration of the localvolatility, we use a method developed in Chapter 2. Since we study exotic

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products, we also compare the prices obtained in different models, namelythe local volatility, stochastic volatility and finally the constant volatilitymodel all combined with stochastic interest rates.

Finally, in Chapter 4 we work with Levy type models for the underlyingdynamics. The idea underlying the Levy model is the use of a more generalstochastic process than the standard Brownian motion which allows to bein agreement with the observed market probability distribution at maturity.In a financial crisis period, this model is especially popular since it has theparticularity to allow for jumps in the dynamics. In this chapter, we areinterested specifically in the evaluation of discretely monitored arithmeticAsian type options whose payoff is based on the discrete arithmetic mean ofthe underlying during the life of the option. As for many exotic options, it isnot possible to derive an analytical pricing formula even in the simple caseof the Black-Scholes model. In this case the only way to price such optionsis by using numerical methods. In Chapter 4, we develop a method based onMonte-Carlo simulations and we use two types of control variates to improvethe convergence. We also develop a method based on a conditioning approachto obtain a lower bound for the Asian option price. The efficiency of this lastmethod outperforms the efficiency of the other methods and the results arerelatively close to the Monte-Carlo value of the corresponding Asian.

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Resume

Ces dix dernieres annees, les marches dactions et de produits derives ontfait plus dune fois lobjet des grands titres des journaux. Tres attractifsgrace aux rendements quils peuvent procurer, ils sont egalement tres dan-gereux, pouvant engendrer denormes pertes dargent et etre responsablesde nombreuses faillites, tout particulierement dans le secteur des banques.Parmi les produits derives les plus populaires, on retrouve les options dachatet de vente Europeennes, creees dans un premier temps comme objets fi-nanciers de couverture de risques, mais reconverties rapidement en produitsde speculations pour les investisseurs a la recherche de profits eleves. Uneoption dachat (resp. de vente) est un contrat financier conclu entre unacheteur et un vendeur qui donne a lacheteur le droit dachat (resp. devente) dun actif sous-jacent a un prix dexercice fixe et a une date precisedans le futur. Ce produit derive depend dune valeur aleatoire, le prix delactif sous-jacent a la date de maturite de loption, rendant par consequentle prix du contrat complique a determiner. Pour calculer ce prix, il estessentiel de modeliser de maniere la plus reelle possible la dynamique delactif financier sous-jacent. De nombreux chercheurs ont tente de resoudrece probleme devaluation doptions. Des methodes en temps discret on etedeveloppees comme par exemple la methode binomiale introduite par Coxet al. (1979) qui consiste a modeliser la dynamique du sous-jacent via unarbre binomial, mais les methodes les plus populaires aujourdhui utilisentdes dynamiques en temps continu et consistent a modeliser les dynamiquesde lactif avec des equations differentielles stochastiques. Le modele le pluscelebre est probablement le modele de Black and Scholes (1973) dans lequelle sous-jacent est modelise par un mouvement Brownien geometrique. Cemodele se base malheureusement sur certaines hypotheses trop eloignees dela realite, pouvant engendrer de graves erreurs devaluations et de couver-ture et par consequent, son utilisation par les banques et autres organismesfinanciers se fait de plus en plus rare. Une hypothese rejetee par le marcheest de considerer la volatilite comme constante durant la vie de loption.Vingt ans plus tard, deux celebres types de modeles apportent une solution

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a ce probleme. Le premier est le modele a volatilite stochastique comme parexemple le modele de Heston (1993) et le second est le modele a volatilitelocale connu sous le nom de modele de Dupire (1994) ou Derman and Kani(1994). Actuellement, ce sont probablement les modeles les plus utilises surles marches.

Il existe egalement de nombreux types doptions dites exotiques, parmilesquelles on retrouve les populaires options a barrieres qui peuvent etre ac-tivees ou desactivees si lactif sous-jacent atteint un certain niveau de barrierefixe durant la vie de loption. Dans ces marches exotiques, les modeles avolatilite locale et stochastique vont avoir des comportements differents etseront souvent en desaccord entre eux et meme avec le marche. Dans leChapitre 1, nous presentons une nouvelle approche pour evaluer ce typedoptions exotiques basee sur une methode connue sous le nom de methodeVanna-Volga. Cette nouvelle methode nous permet une calibration simpleet rapide sur le marche des options a barrieres directement ce qui permetdevaluer ces options avec un outil en accord avec le marche. Nous com-parons egalement nos resultats avec ceux provenant du modele de Dupire etde Heston et nous etudions la sensibilite de cette methode par rapport auxdonnees du marche. Nous donnons une nouvelle justification theorique as-sociee a la methode Vanna-Volga comme etant une approximation de Taylordu premier ordre du prix de loption autour de la volatilite dite a la monnaie.

Dans le Chapitre 2 de la these, nous allons developper un modele qui nonseulement tient compte de la volatilite implicite du marche mais egalement dela variabilite des taux dinterets. Ceci relache lhypothese de taux dinteretsconstants presente dans le modele de Black-Scholes ce qui resout le deuxiemeprincipal probleme rencontre dans ce dernier, pouvant avoir de grandes conse-quences lors de levaluation et de la couverture de produits a longue matu-rite. Nous travaillons dans le marche particulier des taux de changes, avecun modele a volatilite locale pour la dynamique du taux de change danslequel les taux dinterets domestiques et etrangers sont egalement supposesstochastiques. Nous derivons lexpression de la volatilite locale et derivonsdivers resultats particulierement utiles pour la calibration du modele. Finale-ment, nous developpons un nouveau modele hybride ou la volatilite du tauxde change possede une composante locale et une composante stochastique etnous derivons une methode de calibration pour ce nouveau modele.

Dans le Chapitre 3, nous allons appliquer le modele a volatilite locale ettaux dinterets stochastiques developpe dans le precedent chapitre mais dansle cadre devaluation de produits derives associes aux assurances vie. Ces

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options ont comme date decheance lage de la retraite et peuvent donc etreconsiderees comme produits a longue maturite. Nous utilisons une methodede calibration developpee dans le Chapitre 2. Les produits etudies etantexotiques, nous allons egalement comparer les prix obtenus dans differentsmodeles, a savoir le modele a volatilite locale, a volatilite stochastique et en-fin a volatilite constante pour le sous-jacent, les trois modeles etant combinesavec des taux dinterets stochastiques.

Finalement, dans le Chapitre 4, nous allons travailler avec un modeledit de Levy pour modeliser le sous-jacent. Lidee sous-jacente au modelede Levy est lutilisation dun mouvement plus general que le mouvementBrownien ce qui permet detre plus general que le modele de Black-Scholeset detre en accord avec la distribution de probabilite du marche. Ce modelea la particularite dautoriser les sauts dans la dynamique ce qui est de plusen plus apprecie et particulierement en temps de crise financiere. Nous nousinteressons plus precisement a levaluation doptions Asiatiques arithmetiquesdont le profit se base sur la moyenne arithmetique du sous-jacent durant lavie de loption. Comme de nombreuses options exotiques, il nest pas pos-sible dobtenir un prix analytique meme dans le cas simple du modele deBlack-Scholes et dans ce cas, seules les methodes numeriques permettent deresoudre le probleme. Dans ce Chapitre 4, nous developpons une methodebasee sur la methode de simulations de Monte-Carlo et nous employons deuxtypes de variables de controle permettant dameliorer la convergence du pro-gramme. Nous developpons egalement une methode permettant dobtenirune borne inferieure au prix de loption avec une efficacite qui surpasse lesautres methodes, donnant des resultats relativement proches de la valeur ex-acte, pouvant donc faire partie des outils devaluation et de gestion du risqueutilises par les fournisseurs du marche.

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Introduction

Derivatives are financial contracts whose value depends on the value of otherfinancial assets, the underlyings, which can be for example stocks, interestrates, commodities, foreign exchange rates. They can be highly risky: in-deed, because of the leverage effect, they can generate large profits but alsocomplete losses when the market goes in the wrong direction. Recently, thefinancial crisis has shown how underlying asset values can dramatically vary,making the derivative market more dangerous. However, this market is stillgrowing. The total notional amount of all the outstanding positions of overthe counter derivatives at the end of June 2004 stood at 220 trillion US dol-lars and by the end of 2011 this figure had risen to more than 700 trillionUS dollars (see BIS Quarterly Review, December 2011). Derivatives haveplayed a negative role in the financial crisis mainly because of wrong pricingof complicated products. Because of the amount of money involved and therisk implied by trading these contracts, banks and insurance company needsuitable models to price and hedge derivatives.

Pricing options and more complex derivatives involves developing a suit-able pricing model able to represent as close as possible the real world be-havior of the underlyings. Nowadays, the most popular approach consists ofusing Stochastic Differential Equation (SDEs) in order to simulate the mar-ket dynamics. The first model based on SDEs appeared in Bachelier (1900)where the stock prices, denoted by S(t), follow an arithmetic Brownian Mo-tion, namely, dS(t) = dB(t), where is the (normal) constant volatilityof the stock price and leading to a normal distribution for the stock pricereturn. Geometric Brownian motions were introduced by Osborne (1959)where the stock price dynamics are defined by an exponential Brownian mo-tion (S(t) = S0e

B(t)) avoiding negative value for stock prices. This modelleads to a log normal distribution for the stock price and has been stud-ied by many authors such as Sprenkle (1964), Boness (1964) and Samuelson(1965), all working on the European options pricing problem which was fi-nally solved by Black and Scholes (1973) and Merton (1973) introducing no

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arbitrage pricing arguments and an explicit connection between the price ofa derivative and a hedging strategy. Unfortunately, this model is still basedon several unrealistic assumptions that render the prices usually inaccurateand therefore is rarely used by actual practitioners for pricing derivatives.Still it remains a reference for quotation.

A European call option is one of the simplest and more liquid derivativeavailable in the market. It is refereed as a plain vanilla option and it givesthe right (but not the obligation) to the buyer to buy an agreed quantity of aparticular financial instrument (the underlying) from the seller of the optionat a certain time (the expiration date T , usually called the maturity) for acertain price (the strike price K). In a Black-Scholes world, the price of acall option is a function of the volatility which is assumed to be constantduring the live of the option. More precisely, if we consider the following(Black-Scholes) geometric Brownian motion for the stock price S, namely,

dS(t) = S(t)dt + S(t)dB(t), (1)

where is the rate of return of the asset commonly referred to as the drift.Applying the risk neutral valuation approach from Black and Scholes (1973),the price (at t = 0) of a European Call with strike K and maturity T ,C(K,T ), is given by

C(K,T ) = S(0)N (d1)KerTN (d2),

where

d1 =ln(S(0)

K) + (r +

2

2)T

T, d2 = d1

T ,

r is the risk-free interest rate and N (x) is the cumulative distribution func-tion of a standard normal variable.

However, in the real market, the volatility does not seem to be con-stant at all. To show this, we use the concept of implied volatility which isdefined by the volatility () one has to plug into the Black-Scholes call priceformula in order to obtain the market value. Plotting the implied volatili-ties in function of the strike and the maturity leads to a surface which hasthe shape of a smile with a skew (see Figure 1) and is commonly called the

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Figure 1: Implied volatility surface of JPY/USD

Smile/Skew implied volatility surface. In a Black-Scholes world this surfaceis flat and therefore the Black-Scholes model is not consistent with the realEuropean call market. The implied volatility Smile is a good information ofthe market behavior. Actual market practices are such that a suitable modelto price derivatives is a model able to fit this market implied volatility surface.

One of the most famous models consistent with the implied volatilitysurface is the Heston (1993) stochastic volatility model. In this model, thespot price S is solution of the following stochastic process:

dS(t) = S(t)dt +

(t)S(t)dBS(t),

where (the square of the asset price volatility) follows a CoxIngersollRoss(CIR) process, namely,

d(t) = k[ (t)] dt +

(t)dB(t)

and where BS(t) and B(t) are correlated Brownian motions, is the longvariance, or long run average price variance. As t tends to infinity, the ex-pected value of (t) tends to . The parameter k is the mean reversion rate,namely, the rate at which (t) reverts to and is the volatility of thevolatility (called the vol of vol) and determines the variance of (t).

Heston (1993) has derived an analytical formula for the price of Euro-pean options. Using such model, it is possible to determine the value of the

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parameters, k, , in order to fit quite closely the market implied volatil-ity surface. This optimization problem is called the calibration of the modeland is probably the most important step when pricing derivatives in practice.

Another famous model able to be calibrated on the market implied volatil-ity surface is the local volatility model introduced by Derman and Kani (1994)and Dupire (1994) where the asset price dynamic is given by the followingSDE,

dS(t) = S(t)dt + (t, S(t))S(t)dB(t).

This model generalizes the Black-Scholes model by making the instanta-neous volatility of the stock returns a deterministic function of the time andthe stock price. This function is called the local volatility function. Dupire(1994) and Derman and Kani (1994) have shown that one may recover thelocal volatility function (t, S(t)) from the prices of traded plain vanilla Eu-ropean call options using the following formula:

(T, K) =

C(K,T )

T+ rK C(K,T )

K

12K2

2C(K,T )K2

.

Both the Heston and the local volatility models are probably, until to-day, the favorite models for practitioners. The main advantage of the localvolatility model is that the volatility is a deterministic function and thereforethe model is Markovian in only one factor. It avoids the problem of work-ing in incomplete market in comparison with stochastic volatility models,which is better for the construction of hedging strategies. Because the localvolatility is calibrated on the whole implied volatility surface, local volatil-ity models capture the whole implied volatility surface generally better thanstochastic volatility models. Local volatility models have the drawback thatthe volatility of the stock returns is not assumed to be stochastic whereasreal market volatilities exhibit stochastic behavior. However, when pricingvanilla options, the Heston and the local volatility models are both able toreturn market consistent prices since they are both able to being calibratedon the implied volatility surface. The problem comes once you deal withpath dependent options. For example, in the case of barrier type options,i.e. European options that can be activated or deactivated if the underlyingreaches a certain barrier level, the payout depends on the entire spot path(S(t), 0 t T ). Therefore, the price is not only a function of the terminal

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spot density probability function (fS(T )(x)) but also a function of the tran-sition probability density functions of the spot at time t conditional on thelevel of the spot at time s, 0 s < t T (fS(T )|S(s)(x), 0 s < t T )). Theimplied volatility surface provides full information about the terminal spot

density (fS(T )(K) = erT

2C(K,T )K2

) but nothing about the transition probabil-ity density functions. For that reason, stochastic volatility models or localvolatility models perfectly calibrated on the vanilla market can yield verydifferent prices for path dependent options (see Schoutens et al. (2004) andBossens et al. (2010)).

Hybrid volatility models that combine both stochastic volatility and localvolatility dynamics could better capture market dynamics. This approachhas been studied by Lipton (2002); Lipton and McGhee (2002); Madan et al.(2007); Tavella et al. (2006)). For example in Madan et al. (2007), the hybridvolatility model is defined by the following risk-neutral dynamics for the spotprices S(t):

dS(t) = (r q)S(t)dt + (t)(t, S(t))S(t)dBQS (t),

using a mean-reverting log-normal model for (t), the stochastic componentof the volatility, namely,

d ln((t)) = k[(t) ln((t))] dt + dBQ (t),

where dBQS (t) and dBQ (t) are uncorrelated

1 Brownian motions under the riskneutral measure Q. The constant risk-free rate and the constant dividendyield are respectively denoted by r and q. They used a long-term determin-istic drift (t), k is the rate of mean reversion and is the vol of vol.

However, such models are known to be complex for the implementation.They require numerical methods which are computationally demanding andtheir calibration is delicate. An alternative solution is the Vanna-Volgamethod which is not a rigorous model but an ad-hoc pricing techniquethat is easy to implement and fast. This method consists in adjusting the

1Madan et al. (2007) assume that the Brownian motion driving the stochastic compo-nent of volatility is uncorrelated with the Brownian motion driving the stock price sincethe dependence of volatility on the stock price is already captured in the local volatilityfunction (t, S(t)). However, it could be extended to a non-zero correlation which is usu-ally observed in the market. In that case, some of the Smile/Skew would come from thecorrelation and from the local volatility function.

15

Black-Scholes price with a smile impact correction analytically derived froma portfolio composed of three vanilla strategies, which zeros out the Vega2,Vanna3 and Volga4 of the option at hand. The Vanna-Volga method arisesfrom the idea that the smile adjustment to an option price is associated withthe costs incurred by hedging its volatility risk. This method has been studiedespecially in the FX options market by Lipton and McGhee (2002), Wystup(2003), Castagna and Mercurio (2007), Fisher (2007) and Shkolnikov (2009).In Chapter 1, we describe this method and give a new intuitive justificationfor the Vanna-Volga correction in the case of plain vanilla options. More pre-cisely, we show that the method can be seen as a first-order Taylor expansionof the Black-Scholes price of a vanilla option around the at-the money im-plied volatility. We develop a new variation of the method in order to pricebarrier options. This method allows for a fast calibration on the barrier op-tions market and therefore it offers a consistent market pricing system forthat type of options. Furthermore we investigate how this method behavesin pricing with respect to more rigorous models, namely, the local volatilitymodel and the Heston stochastic volatility model. Because the Vanna-Volgamethod is strongly sensible to options market data, we present some of therelevant FX conventions.

All the models presented above are based on the assumption that interestrates remain constant throughout the life of the option. This assumption isunfortunately wrong in a real market (Figure 2 is a plot of the overnightEuro LIBOR interest rate5, illustrating the stochastic behavior of interestrates) and can have a significant impact on the evolution of the stock priceas well as on the price of the option and the corresponding hedging strate-gies. Especially when the financial sector is in crisis, Central Banks usuallymodify the level of interest rates and this may have an impact on the stockvalue. For example, if they decrease the level of interest rates, investors maybe interested to invest in stocks because of the poor return on bank account.Market practice seems to avoid to work with stochastic interest rates whendealing with short-dated derivatives (less than one year) since it doesnt leadto significant errors in pricing. However, the market of long dated options

2The Vega is the sensitivity of the price (P ) of a derivative to a change in the volatility, namely, P .

3The Vanna is the sensitivity of the Vega of a derivative to a change in the underlyinginstrument price S, namely, V egaS .

4The Volga is the sensitivity of the Vega of a derivative to a change in the volatility ,namely, V ega .

5The overnight Euro LIBOR interest rate is the interest rate at which a panel of selectedbanks borrow euro funds from one another with a maturity of one day.

16

Figure 2: Plot of the overnight Euro LIBOR interest rates (coming fromwww.homefinance.nl) illustrating the stochastic behavior of the market ofinterest rates.

becomes more and more important and when pricing such options, stochasticinterest rates dynamics should be included in the pricing model.

Interest rates models have been studied by many researchers (see forexample Brigo and Mercurio (2006)). One of the most popular tractableinterest rate model is the Hull and White one-factor Gaussian model (Hulland White (1993)). The interest rates r(t) are modelled by the followingOrnstein-Uhlenbeck process:

dr(t) = [(t) (t)r(t)]dt + r(t)dBQr (t), (2)where BQr (t) is a Brownian motion under the risk neutral measure Q, (t) isa function of time determining the average direction in which r moves andis chosen such that movements in r are consistent with todays zero couponyield curve. The deterministic parameters (t) and r(t) are the mean re-version rate and the volatility of r(t), respectively. Note that in practice and r are usually assumed to be constant rather than time dependentbecause this setting allows for a better calibration to the market of interestrate derivatives (see Hull and White (1995) and Brigo and Mercurio (2006)).

In order to price long-dated options, researchers and practitioners havestudied models which combine both stochastic interest rates and stochasticstock prices. The first approach studied was probably the two-factor modelwhich combines a geometric Brownian motion for the stock price S(t) (see

17

equation (1)) with Gaussian stochastic interest rates using for example theHull and White one-factor Gaussian model given by equation (2). Suchmodels allow for analytical option pricing formulae (see for example Merton(1973), Rabinovitch (1989) and Amin and Jarrow (1991,2)) but do not takeinto account the smile effect. Following the same idea as Heston (1993) butin a stochastic interest rates framework, many researchers as for examplevan Haastrecht et al. (2009) or Grzelak and Oosterlee (2011) have studiedthe three-factor model assuming the volatility of the spot stochastic as well.The local volatility approach including stochastic interest rates has beenfirst studied by Atlan (2006) where the local volatility function (t, S(t)) isderived and is expressed in function of prices of traded plain vanilla EuropeanCall options and an expectation unfortunately not directly related to liquidmarket products, namely,

(T, K) =

C(T,K)

T+ KP (0, T )EQT [r(T )1{S(T )>K} ]

12K2

2C(T,K)K2

, (3)

where P (0, T ) is the price of a zero-coupon bond maturing at time T andwhere QT is the T -forward measure associated to the zero-coupon bond asnumeraire.

When pricing foreign exchange options (FX options), one has to considerdomestic and foreign interest rates since they both have an impact on thedynamics of the spot FX rate. In Chapter 2, we study the three-factor modelwith local volatility which consists of using a local volatility for the volatilityof the spot FX rate and where domestic and foreign interest rates denoted byrd(t) and rf (t) respectively follow a Hull-White one-factor Gaussian model.This model has been considered in Piterbarg (2006) in which the authorderives an approximative formula for the local volatility function. The ap-proximation allows for a fast calibration of the model on vanilla options.However, the approximation is able to capture the slope of the impliedvolatility surface but not the convexity. In Chapter 2, we derive the exactlocal volatility function6:

6The expression of the local volatility (4) is derived under the risk neutral measure Q.However, it is sometimes necessary to be able to calibrate the model under the real worldmeasure as for example in risk management where real world asset dynamics are oftenneeded. The derivation of the local volatility could be done under the real probabilitymeasure using the same approach as in Chapter 2. This derivation and the calibration ofthe model under the real world measure is left for future research.

18

(T, K) =

C(K,T )

T Pd(0, T )EQT [(rd(T )K rf (T )S(T ))1{S(T )>K} ]

12K2

2C(K,T )K2

, (4)

where Pd(0, T ) is the price of a domestic zero-coupon bond maturing at timeT .

The calibration of the local volatility is more complicated than in a con-stant interest rates framework, since the local volatility expression dependson an expectation which is not linked to any tradable products. In Chap-ter 2, we derive several approaches for the calibration of the local volatilityfunction (4). More precisely, we present two numerical approaches basedon respectively Monte-Carlo simulations and PDE numerical resolution. Athird method is based on the link between the local volatility function derivedin a three-factor framework and the one coming from the simple one-factorGaussian model. Finally, one has derived a direct link between stochas-tic volatility and local volatility models both in a stochastic interest ratesframework. Therefore, once the stochastic volatility model with stochasticinterest rates is calibrated, one can use that link for the calibration of thelocal volatility function7.

Recall that local volatility models as well as stochastic volatility modelswith stochastic interest rates will be consistent with the vanilla market afterbeing calibrated on the implied volatility surface but this is by no meansa guarantee that these models will price correctly path dependent optionsin the sense of being market consistent. Following the idea of Madan et al.(2007), we study a hybrid volatility model which combines stochastic volatil-ity and local volatility for the spot FX rate but in a stochastic interest rateframework. One has derived a link between this four-factor hybrid volatilitymodel and the pure local volatility model with stochastic interest rates.Using that link, we have developed a calibration procedure for the localvolatility function associated to this hybrid volatility model which is based

7This link is particularly useful when the market is not liquid. More precisely, thecalibration of the local volatilty (4) give stable results only in a liquid market since youhave to compute C(K,T )T and

2C(K,T )K2 by using market call prices. If the market is not

liquid it is sometimes better to first calibrate a stochastic volatility model then use the linkbetween the local volatility model and the stochastic volatility model for the calibrationof the local volatility function. However, if the calibration of both stochastic volatilityand local volatility is not possible, an alternative is to use a Regime-Switching Model,introduced in Hamilton (1989).

19

on the knowledge of the local volatility derived from the three-factor modelwith local volatility.

Chapter 3 is devoted to the numerical application of local volatility mod-els with stochastic interest rates for the pricing of long-dated life insurance(derivative) contracts. More precisely, we study the pricing behavior of thelocal volatility model against the Schobel-Zhu stochastic volatility modeland the Black-Scholes constant volatility model all coupled with Hull-Whitestochastic interest rates. The long-dated insurance products studied are Vari-able Annuity Guarantees, especially Guaranteed Annuity Options (GAO)and Guaranteed Minimum Income Benefit (GMIB). Variable Annuity prod-ucts are generally based on an investment in a mutual fund composed ofstocks and bonds (see for example Gao (2010) and Schrager and Pelsser(2004)) and they offer a range of options to give minimum guarantees andprotect against negative equity movement.

Before pricing these derivatives and compare the results given by the threedifferent models, one has to calibrate them to the same options market data.The main part of the calibration of the local volatility model with stochas-tic interest rates is the calibration of the local volatility. From Chapter 2,four different methods are offered to us, namely, the Monte-Carlo approach,the PDE approach, the adjustment of the tractable Dupire formula and fi-nally the calibration from a stochastic volatility model. We have preferred toapply the numerical integration method based on Monte-Carlo simulationsrather than using a PDE solver (in order to compute numerically the expec-tation present in the local volatility expression). Note that, the calibrationmethod based on the adjustment of the tractable local volatility surface com-ing from the one-factor model with constant interest rates requires marketdata in order to determine the covariance between interest rates r(t) andthe indicator function 1{S(T )>K}. Unfortunately, we do not dispose of suchmarket data. The last calibration method mentioned consists of calibratingthe local volatility from a stochastic volatility model already calibrated. Thecalibration of the stochastic volatility model especially when interest ratesare stochastic is a difficult task and once the calibration is done, the localvolatility can be calibrated by using the link with stochastic volatility models.The link is given by a conditional expectation which is difficult to computenumerically by using traditional numerical methods. However, Malliavin cal-culus allow for representations of such conditional expectation that can becomputed efficiently by Monte-Carlo simulations (see Fournie et al. (2001)).We have preferred to directly calibrate the local volatility function from theavailable implied volatility by using a Monte-Carlo simulations method. The

20

other calibration methods are left for future research.

In the last part of the thesis, we study the pricing of derivatives in the set-ting of Levy processes. Remember indeed that the log-normality assumptionfor the distribution of the stock price is not consistent with the real worldmarket behavior. Local volatility and stochastic volatility models are able todeal with this reality and fit the distribution exhibit by the market. Anotherfamous solution is to replace the (Black-Scholes) geometric Brownian motionby an exponential non-normal Levy process.

A Levy process {Xt, t 0}, is a stochastic process where increments areindependent (i.e. Xt Xs is independent of Xu, 0 u < s < t < +) andstationary, meaning that the distribution of increments does not depend ontime but only on time distance (i.e. Xt Xs has the same distribution asXts, 0 s < t < +). The process starts at 0 (X0 = 0 almost surely)and is stochastically continuous in the sense that the probability of a jumpat time t is null (lim

h0P (|Xt+h Xt| ) = 0). Finally, the path of a Levy

process is Continue a Droite et Limite a Gauche (Cadlag)8 ensuring thatthe path does not reach infinity and the absence of left continuity allowsthe process to have jumps. This last property is particularly consistent withseveral markets which exhibit jumps especially in a crisis period. A Brown-ian motion is a Levy process where the distribution of increments Xt Xs,0 s < t < + are normally distributed and which doesnt allow for jumpssince the paths of the process are assumed to be almost surely continuous.

An empirical study in Mandelbrot (1963) had already revealed that theassumption of log-normality distribution for the stock prices is not suitableand he was probably one of the first to propose to use an exponential non-normal Levy process. The idea proposed by Mandelbrot was to replace theBrownian motion by a symmetric -stable Levy motion with index < 2which yields to a pure jump process for the stock price9. Afterwards, othertypes of exponential Levy processes appeared where the log price process isa combination of a Brownian motion and an independent jump process givenby for example a compound Poisson process with normally distributed jumps(see for example Press (1967)) and Merton (1976)). These processes are notpure jump stock processes.

8RCLL (right continuous with left limits), or corlol (continuous on (the) right, limiton (the) left)

9A Normal distribution is an -stable distribution where = 2

21

More recently, a new category of pure jump Levy processes has been stud-ied, namely, time-changed Brownian motions which generalized the Brownianmotion by making the time itself stochastic. More precisely, the Brownianmotion B(t) is replaced by B(Gt) where {Gt, t 0} is another positive in-creasing stochastic process commonly referred as the stochastic clock or thesubordinator. For example Generalized Hyperbolic Levy processes form aclass of Levy process that can be represented as a time-changed Brownianmotion where the stochastic clock follows a generalized inverse Gaussian dis-tribution. It has been introduced by Barndorff-Nielsen (1977) in order todescribe a phenomenon in physics (the migration of sand-dunes) and thenintroduced in finance by Eberlein and Keller (1995). The Variance Gamma(VG) model which can be track back to McLeish (1982) and the NormalInverse Gaussian model are two sub-classes of Generalized Hyperbolic Levymodels (see e.g. Schoutens (2003) and Raible (2000)). The VG and theNIG processes are pure jump processes (like -stable Levy processes) whichprovide an excellent fit to empirically observed log-return distributions (seeMadan et al. (1998) and Raible (2000)). In both the VG and the NIG mod-els, the characteristic function of the stock price is known analytically andthe price of a Europen call opion of maturity T and strike K can easily becomputed by using the formula derived in Carr and Madan (1998), namely,

C(K, T ) =e ln(K)

+0

eiv ln(K)(v)dv (5)

where

(v) =erT (v ( + 1)i)

2 + v2 + i(2 + 1)v (6)

and where is the characteristic function of the stock price process and is apositive constant such that the th moment of the stock price exists (usually is set to 0.75). This formula is particularly useful for calibration since itcan easily be computed by using the FFT method which return option pricesfor a whole range of strikes simultaneously.

An Asian option, also called average price option, is an exotic optionwhere the payoff is determined by the average underlying price over thelife of the option. More precisely, the payout of an Asian call is given bymax(A(0, T ) K, 0) where A(0, T ) denotes the average of the underlyingprice between time 0 and time T , and K is the strike. Since the payoff

22

depends of the average trajectory of the underlying, Asian options reducethe risk of market manipulation of the price underlying at maturity10. AnAsian option is also cheaper than the corresponding European option sincethe value is a function of the average of the underlying path which reducethe volatility inherent in the option. There exist two types of averaging,namely, the arithmetic and the geometric average and the average can beconsidered in a continuous or discrete monitoring case. Closed-form solu-tions exists for geometric Asian options even within Levy models. In Fusaiand Meucci (2008) the authors provide closed-form solutions for geometricAsian types in terms of the Fourier transform when the underlying evolvesaccording to a generic Levy process. Arithmetic Asian options are morecomplicated to value. There exists no closed-form formula even not in theBlack-Scholes framework and therefore pricing this type of Asian optionsrequires numerical methods. One of the most popular method is the Monte-Carlo simulation method introduced in finance by Boyle (1977) and used bymany authors for the pricing of Asian options as for example Kemna andVorst (1990) and Fusai and Meucci (2008). Monte-Carlo integration meth-ods converge asymptotically to the true value. Nevertheless, with a squareroot rate of convergence it is often slow and in its basic form, the method isusually computationally inefficient. However, the method can be improvedby using variance reduction methods as for example control variate methods,the use of antithetic paths and the use of quasi-random variables.

In Chapter 4, we focus on the pricing of discrete arithmetically averagedAsian options under the VG and NIG models by using Monte-Carlo simu-lation with control variates as variance reduction method. In this chapterwe make two contributions. The first contribution is that we develop severalmulti-control variate methods where the control variates are based on lowerbounds derived in Albrecher et al. (2008) or in a more straightforward wayin Deelstra et al. (2012). These methods highly increase the speed of conver-gence with respect to the crude Monte-Carlo method and therefore providea useful tool for Asian market dealers. The second contribution consists ofa two steps pricing method which provide a fast and accurate method forthe computation of tight lower bounds in a VG and NIG setting. The lowerbound is calculated by first conditioning on the stochastic clock which reducesto a log-normal distribution for the stock price and allows to compute thelower bound analytically (given in Vanduffel et al. (2009)). Afterwards, thesecond step consists of integrating this analytical formula over the stochastic

10European options owner could have interest to manipulate the price of the underlyingat maturity in order to drive up gain at expiry

23

clock domain. The integration is also done by using multi-control variatesMonte-Carlo simulation methods developed in the first contribution part.The resulting lower bound is quite close to the true Asian price and theMonte-Carlo method with multi-control variates converges so fast that theefficiency of this second approach surpasses all the other methods developedin this chapter. It is well known that financial institutions who deal with op-tions need fast and accurate answers for option prices and associated Greeksin order to hedge the risk linked to these highly risky products. Thereforethe last method which offers an attractive trade-off between accuracy andefficiency answers to market practitioner demands.

The thesis is made of four contributions. In the first chapter, Vanna-Volga Methods Applied to FX Derivatives: From Theory to Market Practiceco-authored with Frederic Bossens, Nikos S. Skantzos and Griselda Deelstra,published in August 2010 in International Journal of Theoretical and Ap-plied Finance, we provide a justification for the Vanna-Volga method forthe case of vanilla options and we show how to adapt the method to thepricing of first generation exotic options in the Foreign Exchange market.We develop a simple calibration method based on one-touch prices that al-lows the Vanna-Volga results to be consistent with the market. We compareour results to a large collection of indicative market prices and to local andstochastic volatility models. Because the Vanna-Volga price is strongly sen-sible to some market data, we provide a summary of the relevant FX marketdata conventions. In the second chapter, Local Volatility Pricing Modelsfor Long-dated FX Derivatives co-authored with Griselda Deelstra, to ap-pear in Applied Mathematical Finance, we study the local volatility functionin the Foreign Exchange market where both domestic and foreign interestrates are stochastic. We derive the local volatility function as well as severalresults that can be used for the calibration of the local volatility on the FXoptions market. Finally, a calibration method is derived for a model whichcombines both local and stochastic volatility in a stochastic interest ratesframework. In the third chapter, Pricing Variable Annuity Guarantees in aLocal Volatility framework, co-authored with Griselda Deelstra, submittedto Insurance: Mathematics and Economics, we study the price of long-datedinsurance derivatives, namely, Variable Annuity Guarantees in a local volatil-ity framework with stochastic interest rates. The local volatility function iscalibrated from the implied volatility surface by using a Monte-Carlo simula-tion method. We compare prices obtained in three different settings, namely,the local volatility, the stochastic volatility and the constant volatility mod-els all combined with stochastic interest rates. This study underly that anappropriate volatility modelling is important for the pricing of these long-

24

dated derivatives. Furthermore, the model choice as well as the market datahave a strong influence on the price. The fourth chapter, Using bounds fora faster pricing of Asian style options co-authored with Griselda Deelstra,Steven Vanduffel and Jing Yao, proposes a Monte-Carlo pricing method withcontrol variates variance reduction techniques for arithmetic Asian options(discretely sampled). We use lower bounds as control variates to improveMonte-Carlo simulations significantly. We also propose a new pricing ap-proach based on conditioning on the clock. Efficiency of both approaches areconfirmed by numerical experiments. Finally, we outline how the results canbe useful for the pricing of Asian related insurance products.

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Chapter 1

Vanna-Volga Methods Appliedto FX Derivatives: FromTheory to Market Practice

Abstract

We study Vanna-Volga methods which are used to price first generation exoticoptions in the Foreign Exchange market. They are based on a rescaling ofthe correction to the Black-Scholes price through the so-called probabilityof survival and the expected first exit time. Since the methods rely heavilyon the appropriate treatment of market data we also provide a summaryof the relevant conventions. We offer a justification of the core techniquefor the case of vanilla options and show how to adapt it to the pricing ofexotic options. Our results are compared to a large collection of indicativemarket prices and to more sophisticated models. Finally we propose a simplecalibration method based on one-touch prices that allows the Vanna-Volgaresults to be in line with our pool of market data.

1.1 Introduction

The Foreign Exchange (FX) options market is the largest and most liquidmarket of options in the world. Currently, the various traded products rangefrom simple vanilla options to first-generation exotics (touch-like options andvanillas with barriers), second-generation exotics (options with a fixing-datestructure or options with no available closed form value) and third-generationexotics (hybrid products between different asset classes). Of all the above thefirst-generation products receive the lions share of the traded volume. This

26

makes it imperative for any pricing system to provide a fast and accuratemark-to-market for this family of products. Although using the Black-Scholesmodel Black and Scholes (1973); Merton (1973) it is possible to derive an-alytical prices for barrier- and touch -options, this model is unfortunatelybased on several unrealistic assumptions that render the price inaccurate. Inparticular, the Black-Scholes model assumes that the foreign/domestic inter-est rates and the FX-spot volatility remain constant throughout the lifetimeof the option. This is clearly wrong as these quantities change continuously,reflecting the traders view on the future of the market. Today the Black-Scholes theoretical value (BS TV) is used only as a reference quotation, toensure that the involved counterparties are speaking of the same option.

More realistic models should assume that the foreign/domestic interestrates and the FX spot volatility follow stochastic processes that are coupledto the one of the spot. The choice of the stochastic process depends, amongother factors, on empirical observations. For example, for long-dated optionsthe effect of the interest rate volatility can become as significant as that ofthe FX spot volatility. On the other hand, for short-dated options (typicallyless than 1 year), assuming constant interest rates does not normally lead tosignificant mispricing. In this article we will assume constant interest ratesthroughout.

Stochastic volatility models are unfortunately computationally demand-ing and in most cases require a delicate calibration procedure in order to findthe value of parameters that allow the model reproduce the market dynamics.This has led to alternative ad-hoc pricing techniques that give fast resultsand are simpler to implement, although they often miss the rigor of theirstochastic siblings. One such approach is the Vanna-Volga (VV) methodthat, in a nutshell, consists in adding an analytically derived correction to theBlack-Scholes price of the instrument. To do that, the method uses a smallnumber of market quotes for liquid instruments (typically At-The-Moneyoptions, Risk Reversal and Butterfly strategies) and constructs an hedgingportfolio which zeros out the Black-Scholes Vega, Vanna and Volga of theoption. The choice of this set of Greeks is linked to the fact that they alloffer a measure of the options sensitivity with respect to the volatility, andtherefore the constructed hedging portfolio aims to take the smile effect intoaccount.

The Vanna-Volga method seems to have first appeared in the literature inLipton and McGhee (2002) where the recipe of adjusting the Black-Scholesvalue by the hedging portfolio is applied to double-no-touch options and in

27

Wystup (2003) where it is applied to the pricing of one-touch options inforeign exchange markets. In Lipton and McGhee (2002), the authors pointout its advantages but also the various pricing inconsistencies that arise fromthe non-rigorous nature of the technique. The method was discussed morethoroughly in Castagna and Mercurio (2007) where it is shown that it canbe used as a smile interpolation tool to obtain a value of volatility for agiven strike while reproducing exactly the market quoted volatilities. It hasbeen further analyzed in Fisher (2007) where a number of corrections aresuggested to handle the pricing inconsistencies. Finally a more rigorous andtheoretical justification is given by Shkolnikov (2009) where, among otherdirections, the method is extended to include interest-rate risk.

A crucial ingredient to the Vanna-Volga method, that is often overlookedin the literature, is the correct handling of the market data. In FX marketsthe precise meaning of the broker quotes depends on the details of the con-tract. This can often lead to treading on thin ice. For instance, there areat least four different definitions for at-the-money strike (resp., spot, for-ward, delta neutral, 50 delta call). Using the wrong definition can lead tosignificant errors in the construction of the smile surface. Therefore, beforewe begin to explore the effectiveness of the Vanna-Volga technique we willbriefly present some of the relevant FX conventions.

The aim of this chapter is twofold, namely (i) to describe the Vanna-Volga method and provide an intuitive justification and (ii) to compare itsresulting prices against prices provided by renowned FX market makers, andagainst more sophisticated stochastic models. We attempt to cover a broadrange of market conditions by extending our comparison tests into two dif-ferent smile conditions, one with a mild skew and one with a very highskew. We also describe two variations of the Vanna-Volga method (used bythe market) which tend to give more accurate prices when the spot is closeto a barrier. We finally describe a simple adjustment procedure that allowsthe Vanna-Volga method to provide prices that are in good agreement withthe market for a wide range of exotic options.

To begin with, in section 1.2 we describe the set of exotic instrumentsthat we will use in our comparisons throughout. In section 1.3 we review themarket practice of handling market data. Section 1.4 lays the general ideasunderlying the Vanna-Volga adjustment, and proposes an interpretation ofthe method in the context of Plain Vanilla Options. In sections 1.5.1 and 1.5.2we review two common Vanna-Volga variations used to price exotic options.The main idea behind these variations is to reduce Vanna-Volga correction

28

through an attenuation factor. The first one consists in weighting the Vanna-Volga correction by some function of the survival probability, while the secondone is based on the so-called expected first exit time argument. Since theVanna-Volga technique is by no means a self-consistent model, no-arbitrageconstraints must be enforced on top of the method. This problem is addressedin section 1.5.4. In section 1.5.5 we investigate the sensitivity of the modelwith respect to the accuracy of the input market data. Finally, Section 1.6is devoted to numerical results. After defining a measure of the model errorin section 1.6.1, section 1.6.2 investigates how the Dupire local vol model(see Dupire (1994)) and the Heston stochastic vol model (see Heston (1993))perform in pricing. Section 1.6.3 suggests a simple adaptation that allows theVanna-Volga method to produce prices reasonably in line with those givenby renowned FX platforms. Conclusions of the study are presented in section1.7.

1.2 Description of first-generation exotics

The family of first-generation exotics can be divided into two main subcat-egories: (i) The hedging options which have a strike and (ii) the treasuryoptions which have no strike and pay a fixed amount. The validity of bothtypes of options at maturity is conditioned on whether the FX-spot has re-mained below/above the barrier level(s) according to the contract termsheetduring the lifetime of the option.

Barrier options can be further classified as either knock-out options orknock-in ones. A knock-out option ceases to exist when the underlying assetprice reaches a certain barrier level; a knock-in option comes into existenceonly when the underlying asset price reaches a barrier level. Following theno-arbitrage principle, a knock-out plus a knock-in option (KI) must equalthe value of a plain vanilla.

As an example of the first category, we will consider up-and-out calls(UO, also termed Reverse Knock-Out), and double-knock-out calls (DKO).The latter has two knock-out barriers (one up-and-out barrier above thespot level and one down-and-out barrier below the spot level). The exactBlack-Scholes price of the UO call can be found in Hull (2006); Reiner andRubinstein (1991,1), while a semi-closed form for double-barrier options isgiven in Kunitomo and Ikeda (1992) in terms of an infinite series (most termsof which are shown to fall to zero very rapidly).

29

As an example of the second category, we will select one-touch (OT) op-tions paying at maturity one unit amount of currency if the FX-rate everreaches a pre-specified level during the options life, and double-one-touch(DOT) options paying at maturity one unit amount of currency if the FX-rate ever reaches any of two pre-specified barrier levels (bracketing the FX-spot from below and above). The Black-Scholes price of the OT option canbe found in Rubinstein and Reiner (1992); Wystup (2006), while the DOTBlack-Scholes price is obtained by means of double-knock-in barriers, namelyby going long a double-knock-in call spread and a double knock-in put spread.

Although these four types of options represent only a very small fractionof all existing first-generation exotics, most of the rest can be obtained bycombining the above. This allows us to argue that the results of this studyare actually relevant to most of the existing first-generation exotics.

1.3 Handling Market Data

The most famous defect of the Black-Scholes model is the (wrong) assump-tion that the volatility is constant throughout the lifetime of the option.However, Black-Scholes remains a widespread model due to its simplicityand tractability. To adapt it to market reality, if one uses the Black-Scholesformula1

Call() = DFd(t, T )[FN(d1)KN(d2)]Put() = DFd(t, T )[FN(d1)KN(d2)] (1.1)

in an inverse fashion, giving as input the options price and receiving as out-put the volatility, one obtains the so-called implied volatility. Plotting theimplied volatility as a function of the strike results typically in a shape thatis commonly termed smile (the term smile has been kept for historicalreasons, although the shape can be a simple line instead of a smile-lookingparabola). The reasons behind the smile effect are mainly that the dynam-ics of the spot process does not follow a geometric Brownian motion andalso that demand for out-of-the-money puts and calls is high (to be used bytraders as e.g. protection against market crashes) thereby raising the price,and thus the resulting implied volatility at the edges of the strike domain.

1for a description of our notation, see 1.7.

30

The smile is commonly used as a test-bench for more elaborate stochasticmodels: any acceptable model for the dynamics of the spot must be able toprice vanilla options such that the resulting implied volatilities match themarket-quoted ones. The smile depends on the particular currency pair andthe maturity of the option. As a consequence, a model that appears suitablefor a certain currency pair, may be erroneous for another.

1.3.1 Delta conventions

FX derivative markets use, mainly for historical reasons, the so-called Delta-sticky convention to communicate smile information: the volatilities arequoted in terms of Delta rather than strike value. Practically this meansthat, if the FX spot rate moves all other things being equal the curveof implied volatility vs. Delta will remain unchanged, while the curve of im-plied volatility vs. strike will shift. Some argue this brings more efficiency inthe FX derivatives markets. For a discussion on the appropriateness of thedelta-sticky hypothesis we refer the reader to Derman (1999). On the otherhand, it makes it necessary to precisely agree upon the meaning of Delta. Ingeneral, Delta represents the derivative of the price of an option with respectto the spot. In FX markets, the Delta used to quote volatilities dependson the maturity and the currency pair at hand. An FX spot St quoted asCcy1Ccy2 implies that 1 unit of Ccy1 equals St units of Ccy2. Some currencypairs, mainly those with USD as Ccy2, like EURUSD or GBPUSD, use theBlack-Scholes Delta, the derivative of the price with respect to the spot:

call = DFf (t, T ) N(d1) put = DFf (t, T ) N(d1) (1.2)Setting up the corresponding Delta hedge will make ones position in-

sensitive to small FX spot movements if one is measuring risks in a USD(domestic) risk-neutral world. Other currency pairs (e.g. USDJPY) use thepremium included Delta convention:

call =K

SDFd(t, T ) N(d2) put = K

SDFd(t, T ) N(d2) (1.3)

The quantities (1.2) and (1.3) are expressed in Ccy1, which is by conven-tion the unit of the quoted Delta. Taking the example of USDJPY, settingup the corresponding Delta hedge (1.3) will make ones position insensitiveto small FX spot movements if one is measuring risks in a USD (foreign)risk-neutral world. Note that (1.2) and (1.3) are linked through the optionspremium (1.1), namely S(call call) = Call and similarly for the put (see

31

1.7 for more details).

With regards to the dependency on maturity, the so-called G11 currencypairs use a spot Delta convention (1.2), (1.3) for short maturities (typicallyup to 1 year) while for longer maturities where the interest rate risk becomessignificant, the forward Delta (or driftless Delta) is used, as the derivative ofthe undiscounted premium with respect to forward:

Fcall = N(d1) Fput = N(d1)

Fcall =KS

DFd(t, T )DFf (t, T )

N(d2) Fput = KS

DFd(t, T )DFf (t, T )

N(d2)(1.4)

where, as before, by tilde we denoted the premium-included convention. TheDeltas in the first row represent the nominals of the forward contracts to besettled if one is to forward hedge the Delta risk in a domestic currency whilethose of the second row consider a foreign risk neutral world. Other currencypairs (typically those where interest-rate risks are substantial, even for shortmaturities) use the forward Delta convention for all maturity pillars.

1.3.2 At-The-Money Conventions

As in the case of the Delta, the at-the-money (ATM) volatilities quoted bybrokers can have various interpretations depending on currency pairs. TheATM volatility is the value from the smile curve where the strike is such thatthe Delta of the call equals, in absolute value, that of the put (this strikeis termed ATM straddle or ATM delta neutral ). Solving this equalityyields two possible solutions, depending on whether the currency pair usesthe Black-Scholes Delta or the premium included Delta convention. The 2solutions respectively are:

KATM = F exp

[1

22ATM

]KATM = F exp

[1

22ATM

](1.5)

Note that these expressions are valid for both spot and forward Deltaconventions.

1.3.3 Smile-related quotes and the brokers Strangle

Let us assume that a smile surface is available as a function of the strike(K). In liquid FX markets some of the most traded strategies include

32

Strangle(Kc, Kp) = Call(Kc, (Kc)) + Put(Kp, (Kp)) (1.6)

Straddle(K) = Call(K,ATM) + Put(K,ATM) (1.7)

Butterfly(Kp, K, Kc) =1

2

[Strangle(Kc, Kp) Straddle(K)

](1.8)

Brokers normally quote volatilities instead of the direct prices of theseinstruments. These are expressed as functions of , for instance a volatilityat 25-call or put refers to the volatility at the strikes Kc, Kp that satisfycall(Kc, (Kc)) = 0.25 and put(Kp, (Kp)) = 0.25 respectively (with theappropriate Delta conventions, see section 1.3.1). Typical quotes for the volsare

at-the-money (ATM) volatility: ATM 25-Risk Reversal (RR) volatility: RR25 1-vol-25-Butterfly (BF) volatility: BF25(1vol) 2-vol-25-Butterfly (BF) volatility: BF25(2vol)

By market convention, the RR vol is interpreted as the difference betweenthe call and put implied volatilities respectively:

RR25 = 25C 25P (1.9)where 25C = (Kc) and 25P = (Kp).

The 2-vol-25-Butterfly can be interpreted through

BF25(2vol) =25C + 25P

2 ATM (1.10)

Associated to the BF25(2vol) is the 2-vol-25-strangle vol defined throughSTG25(2vol) = BF25(2vol) + ATM.

The 2-vol-25-Butterfly value BF25(2vol) is in general not directly observ-able in FX markets. Instead, brokers usually communicate the BF25(1vol), us-ing a brokers strangle or 1vol strangle convention. The exact interpretationof BF25(1vol) can be explained in a few steps:

Define STG25(1vol) = ATM + BF25(1vol).

33

Solve equations (1.2),(1.3) to obtain K25C and K25P , the strikes wherethe Delta of a call is exactly 0.25, and the Delta of a put is exactly-0.25 respectively, using the single volatility value STG25(1vol).

Provided that the smile curve (K) is correctly calibrated to the mar-ket, then the quoted value BF25(1vol) is such that the following equalityholds:

Call(K25C , STG25(1vol)) + Put(K25P , STG25(1vol))

= Call(K25C , (K25C)) + Put(K

25P , (K

25P )) (1.11)

The difference between BF25(1vol) and BF25(2vol) can be at times confusing.Often for convenience one sets BF25(2vol) = BF25(1vol) as this greatly simpli-fies the procedure to build up a smile curve. However it leads to errors whenapplied to a steeply skewed market. Figure1.1 provides a graphical inter-pretation of the quantities STG25(1vol), STG25(2vol), BF25(1vol) and BF25(2vol)in 2 very different market conditions; the lower panel corresponds to theUSDCHF-1Y smile, characterized by a relatively mild skew, the upper panelcorresponding to the extremely skewed smile of USDJPY-1Y. As a rule ofthumb one sets BF25(2vol) = BF25(1vol) when RR25 is small in absolute value(typically < 1%). When this empirical condition is not met, BF25(1vol) andBF25(2vol) represent actually two different quantities, and substituting onefor the other in the context of a smile construction algorithm would yieldsubstantial errors.

Table 1.1 gives more details about the numerical values used to producethe 2 smiles of Figure 1.1.

Various differences are observed between the 2 smiles. In the USDCHFcase, the values BF25(2vol) and BF25(1vol) are close to each other. Similarly,the strikes used in the 1vol-25 Strangle are rather close to those attachedto the 2vol-25 Strangle. On the contrary, in the USDJPY case, large dif-ferences are observed between the parameters of the 1vol-25-Strangle andthose of the 2vol-25-Strangle.

Unfortunately, there is no direct mapping between BF25(1vol) and BF25(2vol).This is mainly due to the fact that these two instruments are attached todifferent points of the implied volatility curve. The relationship betweenBF25(2vol) and BF25(1vol) implicitly depends on the entire smile curve.

34

Figure 1.1: Comparison between STG25(2vol) and STG25(1vol), also called bro-ker strangle in two different smile conditions.

In practice however, one may be interested in finding the value of BF25(1vol)from an existing smile curve; this can be achieved using an iterative proce-dure:

pseudo-algorithm 1

Select an initial guess for BF25(1vol). compute the corresponding strikes K25P and K25C . assess the validity of equality (1.11): compare the value of the Strangle

35

USDCHF USDJPY

date 8 Jan 09 28 Nov 08FX spot rate 1.0902 95.47

maturity 1 yearrd 1.3% 1.74%rf 2.03% 3.74%

ATM 16.85% 14.85%RR25 -1.3% -9.4%

BF25(2vol) 1.1% 1.45%BF25(1vol) 1.04% 0.2%

K25P / K25C 0.9586/1.2132 82.28/101.25K25P / K

25C 0.9630/1.2179 85.24/103.53

Table 1.1: Details of market quotes for the two smile curves of Figure 1.1.

(i) valued with a unique vol BF25(1vol) (ii) valued with 2 implied volcorresponding to K25P , respectively K

25C .

If the difference between the two values exceeds some tolerance level,adapt the value BF25(1vol) and go back to 2.

In case one is given a value of BF25(1vol) from the market, and wants touse it to build an implied smile curve, one may proceed the following way:

pseudo-algorithm 2

Select an initial guess for BF25(2vol). Construct an implied smile curve using BF25(2vol) and market value of

RR25.

Compute the value of BF25(1vol) (for instance following guidelines ofpseudo-algorithm 1).

Compare BF25(1vol) you obtained in 3 to the market-given one. If the difference between the two values exceeds some tolerance, adapt

the value BF25(2vol) and go back to 2.

To close this section on the brokers Strangle issue, let us clarify anotherenigmatic concept of FX markets often used by practitioners, the so-calledVega-weighted Strangle quote. This is in fact an approximation for the value

36

of STG25(1vol). To show this, we start from equality (1.11). First we assumeK25P = K25P and K

25C = K25C . Next, we develop both sides in a first order

Taylor expansion in around ATM . After canceling repeating terms on theleft and right-hand side, we are left with:

(STG25(1vol) ATM) (V(K25P , ATM) + V(K25C , ATM)) (25P ATM) V(K25P , ATM) + (25C ATM) V(K25C , ATM)

(1.12)

where V(K,) represents the Vega of the option, namely the sensitivity ofthe option price P with respect to a change of the implied volatility: V = P

.

Solving this for STG25(1vol) yields:

STG25(1vol) 25P V(K25P , ATM) + 25C V(K25C , ATM)V(K25P , ATM) + V(K25C , ATM) (1.13)

which corresponds to the average (weighted by Vega) of the call and putimplied volatilities.

Note that according to Castagna et al. Castagna and Mercurio (2007)practitioners also use the term Vega-weighted butterfly for a structure wherea strangle is bought and an amount of ATM straddle is sold such that theoverall vega of the structure is zero.

1.4 The Vanna-Volga Method

The Vanna-Volga method consists in adjusting the Black-Scholes TV by thecost of a portfolio which hedges three main risks associated to the volatility ofthe option, the Vega, the Vanna and the Volga. The Vanna is the sensitivityof the Vega with respect to a change in the spot FX rate: Vanna = V

S.

Similarly, the Volga is the sensitivity of the Vega with respect to a change ofthe implied volatility : Volga = V

. The hedging portfolio will be composed

of the following three strategies:

ATM =1

2Straddle(KATM)

RR = Call(Kc, (Kc)) Put(Kp, (Kp))BF =

1

2Strangle(Kc, Kp) 1

2Straddle(KATM) (1.14)

where KATM represents the ATM strike, Kc/p the 25-Delta call/put strikesobtained by solving the equations call(Kc, ATM) =

14

and put(Kp, ATM) =1

4and (Kc/p) the corresponding volatilities evaluated from the smile sur-

face.

37

1.4.1 The general framework

In this section we present the Vanna-Volga methodology.The simplest formulation Wystup (2006) suggests that the Vanna-Volga

price XVV of an exotic instrument X is given by

XVV = XBS +Vanna(X)

Vanna(RR) wRR

RRcost +Volga(X)

Volga(BF) wBF

BFcost (1.15)

where by XBS we denoted the Black-Scholes price of the exotic and the Greeksare calculated with ATM volatility. Also, for any instrument I we defineits smile cost as the difference between its price computed with/withoutincluding the smile effect: Icost = Imkt IBS, and in particular

RRcost = [Call(Kc, (Kc)) Put(Kp, (Kp))] [Call(Kc, ATM) Put(Kp, ATM)]

BFcost =1

2[Call(Kc, (Kc)) + Put(Kp, (Kp))]

12

[Call(Kc, ATM) + Put(Kp, ATM)] (1.16)

The rationale behind (1.15) is that one can extract the smile cost of anexotic option by measuring the smile cost of a portfolio designed to hedgeits Vanna and Volga risks. The reason why one chooses the strategies BFand RR to do this is because they are liquid FX instruments and they carryrespectively mainly Volga and Vanna risks. The weighting factors wRR andwBF in (1.15) represent respectively the amount of RR needed to replicatethe options Vanna, and the amount of BF needed to replicate the optionsVolga. The above approach ignores the small (but non-zero) fraction ofVolga carried by the RR and the small fraction of Vanna carried by theBF. It further neglects the cost of hedging the Vega risk. This has led to amore general formulation of the Vanna-Volga method Castagna and Mercurio(2007) in which one considers that within the BS assumptions the exoticoptions Vega, Vanna and Volga can be replicated by the weighted sum ofthree instruments:

~x = A~w (1.17)

with

38

A =

ATMvega RRvega BFvegaATMvanna RRvanna BFvannaATMvolga RRvolga BFvolga

~w =

wATMwRRwBF

~x =

XvegaXvannaXvolga

(1.18)

The weightings ~w are to be found by solving the systems of equations(1.17).

Given this replication, the Vanna-Volga method adjusts the BS price ofan exotic option by the smile cost of the above weighted sum (note that theATM smile cost is zero by construction):

XVV = XBS + wRR(RRmkt RRBS) + wBF

(BFmkt BFBS)

= XBS + ~xT (AT )1~I = XBS + Xvega vega + Xvanna vanna + Xvolga volga(1.19)

where

~I =

0RRmkt RRBSBFmkt BFBS

vegavannavolga

= (AT )1~I (1.20)

and where the quantities i can be interpreted as the market prices attachedto a unit amount of Vega, Vanna and Volga, respectively. For vanillas thisgives a very good approximation of the market price. For exotics, however,e.g. no-touch options close to a barrier, the resulting correction typicallyturns out to be too large. Following market practice we thus modify (1.19)to

XVV = XBS + pvannaXvanna vanna + pvolgaXvolga volga (1.21)

where we have dropped the Vega contribution which turns out to be severalorders of magnitude smaller than the Vanna and Volga terms in all practicalsituations, and where pvanna and pvolga represent attenuation factors whichare functions of either the survival probability or the expected first-exittime. We will return to these concepts in section 1.5.

39

1.4.2 Vanna-Volga as a smile-interpolation method

In Castagna and Mercurio (2007), Castagna and Mercurio show how Vanna-Volga can be used as a smile interpolation method. They give an elegantclosed-form solution (unique) of system (1.17), when X is a European callor put with strike K.In their paper they adjust the Black Scholes price by using a replicatingportfolio composed of a weighted sum of three vanillas (calls or puts) struckrespectively at K1, K2 and K3, where K1 < K2 < K3. They show that theweights wi associated to the vanillas struck at Ki such that the resultingportfolio hedges the Vega, Vanna and Volga risks of the vanilla of strike Kare unique and given by:

w1(K) =Vega(K)

Vega(K1)

ln K2K

ln K3K

ln K2K1

ln K3K1

w2(K) =Vega(K)

Vega(K2)

ln KK1

ln K3K

ln K2K1

ln K3K2

(1.22)

w3(K) =Vega(K)

Vega(K3)

ln KK1

ln KK2

ln K3K1

ln K3K2

The fact that this solution provides an exact interpolation method is eas-ily verified by noticing that wi(Ki) = 1 and wi(Kj) = 0, i 6= j.

This solution still holds in the case of a replicating portfolio composedof ATM, RR and BF instruments as described in section 1.4 by equations(1.14). Setting K1 = Kp, K2 = KATM and K3 = Kc, a simple coordinatetransform yields:

wATM(K) = w1(K) + w2(K) + w3(K)

wRR(K) =1

2(w3(K) w1(K)) (1.23)

wBF(K) = w1(K) + w3(K)

where the weights wATM, wRR and wBF are defined by (1.17)-(1.18).

We now turn back to the elementary Vanna-Volga recipe (1.15). Unlikethe previously exposed exact solution, it does not reproduce the market priceof RR and BF, a fortiori is it not an interpolation method for plain vanillas.However, this approximation possesses the merit of allowing a qualitative

40

interpretation of the RR and BF correction terms in (1.15).

As we will demonstrate, those two terms directly relate to the slopeand convexity of the smile curve. To start with, we introduce a new smileparametrization variable:

Y = lnK

F exp (122ATM)

= lnK

KATM

Note that the Vega of a Plain Vanilla Option is a symmetric function ofY :

Vega(Y ) = Vega(Y ) = S erf n(

Y

)

where n() denotes the Normal density function.

Let us further assume that the smile curve is a quadratic function of Y :

(Y ) = ATM + bY + cY2 (1.24)

In this way we allow the smile to have a skew (linear term) and a curvature(quadratic term), while keeping an analytically tractable expression. We nowexpress Vanna and Volga of Plain Vanilla Options as functions of Y :

Vanna(Y ) = Vega(Y ) Y + 2

S2

Volga(Y ) = Vega(Y ) Y2 + 2Y

3(1.25)

Working with the plain Black-Scholes Delta (1.2) and the delta-neutralATM definition and defining Yi = ln

KiKATM

we have that YATM and Y25Pand Y25C corresponding respectively to At-The-Money, 25-Delta Put, and25-Delta Call solve

YATM = 0

DFf (t, T ) N

(Y25P 12(225P 2ATM)

25P

)=

1

4

DFf (t, T ) N

(Y25C + 12(225C 2ATM)25C

)=

1

4

41

Under the assumption that 25C 25P ATM we find Y25C Y25P .In this case using equations (1.25) and (1.14), the Vanna of the RR and theVolga of the BF can be expressed as :

Vanna(RR) = Vanna(Y25C) Vanna(Y25P ) = 2Vega(Y25C)Y25CS 2ATM

Volga(BF ) =Volga(Y25C) + Volga(Y25P )

2 Volga(0) = Vega(Y25C)Y

225C

3ATM

(1.26)

To calculate RRcost and BFcost (the difference between the price calculatedwith smile, and that calculated with a constant volatility ATM), we introducethe following convenient approximation:

Call((Y )) Call(ATM) Vega(Y ) ((Y ) ATM) (1.27)using the above, it is straightforward to show that:

RRcost 2b Vega(Y25C)Y25CBF cost cVega(Y25C) Y 225C (1.28)

Substituting expressions (1.25), (1.26) and (1.28) in the simple VV recipe(1.15) yields the following remarkably simple result:

XVV(Y ) = XBS(Y ) +Vanna(Y )

Vanna(RR)RRcost +

Volga(Y )

Volga(BF)BFcost

XBS + Vega(Y)bY + Vega(Y)cY 2 + Vega(Y)2ATM (b + cY )(1.29)

= XBS(Y ) +XBS

(Y ) ((Y ) ATM) + Vega(Y)2ATM (b + cY )

Despite the presence of a residual term, which vanishes as 0 orATM 0, the above expression shows that the Vanna-Volga price (1.15)of a vanilla option can be written as a first-order Taylor expansion of theBS price around ATM. Furthermore, as

Vanna(Y )Vanna(RR)

RRcost Vega(Y)bY andVolga(Y )Volga(BF)

BFcost Vega(Y)cY 2, the RR term (coupled to Vanna) accounts forthe impact of the linear component of the smile on the price, while the BF(coupled to Volga) accounts for the impact of the quadratic component ofthe smile on the price.

42

1.5 Market-adapted variations of Vanna-Volga

In this section we describe two empirical ways of adjusting the weights(pvanna, pvolga

)in (1.21). We will focus our attention on knock-out options,

although the Vanna-Volga approach can be readily generalized to optionscontaining knock-in barriers, as those can always be decomposed into twoknock-out (or vanilla) ones (through the no-arbitrage relation knock-in =vanilla knock-out).

To justify the need for the correction factors to (1.21) we argue as follows:As the knock-out barrier level B of an option is gradually moved towardthe spot level St, the BS price of a KO option must be a monotonicallydecreasing function, co