Foundations of Quantitative Finance, Book VI: Densities, Transformed Distributions, and Limit Theorems

Robert R. Reitano is Professor of the Practice of Finance at the Brandeis International Business School where he specializes in risk management and quantitative finance, and where he previously served as MSF Program Director, and Senior Academic Director. He has a Ph.D. in Mathematics from MIT, is a Fellow of the Society of Actuaries, and a Chartered Enterprise Risk Analyst. He has taught as Visiting Professor at Wuhan University of Technology School of Economics, Reykjavik University School of Business, and as Adjunct Professor in Boston University’s Masters Degree program in Mathematical Finance. Dr. Reitano consults in investment strategy and asset/liability risk management and previously had a 29-year career at John Hancock/Manulife in investment strategy and asset/liability management, advancing to Executive Vice President & Chief Investment Strategist. His research papers have appeared in a number of journals and have won an Annual Prize of the Society of Actuaries and two F.M.

• 1 Density Functions1.1 Density Functions of Measures1.2 Density Functions of Distributions1.2.1 Distribution Functions and Random Vectors1.2.2 Distribution Functions and Probability Measures1.2.3 Existence of Density Functions1.3 Marginal Density Functions1.4 Densities and Independent RVs1.5 Conditional Density Functions2 Transformations of Random Vectors2.1 Cavalieri.s Principle2.2 Sums of Independent Random Vectors2.2.1 Distribution Functions2.2.2 Density Functions2.3 A Result on Convolutions2.4 Ratios of Independent Random Variables2.5 Densities of Transformed Random Vectors3 Weak Convergence of Probability Measures3.1 Portmanteau Theorem on R3.2 Portmanteau Theorem on Rm3.3 Applications3.3.1 The Mapping Theorem3.3.2 Mann-Wald Theorem3.3.3 Cramér-Wold Theorem - Part 13.3.4 Slutsky.s Theorem3.3.5 The Delta Method3.3.6 Sche¤é.s Theorem3.3.7 Prokhorov.s theorem4 Expectations of Random Variables 24.1 Expectations and Moments4.1.1 Expectations of Independent RV Products4.1.2 Moments and the MGF4.1.3 Properties of Moments4.2 Weak Convergence and Moment Limits4.3 Conditional Expectations4.3.1 Conditional Probability Measures4.3.2 Conditional Expectation -An Introduction4.3.3 Conditional Expectation as a Function4.3.4 Existence of Conditional Expectation4.4 Properties of Conditional Expectations4.4.1 Fundamental Properties4.4.2 Conditional Jensen.s Inequality4.4.3 Lp(S)-Space Properties4.5 Conditional Expectations in the Limit4.5.1 Conditional Monotone Convergence4.5.2 Conditional Fatou.s Lemma4.5.3 Conditional Dominated Convergence5 The Characteristic Function5.1 The Moment Generating Function5.2 Integration of Complex-Valued Functions5.3 The Characteristic Function5.4 Examples of Characteristic Functions5.4.1 Discrete Distributions5.4.2 Continuous Distributions5.5 Properties of Characteristic Functions on R5.6 Properties of Characteristic Functions on Rn5.6.1 The Cramér-Wold Theorem5.7 Bochner.s Theorem5.7.1 Positive Semide.nite Functions5.7.2 Bochner.s Theorem5.8 A Uniqueness of Moments Result6 Multivariate Normal Distribution6.1 Derivation and De.nition6.1.1 Density Function Approach6.1.2 Characteristic Function Approach6.1.3 Multivariate Normal De.nition6.2 Existence of Densities6.3 The Cholesky Decomposition6.4 Properties of Multivariate Normal6.4.1 Higher Moments6.4.2 Independent vs. Uncorrelated Normals6.4.3 Sample Mean and Variance7 Applications of Characteristic Functions7.1 Central Limit Theorems7.1.1 The Classical Central Limit Theorem7.1.2 Lindeberg.s Central Limit Theorem7.1.3 Lyapunov.s Central Limit Theorem7.1.4 A Central Limit Theorem on Rn7.2 Distribution Families Related Under Addition7.2.1 Discrete Distributions7.2.2 Continuous Distributions7.3 In.nitely Divisible Distributions7.3.1 De Finetti.s Theorem7.4 Distribution Families Related Under Multiplication8 Discrete Time Asset Models in Finance8.1 Models of Asset Prices8.1.1 Additive Temporal Models8.1.2 Multiplicative Temporal Models8.1.3 Simulating Asset Price Paths8.2 Scalable Asset Models8.2.1 Properties of Scalable Models8.2.2 Scalable Additive Models8.2.3 Scalable Multiplicative Models8.3 Limiting Distributions of Scalable Models8.3.1 Scalable Additive Models8.3.2 Scalable Multiplicative Models9 Pricing of Financial Derivatives9.1 Binomial Lattice Pricing9.1.1 European Derivatives9.1.2 American Options9.2 Limiting Risk Neutral Asset Distribution9.2.1 Analysis of the Probability q(_t)9.2.2 Limiting Asset Distribution Under q (_t)9.3 A Real World Model Under p (_t)9.4 Limiting Price of European Derivatives9.4.1 Black-Scholes-Merton Option Pricing9.5 Properties of Black-Scholes-Merton Prices9.5.1 Price Convergence to Payo¤9.5.2 Put-Call Parity9.5.3 Black-Scholes-Merton PDE9.5.4 Lattice Approximations for "Greeks"9.6 Limiting Price of American Derivatives9.7 Path Dependent Derivatives9.7.1 Path-Based Pricing of European Derivatives9.7.2 Lattice Pricing of European PD Derivatives9.7.3 Lattice Pricing of American PD Derivatives9.7.4 Monte Carlo Pricing of European PD Derivatives9.8 Lognormal Pricing Model9.8.1 European Financial Derivatives9.8.2 European PD Financial Derivatives10 Limits of Binomial Motion10.1 Binomial Paths10.2 Uniform Limits of Bt(_t)10.3 Distributional Limits of Bt(_t)10.4 Nonstandard Binomial Motion10.4.1 Nonstandard Binomial Motion with p 6= 1=2 Fixed10.4.2 Nonstandard Binomial Motion with p = q (_t)10.5 Limits of Binomial Asset ModelsReferences