analysis geometry and modeling in finance advanced methods in option pricing pdf

Rama Cont Center for Financial Engineering Columbia University New York Published Titles American-Style Derivatives; Valuation and Computation, Jerome Detemple Analysis, Geometry, and Mo

Analysis, Geometry, and Modeling in Finance

Advanced Methods in Option Pricing

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Published Titles

American-Style Derivatives; Valuation and Computation, Jerome Detemple

Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,

Pierre Henry-Labordère

Credit Risk: Models, Derivatives, and Management, Niklas Wagner

Engineering BGM, Alan Brace

Financial Modelling with Jump Processes, Rama Cont and Peter Tankov

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Quantitative Fund Management, M A H Dempster, Georg Pflug, and Gautam Mitra

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Ludger Overbeck

Understanding Risk: The Theory and Practice of Financial Risk Management,

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Analysis, Geometry, and

Modeling in Finance

Advanced Methods in

Option Pricing

Pierre Henry-Labordère

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Taylor & Francis Group

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1 Options (Finance) Mathematical models I Title II Series.

List of Tables

2.1 Example of one-factor short-rate models 47

5.1 Example of separable LV models satisfying C(0) = 0 124

5.2 Feller criteria for the CEV model 130

6.1 Example of SVMs 151

6.2 Example of metrics for SVMs 155

8.1 Example of stochastic (or local) volatility Libor market models 208 8.2 Libor volatility triangle 214

9.1 Feller boundary classification for one-dimensional Itˆo processes 267 9.2 Condition at z = 0 276

9.3 Condition at z = 1 276

9.4 Condition at z = ∞ 278

9.5 Example of solvable superpotentials 278

9.6 Example of solvable one-factor short-rate models 279

9.7 Example of Gauge free stochastic volatility models 281

9.8 Stochastic volatility models and potential J (s) 284

10.1 Example of potentials associated to LV models 295

11.1 A dictionary from Malliavin calculus to QFT 310

B.1 Associativity diagram 360

B.2 Co-associativity diagram 360

List of Figures

(03-09-2007) The two axes represent the strikes and the maturity

dates Spot S0= 4296 2

4.1 Manifold 80

4.2 2-sphere 82

4.3 Line bundle 89

5.1 Comparison of the asymptotic solution at the first-order (resp second-order) against the exact solution (5.23) f0 = 1, σ = 0.3, β = 0.33, τ = 10 years 133

5.2 Comparison of the asymptotic solution at the first-order (resp second-order) against the exact solution (5.23) f0 = 1, σ = 0.3, β = 0.6, τ = 10 years 134

5.3 Market implied volatility (SP500, 3-March-2008) versus Dupire local volatility (multiplied by ×100) T = 1 year Note that the local skew is twice the implied volatility skew 137

5.4 Comparison of the asymptotic implied volatility (5.41) at the zero-order (resp first-order) against the exact solution (5.42) f0= 1, σ = 0.3, τ = 10 years, β = 0.33 142

5.5 Comparison of the asymptotic implied volatility (5.41) at the zero-order (resp first-order) against the exact solution (5.42) f0= 1, σ = 0.3%, τ = 10 years, β = 0.6 143

6.1 Poincar´e disk D and upper half-plane H2with some geodesics In the upper half-plane, the geodesics correspond to vertical lines and to semi-circles centered on the horizon =(z) = 0 and in D the geodesics are circles orthogonal to D 169

6.2 Probability density p(K, T |f0) = ∂2C(T,K)∂2K Asymptotic solu-tion vs numerical solusolu-tion (PDE solver) The Hagan-al formula has been plotted to see the impact of the mean-reverting term Here f0is a swap spot and α has been fixed such that the Black volatility αf0β−1= 30% 172

6.3 Implied volatility for the SABR model τ = 1Y α = 0.2, ρ = −0.7, ν2 2τ = 0.5 and β = 1 174

7.1 Basket implied volatility with constant volatilities ρij = e−0.3|i−j|,

0.5, 5 × 15 and 10 × 10 using our asymptotic formula and the

5 × 15 and 10 × 10 using our asymptotic formula and a MC

(EUR, JPY, USD curves-February 17th 2007) using our

(EUR, JPY-February 17th 2007) using our asymptotic formula

10 (EUR-February 17th 2007) using different value of (ν, ρ).The LMM has been calibrated to caplet smiles and the ATM

P Probability measure,

usu-ally the risk-neutral

mea-sure

the dates t and T

and Tβ

dates Ti−1and Ti

Mo-tion

2.1 Derivative products 7

2.2 Back to basics 9

2.2.1 Sigma-algebra 9

2.2.2 Probability measure 10

2.2.3 Random variables 10

2.2.4 Conditional probability 12

2.2.5 Radon-Nikodym derivative 13

2.3 Stochastic processes 13

2.4 Itˆo process 15

2.4.1 Stochastic integral 15

2.4.2 Itˆo’s lemma 19

2.4.3 Stochastic differential equations 21

2.5 Market models 24

2.6 Pricing and no-arbitrage 25

2.6.1 Arbitrage 26

2.6.2 Self-financing portfolio 26

2.7 Feynman-Kac’s theorem 32

2.8 Change of num´eraire 34

2.9 Hedging portfolio 41

2.10 Building market models in practice 43

2.10.1 Equity asset case 43

2.10.2 Foreign exchange rate case 45

2.10.3 Fixed income rate case 46

2.10.4 Commodity asset case 49

2.11 Problems 50

3 Smile Dynamics and Pricing of Exotic Options 55 3.1 Implied volatility 55

3.2 Static replication and pricing of European option 57

3.3 Forward starting options and dynamics of the implied volatility 62 3.3.1 Sticky rules 62

3.3.2 Forward-start options 63

3.3.3 Cliquet options 63

3.3.4 Napoleon options 64

3.4.1 Bond 64

3.4.2 Swap 65

3.4.3 Swaption 66

3.4.4 Convexity adjustment and CMS option 68

3.5 Problems 70

4 Differential Geometry and Heat Kernel Expansion 75 4.1 Multi-dimensional Kolmogorov equation 75

4.1.1 Forward Kolmogorov equation 76

4.1.2 Backward Kolmogorov’s equation 78

4.2 Notions in differential geometry 80

4.2.1 Manifold 80

4.2.2 Maps between manifolds 81

4.2.3 Tangent space 82

4.2.4 Metric 83

4.2.5 Cotangent space 84

4.2.6 Tensors 85

4.2.7 Vector bundles 88

4.2.8 Connection on a vector bundle 90

4.2.9 Parallel gauge transport 93

4.2.10 Geodesics 94

4.2.11 Curvature of a connection 101

4.2.12 Integration on a Riemannian manifold 102

4.3 Heat kernel on a Riemannian manifold 103

4.4 Abelian connection and Stratonovich’s calculus 107

4.5 Gauge transformation 108

4.6 Heat kernel expansion 110

4.7 Hypo-elliptic operator and H¨ormander’s theorem 116

4.7.1 Hypo-elliptic operator 116

4.7.2 H¨ormander’s theorem 117

4.8 Problems 119

5 Local Volatility Models and Geometry of Real Curves 123 5.1 Separable local volatility model 123

5.1.1 Weak solution 124

5.1.2 Non-explosion and martingality 126

5.1.3 Real curve 131

5.2 Local volatility model 134

5.2.1 Dupire’s formula 134

5.2.2 Local volatility and asymptotic implied volatility 136

5.3 Implied volatility from local volatility 145

Complex Curves 149

6.1 Stochastic volatility models and Riemann surfaces 149

6.1.1 Stochastic volatility models 149

6.1.2 Riemann surfaces 153

6.1.3 Associated local volatility model 157

6.1.4 First-order asymptotics of implied volatility 159

6.2 Put-Call duality 162

6.3 λ-SABR model and hyperbolic geometry 164

6.3.1 λ-SABR model 165

6.3.2 Asymptotic implied volatility for the λ-SABR 165

6.3.3 Derivation 167

6.4 Analytical solution for the normal and log-normal SABR model 176 6.4.1 Normal SABR model and Laplacian on H2 176

6.4.2 Log-normal SABR model and Laplacian on H3 178

6.5 Heston model: a toy black hole 181

6.5.1 Analytical call option 181

6.5.2 Asymptotic implied volatility 183

6.6 Problems 185

7 Multi-Asset European Option and Flat Geometry 187 7.1 Local volatility models and flat geometry 187

7.2 Basket option 189

7.2.1 Basket local volatility 191

7.2.2 Second moment matching approximation 195

7.3 Collaterized Commodity Obligation 196

7.3.1 Zero correlation 200

7.3.2 Non-zero correlation 201

7.3.3 Implementation 203

8 Stochastic Volatility Libor Market Models and Hyperbolic Geometry 205 8.1 Introduction 205

8.2 Libor market models 207

8.2.1 Calibration 208

8.2.2 Pricing with a Libor market model 216

8.3 Markovian realization and Frobenius theorem 220

8.4 A generic SABR-LMM model 224

8.5 Asymptotic swaption smile 226

8.5.1 First step: deriving the ELV 226

8.5.2 Connection 230

8.5.3 Second step: deriving an implied volatility smile 233

8.5.4 Numerical tests and comments 234

8.6 Extensions 237

8.7 Problems 239

9.1 Introduction 247

9.2 Reduction method 249

9.3 Crash course in functional analysis 251

9.3.1 Linear operator on Hilbert space 252

9.3.2 Spectrum 255

9.3.3 Spectral decomposition 256

9.4 1D time-homogeneous diffusion models 262

9.4.1 Reduction method 262

9.4.2 Solvable (super)potentials 269

9.4.3 Hierarchy of solvable diffusion processes 273

9.4.4 Natanzon (super)potentials 274

9.5 Gauge-free stochastic volatility models 279

9.6 Laplacian heat kernel and Schr¨odinger equations 284

9.7 Problems 287

10 Schr¨odinger Semigroups Estimates and Implied Volatility Wings 289 10.1 Introduction 289

10.2 Wings asymptotics 290

10.3 Local volatility model and Schr¨odinger equation 293

10.3.1 Separable local volatility model 293

10.3.2 General local volatility model 295

10.4 Gaussian estimates of Schr¨odinger semigroups 296

10.4.1 Time-homogenous scalar potential 296

10.4.2 Time-dependent scalar potential 298

10.5 Implied volatility at extreme strikes 300

10.5.1 Separable local volatility model 300

10.5.2 Local volatility model 302

10.6 Gauge-free stochastic volatility models 303

10.7 Problems 307

11 Analysis on Wiener Space with Applications 309 11.1 Introduction 309

11.2 Functional integration 310

11.2.1 Functional space 310

11.2.2 Cylindrical functions 310

11.2.3 Feynman path integral 311

11.3 Functional-Malliavin derivative 313

11.4 Skorohod integral and Wick product 317

11.4.1 Skorohod integral 317

11.4.2 Wick product 319

11.5 Fock space and Wiener chaos expansion 322

11.5.1 Ornstein-Uhlenbeck operator 324

11.6 Applications 325

11.6.2 Sensitivities 327

11.6.3 Local volatility of stochastic volatility models 332

11.7 Problems 337

12 Portfolio Optimization and Bellman-Hamilton-Jacobi Equa-tion 339 12.1 Introduction 339

12.2 Hedging in an incomplete market 340

12.3 The feedback effect of hedging on price 343

12.4 Non-linear Black-Scholes PDE 345

12.5 Optimized portfolio of a large trader 345

A Saddle-Point Method 351 B Monte-Carlo Methods and Hopf Algebra 353 B.1 Introduction 353

B.1.1 Monte Carlo and Quasi Monte Carlo 354

B.1.2 Discretization schemes 354

B.1.3 Taylor-Stratonovich expansion 356

B.2 Algebraic Setting 358

B.2.1 Hopf algebra 359

B.2.2 Chen series 363

B.3 Yamato’s theorem 365

in recent physics and mathematics to the financial field This means thatnew results are obtained when only approximate and partial solutions werepreviously available.

We present powerful tools and methods (such as differential geometry, spectraldecomposition, supersymmetry) that can be applied to practical problems

in mathematical finance Although encountered across different domains intheoretical physics and mathematics (for example differential geometry ingeneral relativity, spectral decomposition in quantum mechanics), they remainquite unheard of when applied to finance and allow to obtain new resultsreadily We introduce these methods through the problem of option pricing

An option is a financial contract that gives the holder the right but not theobligation to enter into a contract at a fixed price in the future The simplestexample is a European call option that gives the right but not the obligation

to buy an asset at a fixed price, called strike, at a fixed future date, calledmaturity date Since the work by Black, Scholes [65] and Merton [32] in 1973,

a general probabilistic framework has been established to price these options

In this framework, the financial variables involved in the definition of an tion are random variables and their dynamics follow stochastic differentialequations (SDEs) For example, in the original Black-Scholes-Merton model,the traded assets are assumed to follow log-normal diffusion processes withconstant volatilities The volatility is the standard deviation of a probabil-ity density in mathematical finance The option price satisfies a (parabolic)partial differential equation (PDE), called the Kolmogorov-Black-Scholes pric-ing equation, depending on the stochastic differential equations introduced tomodel the market

op-The market model depends on unobservable or observable parameters such

as the volatility of each asset They are chosen, we say calibrated, in order

to reproduce the price of liquid options quoted on the market such as

Euro-pean call options If liquid options are not available, historical data for themovement of the assets can be used.

(03-09-2007) The two axes represent the strikes and the maturity dates Spot

This book mainly focuses on the study of the calibration and the dynamics ofthe implied volatility (commonly called smile) The implied volatility is thevalue of the volatility that, when put in the Black-Scholes formula, reproducesthe market price for a European call option In the original log-normal Black-Scholes model, regarding the hypothesis of constant volatility, the impliedvolatility is flat as a function of the maturity date and the strike However,the implied volatility observed on the market is not flat and presents a U-shape (see Fig 1.1) This is one of the indications that the Black-Scholesmodel is based on several unrealistic assumptions which are not satisfied un-der real market conditions For example, assets do not follow log-normal

processes with constant volatilities as they possess “fat tail” probability sities These unrealistic hypotheses, present in the Black-Scholes model, can

den-be relaxed by introducing more elaborate models called local volatility modelsand stochastic volatility models that, in most cases, can calibrate the form ofthe initially observed market smile and give dynamics for the smile consistentwith the fair value of exotic options To be thorough, we should mentionjump-diffusion models Although interesting and mathematically appealing,

we will not discuss these models in this book as they are extensively discussed

in [9]

On the one hand, local volatility models (LVMs) assume that the volatility

derivatives of European call options with respect to the time to maturity andthe strike, which can exactly reproduce the current market of European calloption prices Dupire then shows how to get the calibration solution However,the dynamics is not consistent with what is observed on the market In thiscontext, a better alternative is to introduce (time-homogeneous) stochasticvolatility models

On the other hand, stochastic volatility models (SVMs) assume that thevolatility itself follows a stochastic process LVMs can be seen as a degen-erate example of SVMs as it can be shown that the local volatility functionrepresents some kind of average over all possible instantaneous volatilities in

a SVM

For LVMs or SVMs, the implied volatility can be obtained by various methods.For LVMs (resp SVMs) European call options satisfy the Kolmogorov andBlack-Scholes equation which is a one-dimensional (resp two-dimensional ormore) parabolic PDE This PDE can be traditionally solved by a finite differ-ence scheme (for example Crank-Nicholson) or by a Monte-Carlo simulationvia the Feynman-Kac theorem These pricing methods turn out to be fairlytime-consuming on a personal computer and they are generally not appropri-ate when one tries to calibrate the model to a large number of (European)options In this context, it is much better to use analytical or asymptoticalmethods This is what the book is intended for

For both local and stochastic volatility models, the resulting Black-ScholesPDE is complicated and only a few analytical solutions are known Whenexact solutions are not available, singular perturbative techniques have beenused to obtain asymptotic expressions for the price of European-style options

We will explain how to obtain these asymptotic expressions, in a unified ner, with any kind of stochastic volatility models using the heat kernel expan-sion technique on a Riemannian manifold

man-In order to guide the reader, here is the general description of the book’schapters:

• A quick introduction to the theory of option pricing: The purpose is toallow the reader to acquaint with the main notions and tools useful for pricing

martingales and change of measures Problems have been added at the end

so that the reader can check his understanding

• A recall of a few definitions regarding the dynamics of the implied ity: In particular, we review the different forward starting options (cliquet,Napoleon ) and options on volatility that give a strong hint on the dynamics

volatil-of the smile (in equity markets) A similar presentation is given for the study

of the dynamics of swaption implied volatilities using specific interest rateinstruments

• A review of the heat kernel expansion on a Riemannian manifold endowedwith an Abelian connection: thanks to the rewriting of the Kolmogorov-Black-Scholes equation as a heat kernel equation, we give a general asymptotic so-lution in the short-time limit to the Kolmogorov equation associated to a

differen-tial geometry, useful to grasp the heat kernel expansion, are reviewed carefully.Therefore no prerequisite in geometry is needed

• A focus on the local and stochastic volatility models: In the geometricalframework introduced in the previous chapter, the stochastic (resp local)volatility model corresponds to the geometry of complex (resp real) curves(i.e., Riemann surfaces) For example, the SABR stochastic volatility model,particularly used in the fixed-income market, can be associated to the geom-

we obtain a general asymptotic implied volatility for any SVM, in particularthe SABR model with a mean-reverting drift

• Applications to the pricing of multi-asset European options such as equitybaskets, Collateralized Commodity Obligations (CCO) and swaptions: In par-ticular, we find an asymptotic implied volatility formula for a European basketwhich is valid for a general multi-dimensional LVM and an asymptotic swap-tion implied volatility valid for a stochastic volatility Libor market model Inthis chapter, we review the main issues on the construction and calibration of

a Libor market model

• A classification of solvable LVMs and SVMs: Solvable means that the price

of a European call option can be written in terms of hypergeometric tions Recasting the Black-Scholes-Kolmogorov PDE in a geometrical setting

func-as described in chapter 4, we show how to reduce the complexity of thisequation The three main ingredients are the group of diffeomorphisms, thegroup of gauge transformations and the supersymmetry For LVMs, thesethree transformations allow to reduce the backward Kolmogorov equation to

classification of solvable potentials is already known This chapter illustratesthe power of the differential geometry approach as it reproduces and enlargesthe classification of solvable LVMs and SVMs In addition, a new useful tool

is introduced: the spectral decomposition of unbounded linear operators Areview of the theory of unbounded operators on a Hilbert space will be given

in this context

• Chapter 10 studies the large-strike behavior of the implied volatility for

LVMs and SVMs We use two-sided Gaussian estimates of Schr¨odinger tions with scalar potentials belonging to the Kato class.

equa-• In chapter 11, we give a brief overview of the Malliavin calculus We focus

on two applications: Firstly, we obtain probabilistic representations of tivities of derivatives products according to model parameters Secondly, weshow how to compute by Monte-Carlo simulation the local volatility functionassociated to SVMs

sensi-• In the last chapter, our asymptotic methods are applied to non-linear PDEs,mainly the Bellman-Hamilton-Jacobi equation We focus on the problem ofpricing options when the market is not complete or is illiquid In the lattercase, the hedging strategy of a large trader has an impact on the market Theresulting Black-Scholes equation becomes a non-linear PDE

• Two appendices are included The first one summarizes the saddle-pointmethod and the second one briefly explains Monte-Carlo methods In thispart, we highlight the Hopf algebra structure of Taylor-Stratonovich expan-sions of SDEs This allows to prove the Yamato theorem giving a necessaryand sufficient condition to represent asset prices as functionals of Brownianmotions

Throughout this book, we have tried to present not only a list of theoreticalresults but also as many as possible numerical ones The numerical imple-

have been added at the end of most chapters They are based on recentlypublished research papers and allow the reader to check his understandingand identify the main issues arising about the financial industry

Book audience

In the derivatives finance field, one can mainly identify four types of sionals

profes-• The structurer who designs the derivatives products

• The trader who is directly in contact with the market and daily brates the computer models He determines the current prices of theproducts and the necessary hedging

cali-• The salespeople who are in charge of selling the products at the tions decided by the trader

condi-• The quantitative analyst who is someone with a strong background inmathematics and is in charge of developing the mathematical models

& methods and computer programs aiming at the optimum pricing andhedging of the financial risks by the traders

Among the above mentioned professionals, this book will be useful to tative analysts and highly motivated traders to better understand the modelsthey are using.

quanti-Also, many Ph.D students in mathematics and theoretical physics move tofinance as quantitative analysts A purpose of this book is to explain the newapplications of some advanced mathematics to better solve practical problems

in finance

It is however not intended to be a full monograph on stochastic differentialgeometry Detailed proofs are not included and are replaced by relevant ref-erences

Acknowledgments

writing of this book I would like to thank my colleagues in the Equity tives Quantitative research team at Soci´et´e G´enerale for useful discussions andfeedback on the contents of this book

Deriva-Finally, this book could not have been written without the support from my

About the author

research team at Soci´et´e G´en´erale as a quantitative analyst After receiving

he worked in the theoretical physics department at Imperial College (London)before moving to finance in 2004 He also graduated from Ecole Centrale Parisand holds DEAs in theoretical physics and mathematics

Chapter 2

A Brief Course in Financial

Mathematics

The enormous usefulness of Mathematics in the natural sciences

is something bordering on the mysterious and that there is norational explanation for it.1

—E Wigner

appear naturally In this chapter, we review the main ideas in mathematicalfinance and motivate the mathematical concepts

Instead of giving a general definition of a derivative product, let us present

a few classical examples The first one is a European call option on a singleasset which is a contract which gives to the holder the opportunity but notthe obligation to buy an asset at a given price (called strike) at a fixed futuredate (called maturity) Let us suppose a call option is bought with a 100 $strike and a five year maturity T = 5 The initial price of the asset, calledthe spot, is 100 $ If the stock price is 150 $ in five years, the option can beexercised and the buyer is free to purchase the 100 $ stock as the contractallows The counterpart which has sold the contract must then buy the 150 $stock on the market (if he does not have it) and sell it at 100 $ When thebuyer has received the stock for 100 $, he can sell it directly on the market

at 150 $ (we assume that there is no transaction cost) So his net earning is

50 − C$ and the bank has lost −50 + C$, C being the price of the contract

In practice, we don’t have a two-player zero sum game as the bank hedges itsrisk by investing in a portfolio with traded assets and options The earning

1 “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” in tions in Pure and Applied Mathematics, vol 13, No I (February 1960).

Communica-of the bank at the maturity date T is therefore −50 + C + πT where πT is theportfolio value at T As a result, both the option buyer and the seller canhave a positive earning at T

We define the payoff representing the potential gain at maturity date T as

the buyer will be

f (ST) − Cwhether he calls the option or not because he purchased the contract C Abank, after defining the characteristics of the option, needs to compute thefair value C and to build a hedging strategy to cover its own risk

More generally, each derivative product is characterized by a payoff depending

on financial (random) variables such as equity stocks, commodity assets, income rates and foreign exchange rates between various currencies

fixed-In practice, the derivative products are more complex than a European calloption Several characteristic features can however be outlined A derivativecan be

• European: meaning that the option can be exercised by the holder at

a specified maturity date The simplest example is the European calloption that we have presented above Another example most straightfor-ward would be given by a European put option which gives the holderthe right but not the obligation to sell an asset at a fixed price (i.e.,strike) at the maturity date T The payoff at T is

• American: meaning that the option can be exercised either within aspecific time frame or at set dates (in the latter case the option is calledBermudan) The simplest example is an American put option whichgives to the holder the right but not the obligation to sell an asset at afixed price (i.e., strike) at any time up to the maturity date

• Asian: meaning that the option depends on the path of some assets.For example, a European Asian call option on a single stock with strike

K is defined by the following payoff at the maturity date T

1

N

average of the stock price from t up to the maturity date T

• Barrier: meaning that a European option is activated if an asset hasnot or has reached a barrier level up to the maturity date.

Moreover, an option does not necessarily depend on a single asset but candepend on many financial products In this case, one says that we have amulti-asset option For example, a classical multi-asset option is a Europeanbasket option whose payoff at maturity T with strike K depends on the value

T the price of the asset i

at maturity T Increasing the complexity level, one can also consider optionsdepending on equity assets, fixed income rates and foreign exchange rates(FX) One says in this case that the option is hybrid For example, let us

coupon ci) at some specified date Ti if a stock price STi at Ti is greater (resp

F XTi1(STi− U ) + ci1(U − STi)

In the following sections, we model the financial random variables that enter

reader with no familiarity with probability theory, we have included in thenext section a reminder of definitions and results in probability Details andproofs can be found in [26]

2.2.1 Sigma-algebra

In probability, the space of events that can appear are formalized by the notion

of a σ-algebra If Ω is a given set, then a σ-algebra for Ω is a family F ofsubsets of Ω with the following properties:

The pair (Ω, F ) is called a measurable space In the following, an element of Ω

is noted ω If we replace the third condition by finite union (and intersection),

we have an algebra

Example 2.1 Borel σ-algebra

When we have a topological space X, the smallest σ-algebra generated by theopen sets is called the Borel σ-algebra that we note below B(X)

We can endow a measurable space with a probability measure

prob-In the following, all probability spaces will be assumed to be complete It

is often simpler to construct a measure on an algebra which generates theσ-algebra The issue is then to extend this measure to the σ-algebra itself

theorem

Let A be an algebra and P : A → [0, 1] be a probability measure on A There

2.2.3 Random variables

for all open sets U ∈ Rn.

Example 2.2 Continuous function

function Note that the terminology is not suitable as a r.v is actually not avariable but a function

F by definition of the measurability of X

At this stage we can introduce the expectation of a r.v First, we restrainthe definition to a particular class of r.v., the simple r.v.s (also called stepfunctions) which take only a finite number of values and hence can be written

r.v by

Note that the expectation above can be ∞ Finally, for an arbitrary r.v, X,

3 1 Ai(x) = 0 if x ∈ A i , zero otherwise.

and we set

] and EP[X−] are finite,

(Ω, F , P)denotes the set of all integrable r.v Moreover, the set of all k times integrable

Note that when the random variable X has a density p(x) on R, it is integrable

(Ω, F , P) andlet G be a sub σ-algebra of F Then the conditional expectation of X given G,

If we have two r.v X and Y admitting a probability density, the conditional

The probability to have X ∈ [x, x + dx] and Y ∈ [y, y + dy] is by definitionp(x, y)dxdy Then, we can prove that

2.2.5 Radon-Nikodym derivative

This brief review of probability theory will conclude with the Radon-Nikodymderivative which will be used when we discuss the Girsanov theorem in section2.8

Let P be a probability space on (Ω, F) and let Q be a finite measure on (Ω, F)

We say that Q is equivalent to P (denoted Q ∼ P) if Q(A) = 0 if and only ifP(A) = 0 for every A ∈ F The Radon-Nikodym theorem states that Q ∼ P

if and only if there exists a non-negative r.v X such that

Moreover X is unique P-almost surely and we note

dP

X is called the Radon-Nikodym derivative of Q with respect to P

At this stage, we can define a stochastic process Complements can be found

in [34] and [27]

In chapter 4, we will consider stochastic processes that take their values in aRiemannian manifold

In finance, an asset price is modeled by a stochastic process The past values

of the price are completely known (historical data) The information that

we have about a stochastic process up to a certain time (usually today) isformalized by the notion of filtration

is a sub σ-algebra of F , we can define the conditional expectation of the r.v

Xsaccording to the filtration {Ft}t, with s > t

EP[Xs|Ft]

By definition, this r.v is Ft-measurable This is not necessarily the case for

for every t, we will say that the process X is adapted to the filtration {Ft}t≥0.The most important example of stochastic processes is the Brownian motionwhich will be our building block for elaborating more complex stochastic pro-cesses From this Brownian motion, we can generate a natural filtration

Example 2.3 Brownian motion

defined on a probability space (Ω, F , P) with the properties

• All increments on non-overlapping time intervals are independent: that

is Wtn− Wt n−1, , Wt2− Wt 1 are independent for all 0 ≤ t1≤ · · · ≤ tn

EP[Wti− Wti−1] = 0 and variance

from which we can reproduce the variance (2.5)

Note that EP[WtWs] = min(t, s)

By definition of the filtration FtW, the process Wtis FtW-adapted This is not

In the following, without any specification, we denote (Ω, F , P) theprobability space where our Brownian motion is defined and the

diffusion process is formally defined by considering the limit ∆t → 0 in (2.7)

We now give a precise meaning to this limit under the integral form

defined on the interval [0, t] by

The Itˆo integral is thus defined as

([S, T ] × P) ∀S < T meaning that for each f ∈ Υ,

The limit does not depend on the actual choice {φn} as long as (2.11) holds.Finally, we have given a meaning to the equation (2.8) which is usually writtenformally as

to the class Υ and each process bi(t, x) is Ft-adapted and4

P[

Z t

0

|b(s, xs(ω))|ds < ∞ ∀ t ≥ 0] = 1then the process

Let us introduce the simple function

with t0= 0 and tn= t This simple function satisfies condition (2.11):

Note that the Stratonovich integral looks like a conventional Riemann integral

In the Itˆo integral, there is an extra term −12t

In the same way the computation of a Riemann integral is done without using

explain in the next section

2.4.2 Itˆ o’s lemma

As we will see in the following section, the fair value C of a European calloption on a single asset at time t depends implicitly on the time t and the

∆Ct≡ C(t + ∆t, St+ ∆St) − C(t, St)

([0, ∞) × R) (i.e., (resp.twice) continuously differentiable on [0, ∞) (resp R)) then

2∂

2

SC(∆St)2+ Rwith R the rest of the Taylor expansion The arguments of the function C

its variation (2.7), we obtain

(P)X

To prove this, we set vi= σ(ti, Sti)2 and consider





∆Wtj
2

− ∆tj

]For i < j (similarly j > i), the r.v (with zero mean) vivj (∆Wti)2− ∆ti

the sum reduces to

EP[v2i((∆Wti)2− ∆ti)2] = EP[v2i]EP[(∆Wti)4− 2∆ti(∆Wti)2+ (∆ti)2]According to the definition of a Brownian motion, we have EP[(∆Wti)2] = ∆ti,

EP[(∆Wti)4] = 3(∆ti)2 (see exercise 2) and we obtain

EP[ (∆Wti)2− ∆ti
2

] = 2(∆ti)2Finally

dCt is therefore an Itˆo diffusion process with a drift (∂tC + σ(t,St ) 2

SC +b(t, St)∂SC) and a diffusion term σ(t, St)∂SC Similarly for a function of n-dimensional Itˆo diffusion processes xi

([0, ∞) × Rn) real function Then the process Ct≡ C(t, xt)

a SDE admits a unique (strong or weak) solution

2.4.3 Stochastic differential equations

Example 2.6 Geometric Brownian motion

A Geometric Brownian motion (GBM) is the core of the Black-Scholes marketmodel A GBM is given by the following SDE

dXt= µ(t)Xtdt + σ(t)XtdWt