Introduction to Option Pricing Theory

1 Stochastic Integration.- 1.1 Notation and definitions.- 1.2 The predictable ? field.- 1.3 The Itô integral.- 1.4 Quadratic variation of a continuous martingale.- 1.5 The stochastic integral w.r.t. continuous local martingales.- 1.6 Stochastic integral w.r.t. continuous semimartingales.- 1.7 Integration w.r.t. semimartingales.- 2 Itô's Formula and its Applications.- 2.1 Preliminaries.- 2.2 Itô's formula for continuous semimartingales.- 2.3 Itô's formula for r.c.l.l. semimartingales.- 2.4 Applications.- 2.5 Application to geometric Brownian motion.- 2.6 Local time and the Tanaka formula.- 2.7 Brownian motion and the heat equation.- 3 Representation of Square Integrable Martingales.- 3.1 The Itô representation.- 3.2 The Kunita-Watanabe representation.- 4 Stochastic Differential Equations.- 4.1 Preliminaries.- 4.2 Existence and uniqueness of solutions.- 4.3 The Feynman-Kac formula.- 4.4 The Ornstein-Uhlenbeck process (O.U.P).- 5 Girsanov's Theorem.- 5.1 Auxiliary results.- 5.2 Girsanov's Theorem.- 6 Option Pricing in Discrete Time.- 6.1 Arbitrage opportunities.- 6.2 Option pricing: an example.- 6.3 European call option.- 6.4 Complete markets.- 6.5 The American option.- 7 Introduction to Continuous Time Trading.- 7.1 Introduction.- 7.2 A general model.- 7.3 Trading strategies and arbitrage opportunities.- 7.4 Examples.- 7.5 Contingent claims and complete markets.- 8 Arbitrage and Equivalent Martingale Measures.- 8.1 Introduction.- 8.2 Necessary and sufficient conditions for NA.- 8.3 A general model of stock prices.- 8.4 The separation theorem.- 8.5 Orlicz spaces.- 8.6 No arbitrage with controlled risk.- 8.7 Fractional Brownian motion (1/2 9.1 Definition.- 9.2 Representation of martingales.- 9.3 Examples of complete markets.- 9.3.1 Geometric Brownian motion (GBM).- 9.3.2 Diffusion model for stock prices.- 9.4 Equivalent martingale measures.- 9.5 Incomplete markets.- 9.6 Completeness and underlying filtration.- 10 Black and Scholes Theory.- 10.1 Preliminaries.- 10.2 The Black-Scholes PDE.- 10.3 Explicit solution of the Black-Scholes PDE.- 10.4 The Black-Scholes formula.- 10.5 Diffusion model.- 11 Discrete Approximations.- 11.1 The binomial model.- 11.2 A binomial Feynman-Kac formula.- 11.3 Approximation of the Black-Scholes PDE.- 11.4 Approximation to the Black-Scholes fonnula.- 12 The American Options.- 12.1 Model.- 12.2 Upper and lower bounds.- 12.3 American claims in complete markets.- 13 Asset Pricing with Stochastic Volatility.- 13.1 Introduction.- 13.2 Incompleteness of the market.- 13.3 Asymptotic analysis for models with two scales.- 13.4 Filtering of the stochastic volatility.- 13.5 PDE whenSis observed.- 14 The Russian Options.- 14.1 Introduction and background.- 14.2 The Russian put option.- 14.3 A free boundary problem for the put option.- 14.4 Proofs of the lemmas.- 14.5 The Russian call option (or the option for selling short).- 14.6 The F.B.P. for the call option.- References.