Preface
Acknowledgements
Part I: Theory
1 SinglcbPeriod Option Pricing
1.1 0ption pricing in a nutsheU
1.2 The simplest setting
1,3 General one-period economy
1.3.1 Pricing
1.3,2 Conditions for no arbitrage: existence of Z
1.3.3 Completeness: uniqueness of Z
1.3.4 Probabilistic formulation
1.3.5 Units and numeraires
1.4 A two-period example
2 Brownian Motion
2.1 Introduction
2.2 Defirution and existence
2.3 Basic properties of Brownian motion
2.3,1 Linut of a random walk
2.3.2 Deterministic transformations of Brownian motion
2.3.3 Some basic sample path properties
2.4 Strong Markov property
2.4.1 R,eflection principle
3 Martingales
3.1 Definition and basic propert,ies
3.2 Classes of martingales
3.2.1 Martingales bounded in Cl
3.2.2 Uniformly integrable martingales
3.2.3 Square-integrable martingales
3,3 Stopping times and the optional sampling theorem
3.3.1 Stopping times
3.3.2 0ptional sampling theorem
3.4 Variation, quadratic variation and integration
3.4.1 Total variation and Stieltjes integration
3.4.2 Quadratic variation
3.4,3 Quadratic covariation
3.5 Local martingales and semimartingales
3.5.1 The space cA-tioc
3.5.2 Semimartingales
3.6 Supermartingales and the Doob-Meyer decomposition
4 Stochastic Integration
4.1 0utline
4.2 Predictable processes
4.3 Stochastic integrals: the L2 theory
4.3.1 The simplest integral
4.3.2 The Hilbert space L2(M)
4.3.3 The L2 integral
4.3.4 Modes of convergence to H *M
4.4 Properties of the stochastic integral
4.5 Extensions via localization
4.5.1 Continuous local martingales as integrators
4.5,2 Semimartingales as integrators
4.5.3 The end of the road!
4.6 Stochastic calculus: Ito's formula
4.6.1 Integration by parts and Ito's formula
4.6.2 Differential notation
4.6.3 Multidimensional version oflto's formula
4.6.4 Levy's theorem
5 Girsanov and Martingale Representation
5.1 Equivalent probability measures and the R,adon-Nikodym derivative
5.1.1 Basic results and properties
5.1.2 Equivalent and locally equivalent measures on a filtered space
5.1.3 Novikov's condition
5.2 Girsanov's theorem
5.2.1 Girsanov's theorem for continuous semimartingales
5.2.2 Girsanov's theorem for Brownian motion
5.3 Martingale representation theorem
5.3.1 The space于(M) and its orthogonal complement
5.3.2 Martingale measures and the martingale representation theorem
5.3.3 Extensions and the Brownian case
6 Stochastic Differential Equations
6.1 Introduction
6.2 Formal definition of an SDE
6.3 An aside on the canorucal set-up
6.4 Weak and strong solutions
6.4.1 Weak solutions
6.4.2 Strong solutions
6.4.3 Tying together strong and weak
6.5 Establishing existence and uniqueness: Ito theory
6,5.1 Picard-Lindelof iteration and ODEs
6.5.2 A technical lemma
6.5.3 Existence and uniqueness for Lipscfutz coefficients
6.6 Strong Markov property
6.7 Martingale representation revisited
7 0ption Pricing in Continuous Time
7.1 Asset price processes and trading strategies
7.1.1 A model for asset prices
7.1.2 Self-financing trading strategies
7.2 Pricing European options
7.2.1 0ption value as a solution to a PDE
7.2.2 0ption pricing via an eqLuvalent martingale measure
7.3 Continuous time theory
7.3.1 Information within the economy
7,3.2 Units, numeraires and martingale measures
7.3.3 Arbitrage and admissible strategies
7.3.4 Derivative pricing in an arbitrage-free economy
7.3.5 Completeness
7.3.6 Pricing kernels
7.4 Extensions
7.4.1 General payout schedules
7.4.2 Controlled derivative payouts
7.4.3 More general asset price processes
7.4,4 Infinite trading horizon
8 Dynamic Term Structure Models
8.1 Introduction
8.2 An economy of pure discount bonds
8.3 Modelling the term structure
8.3.1 Pure discount bond models
8.3.2 Pricing kernel approach
8.3.3 Numeraire models
8.3.4 Finite variation kernel modeLs
8.3.5 Absolutely continuous (FVK) models
8.3.6 Short-rate models
8.3.7 Heath-Jarrow-Morton models
8.3.8 Flesaker-Hughston models |
