DISCRETE AND CONTINUOUS BOUNDARY PROBLEMS

By B. GASAM

CONTENTS REVIEW OF PRELIMINARIES RISKY ASSETS SINGLE AND MULTIPLE DISCRETE TIME PERIODS CONTINUOUS-TIME PROCESSES MARTINGALES RISK AVERSION AND PORTFOLIOS OF ASSETS RISK AVERSION CONSTANT THE PORTFOLIO PROBLEM EXPECTATION, VARIANCE, AND COVARIANCE ONE VARIABLE EXPECTATION EXPECTATION FOR MULTIPLE RANDOM VARIABLES VARIANCE OF A LINEAR COMBINATION SIMPLE PORTFOLIO OPTIMIZATION DERIVATIVE ASSETS AND ARBITRAGE FUTURES ARBITRAGE AND FUTURES OPTIONS VALUATION OF DERIVATIVES IN A SINGLE TIME PERIOD REPLICATING PORTFOLIOS RISK-NEUTRAL VALUATION PORTFOLIO SELECTION AND CAPM THEORY PORTFOLIO OPTIMIZATION WITH MULTIPLE ASSETS LAGRANGE MULTIPLIERS QUALITATIVE BEHAVIOR CORRELATED ASSETS PORTFOLIO SEPARATION AND THE MARKET PORTFOLIO CAPITAL MARKET THEORY, PART I LINEAR ALGEBRAIC APPROACH E?CIENT MEAN-STANDARD DEVIATION FRONTIER CAPITAL MARKET THEORY, PART II CAPITAL MARKET LINE CAPM FORMULA; ASSET ? SYSTEMATIC AND NON-SYSTEMATIC RISK; PRICING USING CAPM UTILITY THEORY SECURITIES AND AXIOMS FOR INVESTOR BEHAVIOR INDI?ERENCE CURVES, CERTAINTY EQUIVALENT, RISK AVER- SION EXAMPLES OF UTILITY FUNCTIONS ABSOLUTE AND RELATIVE RISK AVERSION UTILITY MAXIMIZATION MULTIPLE PERIOD PORTFOLIO PROBLEMS PROBLEM DESCRIPTION AND DYNAMIC PROGRAMMING AP- PROACH EXAMPLES OPTIMAL PORTFOLIOS AND MARTINGALES DISCRETE-TIME DERIVATIVES VALUATION OPTIONS PRICING FOR MULTIPLE TIME PERIODS INTRODUCTION VALUATION BY CHAINING VALUATION BY MARTINGALES KEY IDEAS OF DISCRETE PROBABILITY, PART I ALGEBRAS AND MEASURABILITY INDEPENDENCE KEY IDEAS OF DISCRETE PROBABILITY, PART II CONDITIONAL EXPECTATION APPLICATION TO PRICING MODELS FUNDAMENTAL THEOREMS OF OPTIONS PRICING THE MARKET MODEL GAIN, ARBITRAGE, AND ATTAINABILITY MARTINGALE MEASURES AND THE FUNDAMENTAL THEOREMS VALUATION OF NON-VANILLA OPTIONS AMERICAN AND BERMUDAN OPTIONS BARRIER OPTIONS ASIAN OPTIONS TWO-ASSET DERIVATIVES DERIVATIVES PRICING BY SIMULATION SETUP AND ALGORITHM FROM DISCRETE TO CONTINUOUS TIME (A PREVIEW) CONTINUOUS PROBABILITY MODELS CONTINUOUS DISTRIBUTIONS AND EXPECTATION DENSITIES AND CUMULATIVE DISTRIBUTION FUNCTIONS EXPECTATION NORMAL AND LOGNORMAL DISTRIBUTIONS JOINT DISTRIBUTIONS BASIC IDEAS MARGINAL AND CONDITIONAL DISTRIBUTIONS COVARIANCE AND CORRELATION BIVARIATE NORMAL DISTRIBUTION MEASURABILITY AND CONDITIONAL EXPECTATION SIGMA ALGEBRAS RANDOM VARIABLES AND MEASURABILITY CONTINUOUS CONDITIONAL EXPECTATION BROWNIAN MOTION AND GEOMETRIC BROWNIAN MOTION RANDOM WALK PROCESSES STANDARD BROWNIAN MOTION NON-STANDARD BROWNIAN MOTION GEOMETRIC BROWNIAN MOTION BROWNIAN MOTION AND BINOMIAL BRANCH PROCESSES INTRODUCTION TO STOCHASTIC DI?ERENTIAL EQUATIONS MEANING OF THE GENERAL SDE ITO’S FORMULA GEOMETRIC BROWNIAN MOTION AS SOLUTION DERIVATIVE VALUATION IN CONTINUOUS TIME BLACK-SCHOLES VIA LIMITS BLACK-SCHOLES FORMULA LIMITING APPROACH PUT OPTIONS AND PUT-CALL PARITY BLACK-SCHOLES VIA MARTINGALES TRADING STRATEGIES AND MARTINGALE VALUATION MARTINGALE MEASURES ASSET AND BOND BINARIES OTHER BINARY DERIVATIVES BLACK-SCHOLES VIA DI?ERENTIAL EQUATIONS DERIVING THE PDE BOUNDARY CONDITIONS; SOLVING THE PDE CHECKING BLACK-SCHOLES ASSUMPTIONS NORMALITY OF RATES OF RETURN STABILITY OF PARAMETERS INDEPENDENCE OF RATES OF RETURN