APRIMER for the MATHEMATICS of FINANCIAL ENGINEERING Second Edition

Transcription

1 APRIMER for the MATHEMATICS of FINANCIAL ENGINEERING Second Edition DAN STEFANICA Baruch College City University of New York FE Press New York

2 Contents List of Tables Preface to the Second Edition Preface to the First Edition Acknowledgments HowtoUseThisBook xii xiii xv xvii xix 1 Calculus review. Options. Put Call parity Briefreviewofdifferentiation Briefreviewofintegration Differentiating definite integrals Limits L Hôpital srule Multivariablefunctions Functionsoftwovariables Plain vanilla European call and put options Arbitrage freepricing ThePut CallparityforEuropeanoptions Arbitrage opportunities Forwardandfuturescontracts ForwardcontractsandthePut Callparity References Exercises Numerical integration. Interest Rates. Bonds Improperintegrals Differentiating improper integrals Midpoint, Trapezoidal, and Simpson s rules Convergenceofnumericalintegrationmethods vii

3 viii CONTENTS Implementationofnumericalintegrationmethods A concrete example InterestRateCurves Constantinterestrates Forwardrates Bonds.Yieldofabond Bonddurationandconvexity Discretely compounded interest Zerocouponbonds Numerical implementation of bond mathematics References Exercises Probability concepts. Black Scholes formula. Greeks and hedging Discrete probability concepts Continuous probability concepts Variance,covariance,andcorrelation Thestandardnormalvariable Normalrandomvariables TheBlack Scholesformula TheGreeksofEuropeanoptions Explaining the magic of Greeks computations Implied volatility Theconceptofhedging.Δ andγ hedging Implementation of the Black Scholes formula References Exercises Lognormal variables. Risk neutral valuation Changeofpdfsforfunctionsofrandomvariables Lognormalrandomvariables Independentrandomvariables Independentnormalrandomvariables Independentlognormalrandomvariables Thelognormalmodelforassetprices Risk neutralderivationofblack Scholes Probability that options expire in the money Financial interpretation of N(d 1 )andn(d 2 )

4 ix 4.8 References Exercises Newton s method. Implied volatility. Bootstrapping Numerical methods for nonlinear problems Bisection Method Newton smethod SecantMethod Numerical methods for N dimensional problems The N dimensional Newton s Method The Approximate Newton s Method Computing bond yields Implied volatility Bootstrapping for finding zero rate curves References Exercises Taylor s formula. Taylor series. Bond portfolio optimization. ATM approximations of Black Scholes formulas Taylor sformulaforfunctionsofonevariable Taylor sformulaformultivariablefunctions Taylor sformulaforfunctionsoftwovariables Taylorseriesexpansions Examples of Taylor series expansions Percentageandlogreturns Assetreturnsandportfolioreturns Parallelshiftsintheyieldcurve Connectionsbetweendurationandconvexity Dollardurationanddollarconvexity DV Bond portfolio immunization Black Scholesformula:ATMapproximations SeveralATMapproximationsformulas DerivingtheATMapproximationsformulas The precision of the ATM approximation of the Black Scholes formula GreeksandTaylor sformula References Exercises

5 x CONTENTS 7 Finite Differences. Black Scholes PDE Forward, backward, central finite differences Finite difference solutions of ODEs Finite difference approximations of the Greeks Numerical accuracy of the finite difference approximations of thegreeks TheBlack ScholesPDE Financial interpretation of the Black Scholes PDE TheBlack ScholesPDEandtheGreeks References Exercises Multivariable calculus: chain rule, double integrals, extremum points. Optimality of early exercise Chainruleforfunctionsofseveralvariables Double integrals Change of variables for double integrals Change of variables to polar coordinates Relative extrema of multivariable functions The Theta of a derivative security Integrating the density function of Z The Box Muller method TheBlack ScholesPDEandtheheatequation Barrieroptions Optimalityofearlyexercise References Exercises Lagrange multipliers. Portfolio optimization Lagrange multipliers Examples Optimalinvestmentportfolios Minimum variance portfolios Maximumreturnportfolios References Exercises Mathematical Appendix Evenandoddfunctions

6 xi 10.2Polynomialinterpolation Usefulsumswithinterestingproofs Sums of the form n k=1 ki Sums of the form n k=1 kj x k Sequences satisfying linear recursions The BigO and littleo notations Powerseries Stirling s formula ConvergenceresultsforTaylorexpansions References Exercises Bibliography 325 Index 329