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Financial Mathematics

by Giuseppe Campolieti, Roman N. Makarov

October 2018

Intermediate to advanced

829 pages

23h 51m

English

Chapman and Hall/CRC

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Preliminaries
Preface
Objectives and AudienceGuide to MaterialAcknowledgements
Part I Introduction to Pricing and Management of Financial Securities
Chapter 1 Mathematics of Compounding
1.1 Interest and Return1.1.1 Amount Function and Return1.1.2 Simple Interest1.1.3 Periodic Compound Interest1.1.4 Continuous Compound Interest1.1.5 Equivalent Rates1.1.6 Continuously Varying Interest Rates1.2 Time Value of Money and Cash Flows1.2.1 Equations of Value1.2.2 Deterministic Cash Flows1.3 Annuities1.3.1 Simple Annuities1.3.1.1 Ordinary Annuities1.3.1.2 Annuities Due1.3.1.3 Deferred Annuities1.3.2 Determining the Term of an Annuity1.3.3 General Annuities1.3.4 Perpetuities1.3.5 Continuous Annuities1.4 Bonds1.4.1 Introduction and Terminology1.4.2 Zero-Coupon Bonds1.4.3 Coupon Bonds1.4.4 Serial Bonds, Strip Bonds, and Callable Bonds1.5 Yield Rates1.5.1 Internal Rate of Return and Evaluation Criteria1.5.2 Determining Yield Rates for Bonds1.5.2.1 Zero-Coupon Bonds1.5.2.2 Coupon Bonds1.5.3 Approximation Methods1.5.3.1 The Method of Averages1.5.3.2 The Method of Interpolation1.5.3.3 Numerical Methods1.5.4 The Yield Curve1.6 ExercisesFigure 1.1Figure 1.2Figure 1.4Figure 1.5Figure 1.7Figure 1.8Figure 1.9Figure 1.12Figure 1.13Figure 1.14Figure 1.15Figure 1.17Table 1.3Table 1.6Table 1.10Table 1.11Table 1.16
Chapter 2 Primer on Pricing Risky Securities
2.1 Stocks and Stock Price Models2.1.1 Underlying Assets and Derivative Securities2.1.2 Basic Assumptions for Asset Price Models2.2 Basic Price Models2.2.1 A Single-Period Binomial Model2.2.2 A Discrete-Time Model with a Finite Number of States2.2.2.1 Asset Returns2.2.3 Introducing the Binomial Tree Model2.2.4 Self-Financing Investment Strategies in the Binomial Model2.2.5 Log-Normal Pricing Model2.3 Arbitrage and Risk-Neutral Pricing2.3.1 The Law of One Price2.3.2 A First Look at Arbitrage in the Single-Period Binomial Model2.3.3 Arbitrage in the Binomial Tree Model2.3.4 Risk-Neutral Probabilities2.3.5 Martingale Property2.3.6 Risk-Neutral Log-Normal Model2.4 Value at Risk2.5 Dividend Paying Stock2.6 ExercisesFigure 2.1Figure 2.2Figure 2.3Figure 2.4Figure 2.5
Chapter 3 Portfolio Management
3.1 Expected Utility Functions3.1.1 Utility Functions3.1.1.1 Risk Aversion3.1.2 Mean-Variance Criterion3.2 Portfolio Optimization for Two Assets3.2.1 Portfolio of Two Assets3.2.2 Portfolio Lines3.2.2.1 Case with |ρ12| = 13.2.2.2 Case with |ρ12| < 1| < 13.2.2.3 Case with a Risk-Free Asset3.2.3 The Minimum Variance Portfolio3.2.3.1 Case without Short Selling3.2.4 Selection of Optimal Portfolios3.2.4.1 Minimum Variance Portfolio3.2.4.2 Maximum Expected Utility Portfolio3.2.4.3 Minimum Loss-Probability Portfolio3.3 Portfolio Optimization for N Assets3.3.1 Portfolios of Several Assets3.3.2 The Minimum Variance Portfolio3.3.3 The Minimum Variance Portfolio Line3.3.4 Case without Short Selling3.3.5 Efficient Frontier and Capital Market Line3.4 The Capital Asset Pricing Model3.5 ExercisesFigure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Figure 3.7Figure 3.8Figure 3.9Figure 3.10Figure 3.11Figure 3.12Figure 3.13
Chapter 4 Primer on Derivative Securities
4.1 Forward Contracts4.1.1 No-Arbitrage Evaluation of Forward Contracts4.1.1.1 Forward Price for a Stock Paying No Dividends4.1.1.2 Forward Price for a Stock Paying Dividends4.1.2 Value of a Forward Contract4.2 Basic Options Theory4.2.1 Concept of an Option Contract4.2.2 Put-Call Parities4.2.3 Properties of European Options4.2.4 Early Exercise and American Options4.2.4.1 Relation to European Option Prices4.2.4.2 Put-Call Parity Estimate4.2.4.3 Monotonicity Properties of American Option Prices4.2.5 Nonstandard European Options4.3 Basics of Option Pricing4.3.1 Pricing of European-Style Derivatives in the Binomial Tree Model4.3.1.1 Replication and Pricing of Options in the Single-Period Binomial Model4.3.1.2 Pricing in the Binomial Tree Model4.3.2 Pricing of American Options in the Binomial Tree Model4.3.3 Option Pricing in the Log-Normal Model: The Black–Scholes– Merton Formula4.3.4 Greeks and Hedging of Options4.3.4.1 Delta of a Derivative in the Binomial Model4.3.4.2 Delta of a Derivative in the Log-Normal Model4.3.4.3 Delta Hedging4.3.4.4 Greeks4.3.5 Black–Scholes Equation4.4 ExercisesFigure 4.1Figure 4.2Figure 4.3Figure 4.4Figure 4.5Figure 4.6Figure 4.7Figure 4.8Figure 4.9Figure 4.10Figure 4.11Figure 4.12Figure 4.13Figure 4.14Figure 4.15Figure 4.16Figure 4.17Figure 4.18
Part II Discrete-Time Modelling
Chapter 5 Single-Period Arrow–Debreu Models
5.1 Specification of the Model5.1.1 Finite-State Economy. Vector Space of Payoffs. Securities5.1.2 Initial Price Vector and Payoff Matrix5.1.3 Portfolios of Base Securities5.2 Analysis of the Arrow–Debreu Model5.2.1 Redundant Assets and Attainable Securities5.2.2 Completeness of the Model5.3 No-Arbitrage Asset Pricing5.3.1 The Law of One Price5.3.2 Arbitrage5.3.3 The First Fundamental Theorem of Asset Pricing5.3.3.1 The First FTAP: Sufficiency Part5.3.3.2 The First FTAP: Necessity Part5.3.3.3 A Geometric Interpretation of the First FTAP5.3.4 Risk-Neutral Probabilities5.3.5 The Second Fundamental Theorem of Asset Pricing5.3.6 Investment Portfolio Optimization5.4 Pricing in an Incomplete Market5.4.1 A Trinomial Model of an Incomplete Market5.4.2 Pricing Nonattainable Payoffs: The Bid-Ask Spread5.5 Change of Numéraire5.5.1 The Concept of a Numéraire Asset5.5.2 Change of Numéraire in a Binomial Model5.5.3 Change of Numéraire in a Multinomial Model5.6 ExercisesFigure 5.1Figure 5.2Figure 5.3
Chapter 6 Introduction to Discrete-Time Stochastic Calculus
6.1 A Multi-Period Binomial Probability Model6.1.1 The Binomial Probability Space6.1.1.1 A Sample Space6.1.1.2 Random Variables6.1.1.3 Probability Measure6.1.2 Random Processes6.1.2.1 Binomial Price Process and Path Probabilities6.1.2.2 Random Walk6.2 Information Flow6.2.1 Partitions and Their Refinements6.2.1.1 Partition Generated by a Random Variable6.2.1.2 Refinements of Partitions6.2.2 Sigma-Algebras6.2.2.1 Construction of a Sigma-Algebra from a Partition6.2.2.2 Sigma-Algebra Generated by a Random Variable6.2.3 Filtration6.2.3.1 Construction of a Filtration from an Information Structure6.2.3.2 Construction of a Filtration from a Stochastic Process: Natural Filtration6.2.4 Filtered Probability Space6.3 Conditional Expectation and Martingales6.3.1 Measurability of Random Variables and Processes6.3.2 Conditional Expectations6.3.2.1 Conditioning on an Event6.3.2.2 Conditioning on a Sigma-Algebra6.3.2.3 Conditioning on a Random Variable6.3.3 Properties of Conditional Expectations6.3.3.1 Linearity6.3.3.2 Independence6.3.3.3 Taking out What Is Known6.3.3.4 Tower Property (Iterated Conditioning)6.3.4 Conditioning in the Binomial Model6.3.5 Sub-, Super-, and True Martingales6.3.5.1 Examples6.3.6 Classification of Stochastic Processes6.3.7 Stopping Times6.4 ExercisesFigure 6.1Figure 6.2Figure 6.3Figure 6.4

Chapter 7 Replication and Pricing in the Binomial Tree Model
7.1 The Standard Binomial Tree Model7.2 Self-Financing Strategies and Their Value Processes7.2.1 Equivalent Martingale Measures for the Binomial Model7.3 Dynamic Replication in the Binomial Tree Model7.3.1 Dynamic Replication of Payoffs7.3.2 Replication and Valuation of Random Cash Flows7.4 Pricing and Hedging Non-Path-Dependent Derivatives7.5 Pricing Formulae for Standard European Options7.6 Pricing and Hedging Path-Dependent Derivatives7.6.1 Average Asset Prices and Asian Options7.6.2 Extreme Asset Prices and Lookback Options7.6.3 Recursive Evaluation of Path-Dependent Options7.6.3.1 Pricing Lookback Options on a Two-Dimensional Lattice7.7 American Options7.7.1 Writer's Perspective: Pricing and Hedging7.7.2 Buyer's Perspective: Optimal Exercise7.7.3 Early-Exercise Boundary7.7.4 Pricing American Options: The Case with Dividends7.8 ExercisesFigure 7.1Figure 7.2Figure 7.3Figure 7.4Figure 7.5Figure 7.6Figure 7.7Figure 7.8Figure 7.9Figure 7.10Figure 7.13Figure 7.14Figure 7.15Figure 7.16Table 7.11Table 7.12
Chapter 8 General Multi-Asset Multi-Period Model
8.1 Main Elements of the Model8.2 Assets, Portfolios, and Strategies8.2.1 Payoffs and Assets8.2.2 Static and Dynamic Portfolios8.2.3 Self-Financing Strategies8.2.4 Replication of Payoffs8.3 Fundamental Theorems of Asset Pricing8.3.1 Arbitrage Strategies8.3.2 Enhancing the Law of One Price8.3.3 Equivalent Martingale Measures8.3.4 Calculation of Martingale Measures8.3.5 The First and Second FTAPs8.3.6 Pricing and Hedging Derivatives8.3.7 Radon–Nikodym Derivative Process and Change of Numéraire8.4 Examples of Discrete-Time Models8.4.1 Binomial Tree Model with Stochastic Volatility8.4.2 Binomial Tree Model for Interest Rates8.5 ExercisesFigure 8.1Figure 8.2Figure 8.3Figure 8.4Figure 8.5Figure 8.6
Part III Continuous-Time Modelling
Chapter 9 Essentials of General Probability Theory
9.1 Random Variables and Lebesgue Integration9.2 Multidimensional Lebesgue Integration9.3 Multiple Random Variables and Joint Distributions9.4 Conditioning9.5 Changing Probability Measures
Chapter 10 One-Dimensional Brownian Motion and Related Processes
10.1 Multivariate Normal Distributions10.1.1 Multivariate Normal Distribution10.1.2 Conditional Normal Distributions10.2 Standard Brownian Motion10.2.1 One-Dimensional Symmetric Random Walk10.2.2 Formal Definition and Basic Properties of Brownian Motion10.2.3 Multivariate Distribution of Brownian Motion10.2.4 The Markov Property and the Transition PDF10.2.5 Quadratic Variation and Nondifferentiability of Paths10.3 Some Processes Derived from Brownian Motion10.3.1 Drifted Brownian Motion10.3.2 Geometric Brownian Motion10.3.3 Brownian Bridge10.3.4 Gaussian Processes10.4 First Hitting Times and Maximum and Minimum of Brownian Motion10.4.1 The Reflection Principle: Standard Brownian Motion10.4.2 Translated and Scaled Driftless Brownian Motion10.4.3 Brownian Motion with Drift10.4.3.1 Translated and Scaled Brownian Motion with Drift10.5 ExercisesFigure 10.1Figure 10.2Figure 10.3Figure 10.4Figure 10.5Figure 10.6
Chapter 11 Introduction to Continuous-Time Stochastic Calculus
11.1 The Riemann Integral of Brownian Motion11.1.1 The Riemann Integral11.1.2 The Integral of a Brownian Path11.2 The Riemann–Stieltjes Integral of Brownian Motion11.2.1 The Riemann–Stieltjes Integral11.2.2 Integrals w.r.t. Brownian Motion11.3 The Itô Integral and Its Basic Properties11.3.1 The Itô Integral for Simple Processes11.3.2 Properties of the Itô Integral11.4 Itô Processes and Their Properties11.4.1 Gaussian Processes Generated by Itô Integrals11.4.2 Itô Processes11.4.3 Quadratic (Co-) Variation11.5 Itô's Formula for Functions of BM and Itô Processes11.5.1 Itô's Formula for Functions of BM11.5.2 Itô's Formula for Itô Processes11.6 Stochastic Differential Equations11.6.1 Solutions to Linear SDEs11.6.2 Existence and Uniqueness of a Strong Solution of an SDE11.7 The Markov Property, Feynman–Kac Formulae, and Transition CDFs and PDFs11.7.1 Forward Kolmogorov PDE11.7.2 Transition CDF/PDF for Time-Homogeneous Diffusions11.8 Radon–Nikodym Derivative Process and Girsanov's Theorem11.8.1 Some Applications of Girsanov’s Theorem11.9 Brownian Martingale Representation Theorem11.10 Stochastic Calculus for Multidimensional BM11.10.1 The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM11.10.2 Multidimensional SDEs, Feynman–Kac Formulae, and Transition CDFs and PDFs11.10.3 Girsanov’s Theorem for Multidimensional BM11.10.4 Martingale Representation Theorem for Multidimensional BM11.11 ExercisesFigure 11.1
Chapter 12 Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock
12.1 Replication (Hedging) and Derivative Pricing in the Simplest Black–Scholes Economy12.1.1 Pricing Standard European Calls and Puts12.1.2 Hedging Standard European Calls and Puts12.1.3 Europeans with Piecewise Linear Payoffs12.1.4 Power Options12.1.5 Dividend Paying Stock12.1.5.1 The Case of Continuous Dividend Paying Stock12.1.5.2 The Case of Discrete-Time Dividends12.2 Forward Starting and Compound Options12.3 Some European-Style Path-Dependent Derivatives12.3.1 Risk-Neutral Pricing under GBM12.3.2 Pricing Single Barrier Options12.3.3 Pricing Lookback Options12.4 ExercisesFigure 12.1Figure 12.2Figure 12.3Figure 12.4Figure 12.5
Chapter 13 Risk-Neutral Pricing in a Multi-Asset Economy
13.1 General Multi-Asset Market Model: Replication and Risk-Neutral Pricing13.2 Black–Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives within a General Diffusion Model13.2.1 Standard European Option Pricing for Multi-Stock GBM13.2.2 Explicit Pricing Formulae for the GBM Model13.2.2.1 Exchange and Other Related Options13.2.2.2 Other Basket Options13.2.3 Cross-Currency Option Valuation13.3 Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets13.4 Exercises
Chapter 14 American Options
14.1 Basic Properties of Early-Exercise Options14.2 Arbitrage-Free Pricing of American Options14.2.1 Optimal Stopping Formulation and Early-Exercise Boundary14.2.2 The Smooth Pasting Condition14.2.3 Put-Call Symmetry Relation14.2.4 Dynamic Programming Approach for Bermudan Options14.3 Perpetual American Options14.3.1 Pricing a Perpetual Put Option14.3.2 Pricing a Perpetual Call Option14.4 Finite-Expiration American Options14.4.1 The PDE Formulation14.4.2 The Integral Equation Formulation14.5 ExercisesFigure 14.1Figure 14.2Figure 14.3Figure 14.4Figure 14.5
Chapter 15 Interest-Rate Modelling and Derivative Pricing
15.1 Basic Fixed Income Instruments15.1.1 Bonds15.1.2 Forward Rates15.1.3 Arbitrage-Free Pricing15.1.4 Fixed Income Derivatives15.1.4.1 Options on Bonds15.1.4.2 Cap and Caplets15.1.4.3 Swap and Swaptions15.2 Single-Factor Models15.2.1 Diffusion Models for the Short Rate Process15.2.2 PDE for the Zero-Coupon Bond Value15.2.3 Affine Term Structure Models15.2.4 The Ho–Lee Model15.2.5 The Vasiček Model15.2.6 The Cox–Ingersoll–Ross Model15.3 Heath–Jarrow–Morton Formulation15.3.1 HJM under Risk-Neutral Measure15.3.2 Relationship between HJM and Affine Yield Models15.3.2.1 The Ho–Lee Model in the HJM Framework15.3.2.2 The Vasiček Model in the HJM Framework15.4 Multifactor Affine Term Structure Models15.4.1 Gaussian Multifactor Models15.4.2 Equivalent Classes of Affine Models15.5 Pricing Derivatives under Forward Measures15.5.1 Forward Measures15.5.2 Pricing Stock Options under Stochastic Interest Rates15.5.3 Pricing Options on Zero-Coupon Bonds15.6 LIBOR Model15.6.1 LIBOR Rates15.6.2 Brace–Gatarek–Musiela Model of LIBOR Rates15.6.3 Pricing Caplets, Caps, and Swaps15.7 Exercises
Chapter 16 Alternative Models of Asset Price Dynamics
16.1 Stochastic Volatility Diffusion Models16.1.1 Local Volatility Models16.1.2 Constant Elasticity of Variance Model16.1.2.1 Definition and Basic Properties16.1.2.2 Transition Probability Law16.1.2.3 Pricing European Options16.1.3 The Heston Model16.2 Models with Jumps16.2.1 The Poisson Process16.2.2 Jump-Diffusion Models with a Compound Poisson Component16.2.3 The Variance Gamma Model16.3 ExercisesFigure 16.1Figure 16.2Figure 16.3
Part IV Computational Techniques
Chapter 17 Introduction to Monte Carlo and Simulation Methods
17.1 Introduction17.1.1 The “Hit-or-Miss” Method17.1.2 The Law of Large Numbers17.1.3 Approximation Error and Confidence Interval17.1.4 Parallel Monte Carlo Methods17.1.5 One Monte Carlo Application: Numerical Integration17.2 Generation of Uniformly Distributed Random Numbers17.2.1 Uniform Probability Distributions17.2.2 Linear Congruential Generator17.3 Generation of Nonuniformly Distributed Random Numbers17.3.1 Transformations of Random Variables17.3.2 Inversion Method17.3.2.1 Inverse Distribution Function17.3.2.2 The Chop-Down Search Method17.3.2.3 The Binomial Search Method17.3.3 Composition Methods17.3.3.1 Mixture of PDFs17.3.3.2 Randomized Gamma Distributions17.3.3.3 The Alias Method by Walker17.3.4 Acceptance-Rejection Methods17.3.5 Multivariate Sampling17.3.5.1 Sampling by Conditioning17.3.5.2 The Box–Müller method17.3.5.3 Simulation of Multivariate Normals17.4 Simulation of Random Processes17.4.1 Simulation of Brownian Processes17.4.1.1 Sequential Sampling17.4.1.2 Bridge Sampling17.4.2 Simulation of Gaussian Processes17.4.3 Diffusion Processes: Exact Simulation Methods17.4.3.1 The Stochastic Calculus Approach17.4.3.2 The PDF Approach17.4.4 Diffusion Processes: Approximation Schemes17.4.4.1 Types of Convergence17.4.4.2 The Euler Scheme17.4.4.3 Extrapolation17.4.4.4 Error Analysis17.4.5 Simulation of Processes with Jumps17.4.5.1 Poisson Processes17.4.5.2 Subordinated Processes17.5 Variance Reduction Methods17.5.1 Numerical Integration by a Direct Monte Carlo Method17.5.2 Importance Sampling Method17.5.3 Change of Probability Measure17.5.4 Control Variate Method17.5.5 Antithetic Variate17.5.6 Conditional Sampling17.5.7 Stratified Sampling17.6 ExercisesReferencesFigure 17.2Figure 17.3Figure 17.4Figure 17.5Figure 17.6Table 17.1Table 17.7
Chapter 18 Numerical Applications to Derivative Pricing
18.1 Overview of Deterministic Numerical Methods18.1.1 Quadrature Formulae18.1.1.1 Newton–Cotes Quadrature Formulae18.1.1.2 Gaussian Quadrature Formulae18.1.1.3 Composite Quadrature Formulae18.1.1.4 Extrapolation and Romberg Integration18.1.2 Finite-Difference Methods18.1.2.1 Finite-Difference Approximations for ODEs18.1.2.2 Second-Order Linear PDEs18.1.2.3 Finite-Difference Approximations for the Heat EquationThe Explicit MethodThe Implicit MethodThe Crank–Nicolson Method18.1.2.4 Stability AnalysisStability Analysis of the Explicit MethodStability Analysis of the Implicit MethodStability Analysis of the Crank–Nicolson Method18.2 Pricing European Options18.2.1 Pricing European Options by Quadrature Rules18.2.2 Pricing European Options by the Monte Carlo Method18.2.3 Pricing European Options by Tree Methods18.2.3.1 Binomial Model18.2.3.2 Multinomial Models18.2.4 Pricing European Options by PDEs18.2.4.1 Pricing by the Heat Equation18.2.4.2 Pricing by the Black–Scholes PDEThe Explicit MethodThe Implicit MethodThe Crank–Nicolson Method18.2.5 Calibration of Asset Price Models to Empirical Data18.2.5.1 Least Squares Method18.2.5.2 Maximum Likelihood Estimation18.3 Pricing Early-Exercise and Path-Dependent Options18.3.1 Pricing American and Bermudan Options18.3.1.1 Pricing American Options by Tree Methods18.3.1.2 Pricing Bermudan Options by the Monte Carlo Method18.3.2 Pricing Asian Options18.3.2.1 Pricing Discrete-Time Asian Options by the Monte Carlo MethodFigure 18.2Figure 18.7Figure 18.9Figure 18.11Figure 18.12Figure 18.14Figure 18.16Figure 18.17Table 18.1Table 18.3Table 18.4Table 18.5Table 18.6Table 18.8Table 18.10Table 18.13Table 18.15Table 18.18
Appendix: Some Useful Integral Identities and Symmetry Properties of Normal Random Variables
Glossary of Symbols and Abbreviations
References
Theory of Probability and Stochastic ProcessesIntroduction to Mathematics of FinanceMathematics of Finance (Discrete-Time)Mathematics of Finance (Continuous-Time)Computational MethodsFinancial Economics

Content preview from Financial Mathematics

Chapter 14

American Options

In this chapter we briefly present the theory for pricing early-exercise (American) options in continuous time. Recall that American options were first introduced in Chapter 4, and the discrete-time case was dealt with in Chapter 7. Let us recall some of the properties of early-exercise options. The key difference between European-style and American-style options is that a holder of an American option can exercise her rights any time before the expiration date. This additional early exercise privilege should not be worthless. Thus, an American option is expected to be worth more than its European analogue. The extra premium paid on top of the price of the European option is called the early-exercise premium. We mainly ...

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American Options

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