Introduction

The following sections are included:The Gaussian Merton-Black-Scholes theoryStrong pointsDrawbacksFat tails and anomalous skewness and kurtosisThe Black-Scholes market and formulaVolatility smile and volatility surfaceRemedies for the MBS-theoryGeneral remarksJump-diffusion models and stochastic volatility modelsLévy processesRegular Lévy Processes of Exponential typeCharacteristic function and exponent, Lévy measure and Lévy-Khintchine formulaDefinition of RLPE in 1DInfinitesimal generators of RLPE as PDOPricing of contingent claimsDiscrete time models with a discrete space of states: No-arbitrage and equivalent martingale measuresDiscrete time models with a discrete space of states: Completeness of the market, and pricing of derivative securitiesAbsence of arbitrage, EMM and completeness in the Gaussian Black-Scholes model marketSufficient condition for no-arbitrage in a Lévy market and incompleteness of a Lévy market. The pricing formula for contingent claims of European type and the problem of a choice of EMMOn pricing based on the utility maximizationThe Generalized Black-Scholes equationThe informal derivationAn outline of the reduction of the pricing of contingent claims to boundary value problems for the generalized Black-Scholes equation: barrier optionsThe case of interest bearing securitiesThe generalized Merton-Black-Scholes theoryOptimal stopping problems and the smooth pasting conditionThe case of a dividend-paying stockAnalytical methods used in the bookThe Fourier transform, and Complex AnalysisThe Wiener-Hopf factorization and the Wiener-Hopf equationThe case of the non-stationary Black-Scholes equation and the constant barrierThe case of the non-constant barrier and multi-asset contractsPseudodifferential operatorsAn overview of the results covered in the bookElements of the theory of Lévy processesOption pricingInvestment under uncertainty and capital accumulationEndogenous default and pricing of the corporate debtNumerical methodsExtensionsBasics of PDO theoryCommentary