TOOLS FOR COMPUTATIONAL FINANCE
CONTENTS CHAPTER MODELING TOOLS FOR FINANCIAL OPTIONS OPTIONS MODEL OF THE FINANCIAL MARKET NUMERICAL METHODS THE BINOMIAL METHOD RISK-NEUTRAL VALUATION STOCHASTIC PROCESSES WIENER PROCESS STOCHASTIC INTEGRAL STOCHASTIC DI?ERENTIAL EQUATIONS ITˆO PROCESS APPLICATION TO THE STOCK MARKET MEAN REVERSION VECTOR-VALUED SDES ITˆO LEMMA AND IMPLICATIONS JUMP PROCESSES NOTES AND COMMENTS EXERCISES CHAPTER GENERATING RANDOM NUMBERS WITH SPECI?ED DISTRIBUTIONS UNIFORM DEVIATES LINEAR CONGRUENTIAL GENERATORS QUALITY OF GENERATORS RANDOM VECTORS AND LATTICE STRUCTURE FIBONACCI GENERATORS TRANSFORMED RANDOM VARIABLES INVERSION TRANSFORMATION IN IRN NORMALLY DISTRIBUTED RANDOM VARIABLES METHOD OF BOX AND MULLER VARIANT OF MARSAGLIA CORRELATED RANDOM VARIABLES MONTE CARLO INTEGRATION SEQUENCES OF NUMBERS WITH LOW DISCREPANCY DISCREPANCY EXAMPLES OF LOW-DISCREPANCY SEQUENCES SIMULATION WITH STOCHASTIC DI?ERENTIAL EQUATIONS APPROXIMATION ERROR STOCHASTIC TAYLOR EXPANSION EXAMPLES OF NUMERICAL METHODS INTERMEDIATE VALUES MONTE CARLO SIMULATION INTEGRAL REPRESENTATION THE BASIC VERSION FOR EUROPEAN OPTIONS BIAS VARIANCE REDUCTION AMERICAN OPTIONS FURTHER HINTS STANDARD METHODS FOR STANDARD OPTIONS PREPARATIONS FOUNDATIONS OF FINITE-DI?ERENCE METHODS DI?ERENCE APPROXIMATION THE GRID EXPLICIT METHOD STABILITY AN IMPLICIT METHOD CRANK-NICOLSON METHOD BOUNDARY CONDITIONS AMERICAN OPTIONS AS FREE BOUNDARY PROBLEMS EARLY-EXERCISE CURVE FREE BOUNDARY PROBLEMS BLACK-SCHOLES INEQUALITY OBSTACLE PROBLEMS LINEAR COMPLEMENTARITY FOR AMERICAN PUT OPTIONS COMPUTATION OF AMERICAN OPTIONS DISCRETIZATION WITH FINITE DI?ERENCES ITERATIVE SOLUTION AN ALGORITHM FOR CALCULATING AMERICAN OPTIONS ON THE ACCURACY ELEMENTARY ERROR CONTROL EXTRAPOLATION ANALYTIC METHODS APPROXIMATION BASED ON INTERPOLATION QUADRATIC APPROXIMATION ANALYTIC METHOD OF LINES METHODS EVALUATING PROBABILITIES FINITE-ELEMENT METHODS WEIGHTED RESIDUALS THE PRINCIPLE OF WEIGHTED RESIDUALS EXAMPLES OF WEIGHTING FUNCTIONS EXAMPLES OF BASIS FUNCTIONS GALERKIN APPROACH WITH HAT FUNCTIONS HAT FUNCTIONS ASSEMBLING A SIMPLE APPLICATION APPLICATION TO STANDARD OPTIONS ERROR ESTIMATES CLASSICAL AND WEAK SOLUTIONS APPROXIMATION ON FINITE-DIMENSIONAL SUBSPACES C´EA’S LEMMA PRICING OF EXOTIC OPTIONS EXOTIC OPTIONS OPTIONS DEPENDING ON SEVERAL ASSETS ASIAN OPTIONS THE PAYO? MODELING IN THE BLACK-SCHOLES FRAMEWORK REDUCTION TO A ONE-DIMENSIONAL EQUATION DISCRETE MONITORING NUMERICAL ASPECTS CONVECTION-DI?USION PROBLEMS VON NEUMANN STABILITY ANALYSIS UPWIND SCHEMES AND OTHER METHODS UPWIND SCHEME DISPERSION HIGH-RESOLUTION METHODS THE LAX-WENDRO?METHOD TOTAL VARIATION DIMINISHING NUMERICAL DISSIPATION APPENDICES A FINANCIAL DERIVATIVES INVESTMENT AND RISK FINANCIAL DERIVATIVES FORWARDS AND THE NO-ARBITRAGE PRINCIPLE THE BLACK-SCHOLES EQUATION B STOCHASTIC TOOLS ESSENTIALS OF STOCHASTICS ADVANCED TOPICS STATE-PRICE PROCESS C NUMERICAL METHODS BASIC NUMERICAL TOOLS ITERATIVE METHODS FOR AX = B FUNCTION SPACES D COMPLEMENTARY MATERIAL BOUNDS FOR OPTIONS APPROXIMATION FORMULA SOFTWARE REFERENCES