Gr´egoryRay´ee EssaysonPricingDerivativesbytakingintoaccountvolatilityandinterestratesrisks

JOINT PHD THESIS Universit´ e Libre de Bruxelles

Faculty of Sciences, Department of Mathematics Vrije Universiteit Brussel

Faculty of Economic, Political and Social Sciences and Solvay Business School

Essays on Pricing Derivatives by taking into account volatility and

interest rates risks

Gr´ egory Ray´ ee

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

Docteur en Sciences (Universit´e Libre de Bruxelles)

and

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

Supervisors:

Prof. dr. Griselda Deelstra (Universit´e Libre de Bruxelles) Prof. dr. Steven Vanduffel (Vrije Universiteit Brussel)

Academic year 2011-2012

Jury:

Prof. dr. Griselda Deelstra, Universit´e Libre de Bruxelles

Prof. dr. Pierre Devolder, Universit´e Catholique de Louvain

Prof. dr. Siegfried H¨ormann, Universit´e 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 five years have been a long and hard process, but today I feel very lucky to have lived such an interesting experience. Also I realize that I have been fortunate to receive help, advice and support from so many people. I wish to thank all these people involved.

There are four people who stand out in terms of contributions to this thesis. I am profoundly thankful to my supervisor Griselda Deelstra for giv- ing me the opportunity to realize this project and for all the support and valuable 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 Steven Vanduffel for the “cotutelle” opportunity and for all the research ideas, in particular with respect to the Chapter 4 of this thesis. I would also like to thank Fr´ed´eric Bossens and Nikos Skantzos for all the research ideas and practical advice they gave me during the first year of this thesis. Thanks for giving me the opportunity to discover the world of “Quantitative Analysts”

and 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 with you during the completion of this thesis. I would particularly like to thank Marc Levy and Xavier De Scheemaekere for all the valuable discussions we had.

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.

Gr´egory Ray´ee

May 2012

Contents

Summary 5

French Summary 8

Introduction 11

1 Vanna-Volga Methods Applied to FX Derivatives: From The-

ory to Market Practice 26

1.1 Introduction . . . . 26

1.2 Description of first-generation exotics . . . . 29

1.3 Handling Market Data . . . . 30

1.3.1 Delta conventions . . . . 31

1.3.2 At-The-Money Conventions . . . . 32

1.3.3 Smile-related quotes and the broker’s Strangle . . . . . 32

1.4 The Vanna-Volga Method . . . . 37

1.4.1 The general framework . . . . 38

1.4.2 Vanna-Volga as a smile-interpolation method . . . . 40

1.5 Market-adapted variations of Vanna-Volga . . . . 43

1.5.1 Survival probability . . . . 44

1.5.2 First exit time . . . . 45

1.5.3 Qualitative differences between γ

and γ

. . . . 46

1.5.4 Arbitrage tests . . . . 48

1.5.5 Sensitivity to market data . . . . 49

1.6 Numerical results . . . . 51

1.6.1 Definition of the model error . . . . 52

1.6.2 Shortcomings of common stochastic models in pricing exotic options . . . . 53

1.6.3 Vanna-Volga calibration . . . . 55

1.7 Conclusion . . . . 59

Appendix . . . . 60

A : Definitions of notation used . . . . 60

B : Premium-included Delta . . . . 60

2 Local Volatility Pricing Models for Long-dated FX Deriva- tives 63 2.1 Introduction . . . . 63

2.2 The three-factor pricing model with local volatility . . . . 68

2.3 The local volatility function . . . . 69

2.3.1 Forward PDE . . . . 69

2.3.2 The local volatility derivation . . . . 71

2.4 Calibrating the Local Volatility . . . . 74

2.4.1 Numerical approaches . . . . 75

2.4.2 Comparison between local volatility with and without stochastic interest rates . . . . 77

2.4.3 Calibrating the local volatility by mimicking sto-chastic volatility models . . . . 79

2.5 Hybrid volatility model . . . . 84

2.5.1 Calibration . . . . 88

2.6 Conclusion . . . . 89

3 Pricing Variable Annuity Guarantees in a Local Volatility framework 91 3.1 Introduction . . . . 91

3.2 The local volatility model with stochastic interest rates . . . . 94

3.3 Calibration . . . . 95

3.3.1 The local volatility function . . . . 95

3.3.2 The Monte-Carlo approach . . . . 97

3.3.3 Comparison between local volatility with and without stochastic interest rates . . . . 100

3.3.4 Calibrating the local volatility by mimicking stochastic volatility models . . . . 100

3.4 Variable Annuity Guarantees . . . . 102

3.4.1 Guaranteed Annuity Options . . . . 102

3.4.2 Guaranteed Minimum Income Benefit (Rider) . . . . . 105

3.4.3 Barrier GAOs . . . . 106

3.5 Numerical results . . . . 108

3.5.1 Calibration to the Vanilla option’s Market . . . . 109

3.5.2 A numerical comparison between local volatility with and without stochastic interest rates . . . . 111

3.5.3 GAO results . . . . 112

3.5.4 GMIB Rider . . . . 116

3.5.5 Barrier GAOs . . . . 118

3.6 Conclusion . . . . 123

Appendix . . . . 124

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

B : Down-and-in GAO results . . . . 127

C : Graphics . . . . 128

4 Using bounds for a faster pricing of Asian style options 135 4.1 Introduction . . . . 135

4.2 Bounds as control variates in L´evy markets . . . . 137

4.2.1 Geometric Lower Bound GLB . . . . 142

4.3 Market settings: subordinated Brownian motion . . . . 144

4.3.1 The Variance Gamma model . . . . 145

4.3.2 The Normal Inverse Gaussian model . . . . 146

4.4 ALB and GLB as an expression of power calls . . . . 148

4.4.1 Calculation of power call options in the Variance Gamma model . . . . 149

4.4.2 Calculation of power call options in the Normal Inverse Gaussian model . . . . 151

4.5 Lower bound derived using the stochastic clock . . . . 152

4.6 Numerical illustrations . . . . 155

4.6.1 Efficiency Measure . . . . 155

4.6.2 Numerical results for Asian options . . . . 156

4.7 Applications to other products . . . . 160

4.7.1 Unit Linked Insurance . . . . 160

4.7.2 Ratchet equity-indexed annuities (EIAs) . . . . 162

4.8 Conclusions . . . . 163

Conclusions 165

Summary

Over the past decade, the stock and derivative markets have often made the Journal Headlines. They are very attractive, thanks to the returns they can provide, but they are also very dangerous since they are able to cause conse- quent losses of money and are responsible of many bankruptcies, especially in the banking sector. Among the most popular derivative products, we find the European call and put options, created initially as hedging tools for fi- nancial risks, but quickly converted into products of speculation for investors interested by high profits. A call option (resp. put) is a financial contract between 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 in the future (the maturity date). This derivative product depends on a random variable, namely the price of the underlying asset at the option’s expiration date, 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. Many

researchers have attempted to solve this option pricing problem. Discrete-

time methods have been developed such as the binomial method introduced

by 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 on

continuous-time dynamics and consist in modelling the dynamics of the asset

with stochastic differential equations. The most famous model is probably

the Black and Scholes (1973) model in which the underlying asset is mod-

elled by a geometric Brownian motion. Unfortunately this model is based on

several assumptions which are not consistent with the reality and can lead to

serious pricing and hedging errors. Therefore, its use by banks and other fi-

nancial institutions is becoming rare. One of the most important problems of

the Black and Scholes model is the assumption of a constant volatility during

the lifetime of the option. Twenty years later, two famous types of models

provide a solution to this problem. The first type are stochastic volatility

models such as the Heston model (see Heston (1993)) and the second type

are the local volatility models introduced by Dupire (1994) and Derman and

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

There are also many types of so-called exotic options, among which we have the popular barrier type options that can be activated or deactivated if the underlying asset reaches a certain fixed barrier level during the lifetime of 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-Volga method. This new approach allows for a fast and easy calibration which is directly done on the barrier options market. It allows to price these options with a tool in accordance with the barrier options market. We also compare our results with those coming from the Dupire and Heston models. Further- more, we study the sensitivity of the Vanna-Volga method with respect to the market data. We give a new theoretical justification for the Vanna-Volga method. More precisely, we show that the Vanna-Volga option’s price can be seen as a first-order Taylor expansion of the Black-Scholes option price around 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 the interest rates. This relaxes the assumption of constant interest rates present in the Black-Scholes model and solves a second main problem encountered in the latter, which can have large consequences in the valuation and hedg- ing strategies especially for long maturity products. More precisely, we work in the foreign exchange (FX) market, with a local volatility model for the dynamics of the foreign exchange spot rates in which the domestic and for- eign interest rates are also assumed stochastic. We derive the expression of the local volatility and various results particularly useful for the calibration of the model. Finally, we derive useful results for the calibration of hybrid volatility models where the volatility of the FX spot rate is a mix of a local volatility and a stochastic volatility and we develop a calibration method for this 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 insurance

derivatives. Since the maturity of such options is the retirement age, they

can be considered as long maturity products. For the calibration of the local

volatility, we use a method developed in Chapter 2. Since we study exotic

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

Finally, in Chapter 4 we work with L´evy type models for the underlying dynamics. The idea underlying the L´evy model is the use of a more general stochastic process than the standard Brownian motion which allows to be in agreement with the observed market probability distribution at maturity.

In a financial crisis period, this model is especially popular since it has the

particularity to allow for jumps in the dynamics. In this chapter, we are

interested specifically in the evaluation of discretely monitored arithmetic

Asian type options whose payoff is based on the discrete arithmetic mean of

the underlying during the life of the option. As for many exotic options, it is

not possible to derive an analytical pricing formula even in the simple case

of the Black-Scholes model. In this case the only way to price such options

is by using numerical methods. In Chapter 4, we develop a method based on

Monte-Carlo simulations and we use two types of control variates to improve

the convergence. We also develop a method based on a conditioning approach

to obtain a lower bound for the Asian option price. The efficiency of this last

method outperforms the efficiency of the other methods and the results are

relatively close to the Monte-Carlo value of the corresponding Asian.

R´ esum´ e

Ces dix derni`eres ann´ees, les march´es d’actions et de produits d´eriv´es ont fait plus d’une fois l’objet des grands titres des journaux. Tr`es attractifs grˆace aux rendements qu’ils peuvent procurer, ils sont ´egalement tr`es dan- gereux, pouvant engendrer d’´enormes pertes d’argent et ˆetre responsables de nombreuses faillites, tout particuli`erement dans le secteur des banques.

Parmi les produits d´eriv´es les plus populaires, on retrouve les options d’achat et de vente Europ´eennes, cr´e´ees dans un premier temps comme objets fi- nanciers de couverture de risques, mais reconverties rapidement en produits de sp´eculations pour les investisseurs `a la recherche de profits ´elev´es. Une option d’achat (resp. de vente) est un contrat financier conclu entre un acheteur et un vendeur qui donne `a l’acheteur le droit d’achat (resp. de vente) d’un actif sous-jacent `a un prix d’exercice fix´e et `a une date pr´ecise dans le futur. Ce produit d´eriv´e d´epend d’une valeur al´eatoire, le prix de l’actif sous-jacent `a la date de maturit´e de l’option, rendant par cons´equent le prix du contrat compliqu´e `a d´eterminer. Pour calculer ce prix, il est essentiel de mod´eliser de mani`ere la plus r´eelle possible la dynamique de l’actif financier sous-jacent. De nombreux chercheurs ont tent´e de r´esoudre ce probl`eme d’´evaluation d’options. Des m´ethodes en temps discret on ´et´e d´evelopp´ees comme par exemple la m´ethode binomiale introduite par Cox et al. (1979) qui consiste `a mod´eliser la dynamique du sous-jacent via un arbre binomial, mais les m´ethodes les plus populaires aujourd’hui utilisent des dynamiques en temps continu et consistent `a mod´eliser les dynamiques de l’actif avec des ´equations diff´erentielles stochastiques. Le mod`ele le plus c´el`ebre est probablement le mod`ele de Black and Scholes (1973) dans lequel le sous-jacent est mod´elis´e par un mouvement Brownien g´eom´etrique. Ce mod`ele se base malheureusement sur certaines hypoth`eses trop ´eloign´ees de la r´ealit´e, pouvant engendrer de graves erreurs d’´evaluations et de couver- ture et par cons´equent, son utilisation par les banques et autres organismes financiers se fait de plus en plus rare. Une hypoth`ese rejet´ee par le march´e est de consid´erer la volatilit´e comme constante durant la vie de l’option.

Vingt ans plus tard, deux c´el`ebres types de mod`eles apportent une solution

`a ce probl`eme. Le premier est le mod`ele `a volatilit´e stochastique comme par exemple le mod`ele de Heston (1993) et le second est le mod`ele `a volatilit´e locale connu sous le nom de mod`ele de Dupire (1994) ou Derman and Kani (1994). Actuellement, ce sont probablement les mod`eles les plus utilis´es sur les march´es.

Il existe ´egalement de nombreux types d’options dites exotiques, parmi lesquelles on retrouve les populaires options `a barri`eres qui peuvent ˆetre ac- tiv´ees ou d´esactiv´ees si l’actif sous-jacent atteint un certain niveau de barri`ere fix´e durant la vie de l’option. Dans ces march´es exotiques, les mod`eles `a volatilit´e locale et stochastique vont avoir des comportements diff´erents et seront souvent en d´esaccord entre eux et mˆeme avec le march´e. Dans le Chapitre 1, nous pr´esentons une nouvelle approche pour ´evaluer ce type d’options exotiques bas´ee sur une m´ethode connue sous le nom de m´ethode Vanna-Volga. Cette nouvelle m´ethode nous permet une calibration simple et rapide sur le march´e des options `a barri`eres directement ce qui permet d’´evaluer ces options avec un outil en accord avec le march´e. Nous com- parons ´egalement nos r´esultats avec ceux provenant du mod`ele de Dupire et de Heston et nous ´etudions la sensibilit´e de cette m´ethode par rapport aux donn´ees du march´e. Nous donnons une nouvelle justification th´eorique as- soci´ee `a la m´ethode Vanna-Volga comme ´etant une approximation de Taylor du premier ordre du prix de l’option autour de la volatilit´e dite `a la monnaie.

Dans le Chapitre 2 de la th`ese, nous allons d´evelopper un mod`ele qui non seulement tient compte de la volatilit´e implicite du march´e mais ´egalement de la variabilit´e des taux d’int´erˆets. Ceci relˆache l’hypoth`ese de taux d’int´erˆets constants pr´esente dans le mod`ele de Black-Scholes ce qui r´esout le deuxi`eme principal probl`eme rencontr´e dans ce dernier, pouvant avoir de grandes cons´e- quences lors de l’´evaluation et de la couverture de produits `a longue matu- rit´e. Nous travaillons dans le march´e particulier des taux de changes, avec un mod`ele `a volatilit´e locale pour la dynamique du taux de change dans lequel les taux d’int´erˆets domestiques et ´etrangers sont ´egalement suppos´es stochastiques. Nous d´erivons l’expression de la volatilit´e locale et d´erivons divers r´esultats particuli`erement utiles pour la calibration du mod`ele. Finale- ment, nous d´eveloppons un nouveau mod`ele hybride o`u la volatilit´e du taux de change poss`ede une composante locale et une composante stochastique et nous d´erivons une m´ethode de calibration pour ce nouveau mod`ele.

Dans le Chapitre 3, nous allons appliquer le mod`ele `a volatilit´e locale et

taux d’int´erˆets stochastiques d´evelopp´e dans le pr´ec´edent chapitre mais dans

le cadre d’´evaluation de produits d´eriv´es associ´es aux assurances vie. Ces

options ont comme date d’´ech´eance l’ˆage de la retraite et peuvent donc ˆetre consid´er´ees comme produits `a longue maturit´e. Nous utilisons une m´ethode de calibration d´evelopp´ee dans le Chapitre 2. Les produits ´etudi´es ´etant exotiques, nous allons ´egalement comparer les prix obtenus dans diff´erents mod`eles, `a savoir le mod`ele `a volatilit´e locale, `a volatilit´e stochastique et en- fin `a volatilit´e constante pour le sous-jacent, les trois mod`eles ´etant combin´es avec des taux d’int´erˆets stochastiques.

Finalement, dans le Chapitre 4, nous allons travailler avec un mod`ele

dit de L´evy pour mod´eliser le sous-jacent. L’id´ee sous-jacente au mod`ele

de L´evy est l’utilisation d’un mouvement plus g´en´eral que le mouvement

Brownien ce qui permet d’ˆetre plus g´en´eral que le mod`ele de Black-Scholes

et d’ˆetre en accord avec la distribution de probabilit´e du march´e. Ce mod`ele

a la particularit´e d’autoriser les sauts dans la dynamique ce qui est de plus

en plus appr´eci´e et particuli`erement en temps de crise financi`ere. Nous nous

int´eressons plus pr´ecis´ement `a l’´evaluation d’options Asiatiques arithm´etiques

dont le profit se base sur la moyenne arithm´etique du sous-jacent durant la

vie de l’option. Comme de nombreuses options exotiques, il n’est pas pos-

sible d’obtenir un prix analytique mˆeme dans le cas simple du mod`ele de

Black-Scholes et dans ce cas, seules les m´ethodes num´eriques permettent de

r´esoudre le probl`eme. Dans ce Chapitre 4, nous d´eveloppons une m´ethode

bas´ee sur la m´ethode de simulations de Monte-Carlo et nous employons deux

types de variables de contrˆole permettant d’am´eliorer la convergence du pro-

gramme. Nous d´eveloppons ´egalement une m´ethode permettant d’obtenir

une borne inf´erieure au prix de l’option avec une efficacit´e qui surpasse les

autres m´ethodes, donnant des r´esultats relativement proches de la valeur ex-

acte, pouvant donc faire partie des outils d’´evaluation et de gestion du risque

utilis´es par les fournisseurs du march´e.

Introduction

Derivatives are financial contracts whose value depends on the value of other financial assets, the underlyings, which can be for example stocks, interest rates, commodities, foreign exchange rates. They can be highly risky: in- deed, because of the leverage effect, they can generate large profits but also complete losses when the market goes in the wrong direction. Recently, the financial crisis has shown how underlying asset values can dramatically vary, making the derivative market more dangerous. However, this market is still growing. The total notional amount of all the outstanding positions of over the 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 trillion US dollars (see BIS Quarterly Review, December 2011). Derivatives have played a negative role in the financial crisis mainly because of wrong pricing of complicated products. Because of the amount of money involved and the risk implied by trading these contracts, banks and insurance company need suitable 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 of

using 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) = σdB(t), where σ is the (normal) constant volatility

of the stock price and leading to a normal distribution for the stock price

return. Geometric Brownian motions were introduced by Osborne (1959)

where the stock price dynamics are defined by an exponential Brownian mo-

tion (S(t) = S

e

) avoiding negative value for stock prices. This model

leads 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

arbitrage pricing arguments and an explicit connection between the price of a derivative and a hedging strategy. Unfortunately, this model is still based on several unrealistic assumptions that render the prices usually inaccurate and 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 derivative available in the market. It is refereed as a plain vanilla option and it gives the right (but not the obligation) to the buyer to buy an agreed quantity of a particular financial instrument (the underlying) from the seller of the option at a certain time (the expiration date T , usually called the maturity) for a certain price (the strike price K). In a Black-Scholes world, the price of a call option is a function of the volatility σ which is assumed to be constant during 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) = µS(t)dt + σS (t)dB(t), (1) where µ 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 (d

) − Ke

N (d

), where

d

= ln(

) + (r +

)T σ √

T , d

= d

− σ √ 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 σ does not seem to be con-

stant at all. To show this, we use the concept of implied volatility which is

defined by the volatility (σ) one has to plug into the Black-Scholes call price

formula 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 has

the shape of a smile with a skew (see Figure 1) and is commonly called the

Figure 1: Implied volatility surface of JPY/USD

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

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

dS(t) = µS (t)dt + p

ν(t)S(t)dB

(t),

where ν (the square of the asset price volatility) follows a Cox–Ingersoll–Ross (CIR) process, namely,

dν(t) = k[λ − ν(t)] dt + ξ p

ν(t)dB

(t)

and where B

(t) and B

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

parameters, k, λ, ξ in order to fit quite closely the market implied volatil- ity surface. This optimization problem is called the calibration of the model and 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 following SDE,

dS(t) = µS (t)dt + σ(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 and the stock price. This function is called the local volatility function. Dupire (1994) and Derman and Kani (1994) have shown that one may recover the local volatility function σ(t, S (t)) from the prices of traded plain vanilla Eu- ropean call options using the following formula:

σ(T, K) = v u

u t

+ rK

1

2

K

.

Both the Heston and the local volatility models are probably, until to-

day, the favorite models for practitioners. The main advantage of the local

volatility model is that the volatility is a deterministic function and therefore

the 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 local

volatility is calibrated on the whole implied volatility surface, local volatil-

ity models capture the whole implied volatility surface generally better than

stochastic volatility models. Local volatility models have the drawback that

the volatility of the stock returns is not assumed to be stochastic whereas

real market volatilities exhibit stochastic behavior. However, when pricing

vanilla options, the Heston and the local volatility models are both able to

return market consistent prices since they are both able to being calibrated

on the implied volatility surface. The problem comes once you deal with

path dependent options. For example, in the case of barrier type options,

i.e. European options that can be activated or deactivated if the underlying

reaches 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

spot density probability function (f

(x)) but also a function of the tran- sition probability density functions of the spot at time t conditional on the level of the spot at time s, 0 ≤ s < t ≤ T (f

(x), 0 ≤ s < t ≤ T )). The implied volatility surface provides full information about the terminal spot density (f

(K) = e

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

Hybrid volatility models that combine both stochastic volatility and local volatility dynamics could better capture market dynamics. This approach has 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 hybrid volatility model is defined by the following risk-neutral dynamics for the spot prices S(t):

dS(t) = (r − q)S(t)dt + ν(t)σ(t, S(t))S(t)dB

(t),

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

d ln(ν(t)) = k[λ(t) − ln(ν(t))] dt + ξdB

(t),

where dB

(t) and dB

(t) are uncorrelated

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

However, such models are known to be complex for the implementation.

They require numerical methods which are computationally demanding and their calibration is delicate. An alternative solution is the Vanna-Volga method which is not a rigorous model but an “ad-hoc” pricing technique that 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 since the dependence of volatility on the stock price is already captured in the local volatility functionσ(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 the correlation and from the local volatility function.

Black-Scholes price with a smile impact correction analytically derived from a portfolio composed of three vanilla strategies, which zeros out the Vega

, Vanna

and Volga

of the option at hand. The Vanna-Volga method arises from the idea that the smile adjustment to an option price is associated with the costs incurred by hedging its volatility risk. This method has been studied especially 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 justification for 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 expansion of 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 price barrier options. This method allows for a fast calibration on the barrier op- tions market and therefore it offers a consistent market pricing system for that type of options. Furthermore we investigate how this method behaves in pricing with respect to more rigorous models, namely, the local volatility model and the Heston stochastic volatility model. Because the Vanna-Volga method is strongly sensible to options market data, we present some of the relevant FX conventions.

All the models presented above are based on the assumption that interest rates remain constant throughout the life of the option. This assumption is unfortunately wrong in a real market (Figure 2 is a plot of the overnight Euro LIBOR interest rate

, illustrating the stochastic behavior of interest rates) and can have a significant impact on the evolution of the stock price as well as on the price of the option and the corresponding hedging strate- gies. Especially when the financial sector is in crisis, Central Banks usually modify the level of interest rates and this may have an impact on the stock value. For example, if they decrease the level of interest rates, investors may be interested to invest in stocks because of the poor return on bank account.

Market practice seems to avoid to work with stochastic interest rates when dealing with short-dated derivatives (less than one year) since it doesn’t lead to 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 σ, namely, ^∂P_∂σ.

3The Vanna is the sensitivity of the Vega of a derivative to a change in the underlying instrument priceS, namely, ^{∂V ega}_∂S .

4The Volga is the sensitivity of the Vega of a derivative to a change in the volatilityσ, namely, ^{∂V ega}_∂σ .

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

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

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

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

dr(t) = [θ(t) − α(t)r(t)]dt + σ

(t)dB

(t), (2) where B

(t) is a Brownian motion under the risk neutral measure Q, θ(t) is a function of time determining the average direction in which r moves and is chosen such that movements in r are consistent with today’s zero coupon yield curve. The deterministic parameters α(t) and σ

(t) are the mean re- version rate and the volatility of r(t), respectively. Note that in practice α and σ

are usually assumed to be constant rather than time dependent because this setting allows for a better calibration to the market of interest rate derivatives (see Hull and White (1995) and Brigo and Mercurio (2006)).

In order to price long-dated options, researchers and practitioners have

studied models which combine both stochastic interest rates and stochastic

stock prices. The first approach studied was probably the two-factor model

which combines a geometric Brownian motion for the stock price S(t) (see

equation (1)) with Gaussian stochastic interest rates using for example the Hull and White one-factor Gaussian model given by equation (2). Such models allow for analytical option pricing formulae (see for example Merton (1973), Rabinovitch (1989) and Amin and Jarrow (1991,2)) but do not take into account the smile effect. Following the same idea as Heston (1993) but in a stochastic interest rates framework, many researchers as for example van Haastrecht et al. (2009) or Grzelak and Oosterlee (2011) have studied the three-factor model assuming the volatility of the spot stochastic as well.

The local volatility approach including stochastic interest rates has been first studied by Atlan (2006) where the local volatility function σ(t, S(t)) is derived and is expressed in function of prices of traded plain vanilla European Call options and an expectation unfortunately not directly related to liquid market products, namely,

σ(T, K) = v u

u t

+ KP (0, T )E

[r(T )1

]

1

2

K

, (3)

where P (0, T ) is the price of a zero-coupon bond maturing at time T and where Q

is the T -forward measure associated to the zero-coupon bond as numeraire.

When pricing foreign exchange options (FX options), one has to consider domestic and foreign interest rates since they both have an impact on the dynamics of the spot FX rate. In Chapter 2, we study the three-factor model with local volatility which consists of using a local volatility for the volatility of the spot FX rate and where domestic and foreign interest rates denoted by r

(t) and r

(t) respectively follow a Hull-White one-factor Gaussian model.

This model has been considered in Piterbarg (2006) in which the author derives 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 implied volatility surface but not the convexity. In Chapter 2, we derive the exact local volatility function

:

6The expression of the local volatility (4) is derived under the risk neutral measureQ.

However, it is sometimes necessary to be able to calibrate the model under the real world measure as for example in risk management where real world asset dynamics are often needed. The derivation of the local volatility could be done under the real probability measure using the same approach as in Chapter 2. This derivation and the calibration of the model under the real world measure is left for future research.

σ(T, K) = v u

u t

− P

(0, T )E

[(r

(T )K − r

(T )S(T ))1

]

1

2

K

, (4) where P

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

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

.

Recall that local volatility models as well as stochastic volatility models with stochastic interest rates will be consistent with the vanilla market after being calibrated on the implied volatility surface but this is by no means a guarantee that these models will price correctly path dependent options in 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 rate framework. One has derived a link between this four-factor hybrid volatility model and the “pure” local volatility model with stochastic interest rates.

Using that link, we have developed a calibration procedure for the local volatility function associated to this hybrid volatility model which is based

7This link is particularly useful when the market is not liquid. More precisely, the calibration of the local volatilty (4) give stable results only in a liquid market since you have 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 link between the local volatility model and the stochastic volatility model for the calibration of the local volatility function. However, if the calibration of both stochastic volatility and local volatility is not possible, an alternative is to use a Regime-Switching Model, introduced in Hamilton (1989).

on the knowledge of the local volatility derived from the three-factor model with 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 the local volatility model against the Sch¨obel-Zhu stochastic volatility model and the Black-Scholes constant volatility model all coupled with Hull-White stochastic 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 of stocks and bonds (see for example Gao (2010) and Schrager and Pelsser (2004)) and they offer a range of options to give minimum guarantees and protect against negative equity movement.

Before pricing these derivatives and compare the results given by the three different 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 to apply the numerical integration method based on Monte-Carlo simulations rather than using a PDE solver (in order to compute numerically the expec- tation present in the local volatility expression). Note that, the calibration method based on the adjustment of the tractable local volatility surface com- ing from the one-factor model with constant interest rates requires market data in order to determine the covariance between interest rates r(t) and the indicator function 1

. Unfortunately, we do not dispose of such market data. The last calibration method mentioned consists of calibrating the local volatility from a stochastic volatility model already calibrated. The calibration of the stochastic volatility model especially when interest rates are stochastic is a difficult task and once the calibration is done, the local volatility can be calibrated by using the link with stochastic volatility models.

The link is given by a conditional expectation which is difficult to compute numerically by using traditional numerical methods. However, Malliavin cal- culus allow for representations of such conditional expectation that can be computed efficiently by Monte-Carlo simulations (see Fourni´e et al. (2001)).

We have preferred to directly calibrate the local volatility function from the

available implied volatility by using a Monte-Carlo simulations method. The

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 assumption for the distribution of the stock price is not consistent with the real world market behavior. Local volatility and stochastic volatility models are able to deal with this reality and fit the distribution exhibit by the market. Another famous solution is to replace the (Black-Scholes) geometric Brownian motion by an exponential non-normal L´evy process.

A L´evy process {X

, t ≥ 0}, is a stochastic process where increments are independent (i.e. X

− X

is independent of X

, 0 ≤ u < s < t < +∞) and stationary, meaning that the distribution of increments does not depend on time but only on time distance (i.e. X

− X

has the same distribution as X

, 0 ≤ s < t < +∞). The process starts at 0 (X

= 0 almost surely) and is stochastically continuous in the sense that the probability of a jump at time t is null (lim

h→0

P (|X

− X

| ≥ ε) = 0). Finally, the path of a L´evy process is “Continue `a Droite et Limite `a Gauche” (C`adl`ag)

ensuring that the path does not reach infinity and the absence of left continuity allows the process to have jumps. This last property is particularly consistent with several markets which exhibit jumps especially in a crisis period. A Brown- ian motion is a L´evy process where the distribution of increments X

− X

, 0 ≤ s < t < +∞ are normally distributed and which doesn’t allow for jumps since the paths of the process are assumed to be almost surely continuous.

An empirical study in Mandelbrot (1963) had already revealed that the assumption of log-normality distribution for the stock prices is not suitable and he was probably one of the first to propose to use an exponential non- normal L´evy process. The idea proposed by Mandelbrot was to replace the Brownian motion by a symmetric α-stable L´evy motion with index α < 2 which yields to a pure jump process for the stock price

. Afterwards, other types of exponential Levy processes appeared where the log price process is a combination of a Brownian motion and an independent jump process given by for example a compound Poisson process with normally distributed jumps (see for example Press (1967)) and Merton (1976)). These processes are not pure jump stock processes.

8RCLL (“right continuous with left limits”), or corlol (“continuous on (the) right, limit on (the) left”)

9A Normal distribution is anα-stable distribution whereα= 2

More recently, a new category of pure jump Levy processes has been stud- ied, namely, time-changed Brownian motions which generalized the Brownian motion by making the time itself stochastic. More precisely, the Brownian motion B(t) is replaced by B(G

) where {G

, t ≥ 0} is another positive in- creasing stochastic process commonly referred as the stochastic clock or the subordinator. For example Generalized Hyperbolic Levy processes form a class of L´evy process that can be represented as a time-changed Brownian motion where the stochastic clock follows a generalized inverse Gaussian dis- tribution. It has been introduced by Barndorff-Nielsen (1977) in order to describe a phenomenon in physics (the migration of sand-dunes) and then introduced in finance by Eberlein and Keller (1995). The Variance Gamma (VG) model which can be track back to McLeish (1982) and the Normal Inverse Gaussian model are two sub-classes of Generalized Hyperbolic L´evy models (see e.g. Schoutens (2003) and Raible (2000)). The VG and the NIG processes are pure jump processes (like α-stable L´evy processes) which provide an excellent fit to empirically observed log-return distributions (see Madan 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 and the price of a Europen call opion of maturity T and strike K can easily be computed by using the formula derived in Carr and Madan (1998), namely,

C(K, T ) = e

π

Z

0

e

ψ(v )dv (5)

where

ψ(v ) = e

φ(v − (α + 1)i)

α

+ α − v

+ i(2α + 1)v (6)

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

An Asian option, also called average price option, is an exotic option

where the payoff is determined by the average underlying price over the

life of the option. More precisely, the payout of an Asian call is given by

max(A(0, T ) − K, 0) where A(0, T ) denotes the average of the underlying

price between time 0 and time T , and K is the strike. Since the payoff

depends of the average trajectory of the underlying, Asian options reduce the risk of market manipulation of the price underlying at maturity

. An Asian option is also cheaper than the corresponding European option since the value is a function of the average of the underlying path which reduce the volatility inherent in the option. There exist two types of averaging, namely, the arithmetic and the geometric average and the average can be considered in a continuous or discrete monitoring case. Closed-form solu- tions exists for geometric Asian options even within Levy models. In Fusai and Meucci (2008) the authors provide closed-form solutions for geometric Asian types in terms of the Fourier transform when the underlying evolves according to a generic L´evy process. Arithmetic Asian options are more complicated to value. There exists no closed-form formula even not in the Black-Scholes framework and therefore pricing this type of Asian options requires numerical methods. One of the most popular method is the Monte- Carlo simulation method introduced in finance by Boyle (1977) and used by many authors for the pricing of Asian options as for example Kemna and Vorst (1990) and Fusai and Meucci (2008). Monte-Carlo integration meth- ods converge asymptotically to the true value. Nevertheless, with a square root rate of convergence it is often slow and in its basic form, the method is usually computationally inefficient. However, the method can be improved by 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 averaged Asian options under the VG and NIG models by using Monte-Carlo simu- lation with control variates as variance reduction method. In this chapter we make two contributions. The first contribution is that we develop several multi-control variate methods where the control variates are based on lower bounds derived in Albrecher et al. (2008) or in a more straightforward way in Deelstra et al. (2012). These methods highly increase the speed of conver- gence with respect to the crude Monte-Carlo method and therefore provide a useful tool for Asian market dealers. The second contribution consists of a two steps pricing method which provide a fast and accurate method for the computation of tight lower bounds in a VG and NIG setting. The lower bound is calculated by first conditioning on the stochastic clock which reduces to a log-normal distribution for the stock price and allows to compute the lower bound analytically (given in Vanduffel et al. (2009)). Afterwards, the second step consists of integrating this analytical formula over the stochastic

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

clock “domain”. The integration is also done by using multi-control variates Monte-Carlo simulation methods developed in the first contribution part.

The resulting lower bound is quite close to the true Asian price and the Monte-Carlo method with multi-control variates converges so fast that the efficiency of this second approach surpasses all the other methods developed in this chapter. It is well known that financial institutions who deal with op- tions need fast and accurate answers for option prices and associated Greeks in order to hedge the risk linked to these highly risky products. Therefore the last method which offers an attractive trade-off between accuracy and efficiency 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 Practice”

co-authored with Fr´ed´eric 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 for the case of vanilla options and we show how to adapt the method to the pricing 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 compare

our results to a large collection of indicative market prices and to local and

stochastic volatility models. Because the Vanna-Volga price is strongly sen-

sible to some market data, we provide a summary of the relevant FX market

data conventions. In the second chapter, “Local Volatility Pricing Models

for Long-dated FX Derivatives” co-authored with Griselda Deelstra, to ap-

pear in Applied Mathematical Finance, we study the local volatility function

in the Foreign Exchange market where both domestic and foreign interest

rates are stochastic. We derive the local volatility function as well as several

results that can be used for the calibration of the local volatility on the FX

option’s market. Finally, a calibration method is derived for a model which

combines both local and stochastic volatility in a stochastic interest rates

framework. In the third chapter, “Pricing Variable Annuity Guarantees in a

Local Volatility framework”, co-authored with Griselda Deelstra, submitted

to Insurance: Mathematics and Economics, we study the price of long-dated

insurance derivatives, namely, Variable Annuity Guarantees in a local volatil-

ity framework with stochastic interest rates. The local volatility function is

calibrated 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 an

appropriate volatility modelling is important for the pricing of these long-

dated derivatives. Furthermore, the model choice as well as the market data

have a strong influence on the price. The fourth chapter, “Using bounds for

a faster pricing of Asian style options” co-authored with Griselda Deelstra,

Steven Vanduffel and Jing Yao, proposes a Monte-Carlo pricing method with

control variates variance reduction techniques for arithmetic Asian options

(discretely sampled). We use lower bounds as control variates to improve

Monte-Carlo simulations significantly. We also propose a new pricing ap-

proach based on conditioning on the clock. Efficiency of both approaches are

confirmed by numerical experiments. Finally, we outline how the results can

be useful for the pricing of Asian related insurance products.

Chapter 1

Vanna-Volga Methods Applied to FX Derivatives: From

Theory to Market Practice

Abstract

We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black-Scholes price through the so-called ‘probability of survival’ and the ‘expected first exit time’. Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data.

1.1 Introduction

The Foreign Exchange (FX) option’s market is the largest and most liquid

market of options in the world. Currently, the various traded products range

from simple vanilla options to first-generation exotics (touch-like options and

vanillas with barriers), second-generation exotics (options with a fixing-date

structure or options with no available closed form value) and third-generation

exotics (hybrid products between different asset classes). Of all the above the

first-generation products receive the lion’s share of the traded volume. This

makes it imperative for any pricing system to provide a fast and accurate mark-to-market for this family of products. Although using the Black-Scholes model Black and Scholes (1973); Merton (1973) it is possible to derive an- alytical prices for barrier- and touch -options, this model is unfortunately based on several unrealistic assumptions that render the price inaccurate. In particular, the Black-Scholes model assumes that the foreign/domestic inter- est rates and the FX-spot volatility remain constant throughout the lifetime of 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, to ensure that the involved counterparties are speaking of the same option.

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

Stochastic volatility models are unfortunately computationally demand- ing and in most cases require a delicate calibration procedure in order to find the value of parameters that allow the model reproduce the market dynamics.

This has led to alternative ‘ad-hoc’ pricing techniques that give fast results and are simpler to implement, although they often miss the rigor of their stochastic siblings. One such approach is the ‘Vanna-Volga’ (VV) method that, in a nutshell, consists in adding an analytically derived correction to the Black-Scholes price of the instrument. To do that, the method uses a small number of market quotes for liquid instruments (typically At-The-Money options, Risk Reversal and Butterfly strategies) and constructs an hedging portfolio which zeros out the Black-Scholes Vega, Vanna and Volga of the option. The choice of this set of Greeks is linked to the fact that they all offer a measure of the option’s sensitivity with respect to the volatility, and therefore the constructed hedging portfolio aims to take the ‘smile’ effect into account.

The Vanna-Volga method seems to have first appeared in the literature in

Lipton and McGhee (2002) where the recipe of adjusting the Black-Scholes

value by the hedging portfolio is applied to double-no-touch options and in

Wystup (2003) where it is applied to the pricing of one-touch options in foreign exchange markets. In Lipton and McGhee (2002), the authors point out its advantages but also the various pricing inconsistencies that arise from the non-rigorous nature of the technique. The method was discussed more thoroughly in Castagna and Mercurio (2007) where it is shown that it can be used as a smile interpolation tool to obtain a value of volatility for a given strike while reproducing exactly the market quoted volatilities. It has been further analyzed in Fisher (2007) where a number of corrections are suggested to handle the pricing inconsistencies. Finally a more rigorous and theoretical justification is given by Shkolnikov (2009) where, among other directions, the method is extended to include interest-rate risk.

A crucial ingredient to the Vanna-Volga method, that is often overlooked in the literature, is the correct handling of the market data. In FX markets the 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 are at least four different definitions for at-the-money strike (resp., ‘spot’, ‘for- ward’, ‘delta neutral’, ‘50 delta call’). Using the wrong definition can lead to significant errors in the construction of the smile surface. Therefore, before we begin to explore the effectiveness of the Vanna-Volga technique we will briefly 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 its resulting prices against prices provided by renowned FX market makers, and against more sophisticated stochastic models. We attempt to cover a broad range of market conditions by extending our comparison tests into two dif- ferent ‘smile’ conditions, one with a mild skew and one with a very high skew. We also describe two variations of the Vanna-Volga method (used by the market) which tend to give more accurate prices when the spot is close to a barrier. We finally describe a simple adjustment procedure that allows the Vanna-Volga method to provide prices that are in good agreement with the market for a wide range of exotic options.

To begin with, in section 1.2 we describe the set of exotic instruments that we will use in our comparisons throughout. In section 1.3 we review the market practice of handling market data. Section 1.4 lays the general ideas underlying the Vanna-Volga adjustment, and proposes an interpretation of the method in the context of Plain Vanilla Options. In sections 1.5.1 and 1.5.2 we review two common Vanna-Volga variations used to price exotic options.

The main idea behind these variations is to reduce Vanna-Volga correction