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Financial Mathematics: A Comprehensive Treatment
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Financial Mathematics: A Comprehensive Treatment
by Roman N. Makarov, Giuseppe Campolieti
Financial Mathematics
Preliminaries
Preface
Objectives and Audience
Guide to Material
Acknowledgements
Part I Introduction to Pricing and Management of Financial Securities
Chapter 1 Mathematics of Compounding
1.1 Interest and Return
1.1.1 Amount Function and Return
1.1.2 Simple Interest
1.1.3 Periodic Compound Interest
1.1.4 Continuous Compound Interest
1.1.5 Equivalent Rates
1.1.6 Continuously Varying Interest Rates
1.2 Time Value of Money and Cash Flows
1.2.1 Equations of Value
1.2.2 Deterministic Cash Flows
1.3 Annuities
1.3.1 Simple Annuities
1.3.1.1 Ordinary Annuities
1.3.1.2 Annuities Due
1.3.1.3 Deferred Annuities
1.3.2 Determining the Term of an Annuity
1.3.3 General Annuities
1.3.4 Perpetuities
1.3.5 Continuous Annuities
1.4 Bonds
1.4.1 Introduction and Terminology
1.4.2 Zero-Coupon Bonds
1.4.3 Coupon Bonds
1.4.4 Serial Bonds, Strip Bonds, and Callable Bonds
1.5 Yield Rates
1.5.1 Internal Rate of Return and Evaluation Criteria
1.5.2 Determining Yield Rates for Bonds
1.5.2.1 Zero-Coupon Bonds
1.5.2.2 Coupon Bonds
1.5.3 Approximation Methods
1.5.3.1 The Method of Averages
1.5.3.2 The Method of Interpolation
1.5.3.3 Numerical Methods
1.5.4 The Yield Curve
1.6 Exercises
Figure 1.1
Figure 1.1
Figure 1.2
Figure 1.4
Figure 1.5
Figure 1.7
Figure 1.8
Figure 1.9
Figure 1.12
Figure 1.13
Figure 1.14
Figure 1.15
Figure 1.17
Table 1.3
Table 1.3
Table 1.6
Table 1.10
Table 1.11
Table 1.16
Chapter 2 Primer on Pricing Risky Securities
2.1 Stocks and Stock Price Models
2.1.1 Underlying Assets and Derivative Securities
2.1.2 Basic Assumptions for Asset Price Models
2.2 Basic Price Models
2.2.1 A Single-Period Binomial Model
2.2.2 A Discrete-Time Model with a Finite Number of States
2.2.2.1 Asset Returns
2.2.3 Introducing the Binomial Tree Model
2.2.4 Self-Financing Investment Strategies in the Binomial Model
2.2.5 Log-Normal Pricing Model
2.3 Arbitrage and Risk-Neutral Pricing
2.3.1 The Law of One Price
2.3.2 A First Look at Arbitrage in the Single-Period Binomial Model
2.3.3 Arbitrage in the Binomial Tree Model
2.3.4 Risk-Neutral Probabilities
2.3.5 Martingale Property
2.3.6 Risk-Neutral Log-Normal Model
2.4 Value at Risk
2.5 Dividend Paying Stock
2.6 Exercises
Figure 2.1
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Chapter 3 Portfolio Management
3.1 Expected Utility Functions
3.1.1 Utility Functions
3.1.1.1 Risk Aversion
3.1.2 Mean-Variance Criterion
3.2 Portfolio Optimization for Two Assets
3.2.1 Portfolio of Two Assets
3.2.2 Portfolio Lines
3.2.2.1 Case with |ρ12| = 1
3.2.2.2 Case with |ρ12| < 1| < 1
3.2.2.3 Case with a Risk-Free Asset
3.2.3 The Minimum Variance Portfolio
3.2.3.1 Case without Short Selling
3.2.4 Selection of Optimal Portfolios
3.2.4.1 Minimum Variance Portfolio
3.2.4.2 Maximum Expected Utility Portfolio
3.2.4.3 Minimum Loss-Probability Portfolio
3.3 Portfolio Optimization for N Assets
3.3.1 Portfolios of Several Assets
3.3.2 The Minimum Variance Portfolio
3.3.3 The Minimum Variance Portfolio Line
3.3.4 Case without Short Selling
3.3.5 Efficient Frontier and Capital Market Line
3.4 The Capital Asset Pricing Model
3.5 Exercises
Figure 3.1
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Figure 3.10
Figure 3.11
Figure 3.12
Figure 3.13
Chapter 4 Primer on Derivative Securities
4.1 Forward Contracts
4.1.1 No-Arbitrage Evaluation of Forward Contracts
4.1.1.1 Forward Price for a Stock Paying No Dividends
4.1.1.2 Forward Price for a Stock Paying Dividends
4.1.2 Value of a Forward Contract
4.2 Basic Options Theory
4.2.1 Concept of an Option Contract
4.2.2 Put-Call Parities
4.2.3 Properties of European Options
4.2.4 Early Exercise and American Options
4.2.4.1 Relation to European Option Prices
4.2.4.2 Put-Call Parity Estimate
4.2.4.3 Monotonicity Properties of American Option Prices
4.2.5 Nonstandard European Options
4.3 Basics of Option Pricing
4.3.1 Pricing of European-Style Derivatives in the Binomial Tree Model
4.3.1.1 Replication and Pricing of Options in the Single-Period Binomial Model
4.3.1.2 Pricing in the Binomial Tree Model
4.3.2 Pricing of American Options in the Binomial Tree Model
4.3.3 Option Pricing in the Log-Normal Model: The Black–Scholes– Merton Formula
4.3.4 Greeks and Hedging of Options
4.3.4.1 Delta of a Derivative in the Binomial Model
4.3.4.2 Delta of a Derivative in the Log-Normal Model
4.3.4.3 Delta Hedging
4.3.4.4 Greeks
4.3.5 Black–Scholes Equation
4.4 Exercises
Figure 4.1
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
Figure 4.18
Part II Discrete-Time Modelling
Chapter 5 Single-Period Arrow–Debreu Models
5.1 Specification of the Model
5.1.1 Finite-State Economy. Vector Space of Payoffs. Securities
5.1.2 Initial Price Vector and Payoff Matrix
5.1.3 Portfolios of Base Securities
5.2 Analysis of the Arrow–Debreu Model
5.2.1 Redundant Assets and Attainable Securities
5.2.2 Completeness of the Model
5.3 No-Arbitrage Asset Pricing
5.3.1 The Law of One Price
5.3.2 Arbitrage
5.3.3 The First Fundamental Theorem of Asset Pricing
5.3.3.1 The First FTAP: Sufficiency Part
5.3.3.2 The First FTAP: Necessity Part
5.3.3.3 A Geometric Interpretation of the First FTAP
5.3.4 Risk-Neutral Probabilities
5.3.5 The Second Fundamental Theorem of Asset Pricing
5.3.6 Investment Portfolio Optimization
5.4 Pricing in an Incomplete Market
5.4.1 A Trinomial Model of an Incomplete Market
5.4.2 Pricing Nonattainable Payoffs: The Bid-Ask Spread
5.5 Change of Numéraire
5.5.1 The Concept of a Numéraire Asset
5.5.2 Change of Numéraire in a Binomial Model
5.5.3 Change of Numéraire in a Multinomial Model
5.6 Exercises
Figure 5.1
Figure 5.1
Figure 5.2
Figure 5.3
Chapter 6 Introduction to Discrete-Time Stochastic Calculus
6.1 A Multi-Period Binomial Probability Model
6.1.1 The Binomial Probability Space
6.1.1.1 A Sample Space
6.1.1.2 Random Variables
6.1.1.3 Probability Measure
6.1.2 Random Processes
6.1.2.1 Binomial Price Process and Path Probabilities
6.1.2.2 Random Walk
6.2 Information Flow
6.2.1 Partitions and Their Refinements
6.2.1.1 Partition Generated by a Random Variable
6.2.1.2 Refinements of Partitions
6.2.2 Sigma-Algebras
6.2.2.1 Construction of a Sigma-Algebra from a Partition
6.2.2.2 Sigma-Algebra Generated by a Random Variable
6.2.3 Filtration
6.2.3.1 Construction of a Filtration from an Information Structure
6.2.3.2 Construction of a Filtration from a Stochastic Process: Natural Filtration
6.2.4 Filtered Probability Space
6.3 Conditional Expectation and Martingales
6.3.1 Measurability of Random Variables and Processes
6.3.2 Conditional Expectations
6.3.2.1 Conditioning on an Event
6.3.2.2 Conditioning on a Sigma-Algebra
6.3.2.3 Conditioning on a Random Variable
6.3.3 Properties of Conditional Expectations
6.3.3.1 Linearity
6.3.3.2 Independence
6.3.3.3 Taking out What Is Known
6.3.3.4 Tower Property (Iterated Conditioning)
6.3.4 Conditioning in the Binomial Model
6.3.5 Sub-, Super-, and True Martingales
6.3.5.1 Examples
6.3.6 Classification of Stochastic Processes
6.3.7 Stopping Times
6.4 Exercises
Figure 6.1
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Chapter 7 Replication and Pricing in the Binomial Tree Model
7.1 The Standard Binomial Tree Model
7.2 Self-Financing Strategies and Their Value Processes
7.2.1 Equivalent Martingale Measures for the Binomial Model
7.3 Dynamic Replication in the Binomial Tree Model
7.3.1 Dynamic Replication of Payoffs
7.3.2 Replication and Valuation of Random Cash Flows
7.4 Pricing and Hedging Non-Path-Dependent Derivatives
7.5 Pricing Formulae for Standard European Options
7.6 Pricing and Hedging Path-Dependent Derivatives
7.6.1 Average Asset Prices and Asian Options
7.6.2 Extreme Asset Prices and Lookback Options
7.6.3 Recursive Evaluation of Path-Dependent Options
7.6.3.1 Pricing Lookback Options on a Two-Dimensional Lattice
7.7 American Options
7.7.1 Writer's Perspective: Pricing and Hedging
7.7.2 Buyer's Perspective: Optimal Exercise
7.7.3 Early-Exercise Boundary
7.7.4 Pricing American Options: The Case with Dividends
7.8 Exercises
Figure 7.1
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7
Figure 7.8
Figure 7.9
Figure 7.10
Figure 7.13
Figure 7.14
Figure 7.15
Figure 7.16
Table 7.11
Table 7.11
Table 7.12
Chapter 8 General Multi-Asset Multi-Period Model
8.1 Main Elements of the Model
8.2 Assets, Portfolios, and Strategies
8.2.1 Payoffs and Assets
8.2.2 Static and Dynamic Portfolios
8.2.3 Self-Financing Strategies
8.2.4 Replication of Payoffs
8.3 Fundamental Theorems of Asset Pricing
8.3.1 Arbitrage Strategies
8.3.2 Enhancing the Law of One Price
8.3.3 Equivalent Martingale Measures
8.3.4 Calculation of Martingale Measures
8.3.5 The First and Second FTAPs
8.3.6 Pricing and Hedging Derivatives
8.3.7 Radon–Nikodym Derivative Process and Change of Numéraire
8.4 Examples of Discrete-Time Models
8.4.1 Binomial Tree Model with Stochastic Volatility
8.4.2 Binomial Tree Model for Interest Rates
8.5 Exercises
Figure 8.1
Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5
Figure 8.6
Part III Continuous-Time Modelling
Chapter 9 Essentials of General Probability Theory
9.1 Random Variables and Lebesgue Integration
9.2 Multidimensional Lebesgue Integration
9.3 Multiple Random Variables and Joint Distributions
9.4 Conditioning
9.5 Changing Probability Measures
Chapter 10 One-Dimensional Brownian Motion and Related Processes
10.1 Multivariate Normal Distributions
10.1.1 Multivariate Normal Distribution
10.1.2 Conditional Normal Distributions
10.2 Standard Brownian Motion
10.2.1 One-Dimensional Symmetric Random Walk
10.2.2 Formal Definition and Basic Properties of Brownian Motion
10.2.3 Multivariate Distribution of Brownian Motion
10.2.4 The Markov Property and the Transition PDF
10.2.5 Quadratic Variation and Nondifferentiability of Paths
10.3 Some Processes Derived from Brownian Motion
10.3.1 Drifted Brownian Motion
10.3.2 Geometric Brownian Motion
10.3.3 Brownian Bridge
10.3.4 Gaussian Processes
10.4 First Hitting Times and Maximum and Minimum of Brownian Motion
10.4.1 The Reflection Principle: Standard Brownian Motion
10.4.2 Translated and Scaled Driftless Brownian Motion
10.4.3 Brownian Motion with Drift
10.4.3.1 Translated and Scaled Brownian Motion with Drift
10.5 Exercises
Figure 10.1
Figure 10.1
Figure 10.2
Figure 10.3
Figure 10.4
Figure 10.5
Figure 10.6
Chapter 11 Introduction to Continuous-Time Stochastic Calculus
11.1 The Riemann Integral of Brownian Motion
11.1.1 The Riemann Integral
11.1.2 The Integral of a Brownian Path
11.2 The Riemann–Stieltjes Integral of Brownian Motion
11.2.1 The Riemann–Stieltjes Integral
11.2.2 Integrals w.r.t. Brownian Motion
11.3 The Itô Integral and Its Basic Properties
11.3.1 The Itô Integral for Simple Processes
11.3.2 Properties of the Itô Integral
11.4 Itô Processes and Their Properties
11.4.1 Gaussian Processes Generated by Itô Integrals
11.4.2 Itô Processes
11.4.3 Quadratic (Co-) Variation
11.5 Itô's Formula for Functions of BM and Itô Processes
11.5.1 Itô's Formula for Functions of BM
11.5.2 Itô's Formula for Itô Processes
11.6 Stochastic Differential Equations
11.6.1 Solutions to Linear SDEs
11.6.2 Existence and Uniqueness of a Strong Solution of an SDE
11.7 The Markov Property, Feynman–Kac Formulae, and Transition CDFs and PDFs
11.7.1 Forward Kolmogorov PDE
11.7.2 Transition CDF/PDF for Time-Homogeneous Diffusions
11.8 Radon–Nikodym Derivative Process and Girsanov's Theorem
11.8.1 Some Applications of Girsanov’s Theorem
11.9 Brownian Martingale Representation Theorem
11.10 Stochastic Calculus for Multidimensional BM
11.10.1 The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM
11.10.2 Multidimensional SDEs, Feynman–Kac Formulae, and Transition CDFs and PDFs
11.10.3 Girsanov’s Theorem for Multidimensional BM
11.10.4 Martingale Representation Theorem for Multidimensional BM
11.11 Exercises
Figure 11.1
Figure 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 Economy
12.1.1 Pricing Standard European Calls and Puts
12.1.2 Hedging Standard European Calls and Puts
12.1.3 Europeans with Piecewise Linear Payoffs
12.1.4 Power Options
12.1.5 Dividend Paying Stock
12.1.5.1 The Case of Continuous Dividend Paying Stock
12.1.5.2 The Case of Discrete-Time Dividends
12.2 Forward Starting and Compound Options
12.3 Some European-Style Path-Dependent Derivatives
12.3.1 Risk-Neutral Pricing under GBM
12.3.2 Pricing Single Barrier Options
12.3.3 Pricing Lookback Options
12.4 Exercises
Figure 12.1
Figure 12.1
Figure 12.2
Figure 12.3
Figure 12.4
Figure 12.5
Chapter 13 Risk-Neutral Pricing in a Multi-Asset Economy
13.1 General Multi-Asset Market Model: Replication and Risk-Neutral Pricing
13.2 Black–Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives within a General Diffusion Model
13.2.1 Standard European Option Pricing for Multi-Stock GBM
13.2.2 Explicit Pricing Formulae for the GBM Model
13.2.2.1 Exchange and Other Related Options
13.2.2.2 Other Basket Options
13.2.3 Cross-Currency Option Valuation
13.3 Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets
13.4 Exercises
Chapter 14 American Options
14.1 Basic Properties of Early-Exercise Options
14.2 Arbitrage-Free Pricing of American Options
14.2.1 Optimal Stopping Formulation and Early-Exercise Boundary
14.2.2 The Smooth Pasting Condition
14.2.3 Put-Call Symmetry Relation
14.2.4 Dynamic Programming Approach for Bermudan Options
14.3 Perpetual American Options
14.3.1 Pricing a Perpetual Put Option
14.3.2 Pricing a Perpetual Call Option
14.4 Finite-Expiration American Options
14.4.1 The PDE Formulation
14.4.2 The Integral Equation Formulation
14.5 Exercises
Figure 14.1
Figure 14.1
Figure 14.2
Figure 14.3
Figure 14.4
Figure 14.5
Chapter 15 Interest-Rate Modelling and Derivative Pricing
15.1 Basic Fixed Income Instruments
15.1.1 Bonds
15.1.2 Forward Rates
15.1.3 Arbitrage-Free Pricing
15.1.4 Fixed Income Derivatives
15.1.4.1 Options on Bonds
15.1.4.2 Cap and Caplets
15.1.4.3 Swap and Swaptions
15.2 Single-Factor Models
15.2.1 Diffusion Models for the Short Rate Process
15.2.2 PDE for the Zero-Coupon Bond Value
15.2.3 Affine Term Structure Models
15.2.4 The Ho–Lee Model
15.2.5 The Vasiček Model
15.2.6 The Cox–Ingersoll–Ross Model
15.3 Heath–Jarrow–Morton Formulation
15.3.1 HJM under Risk-Neutral Measure
15.3.2 Relationship between HJM and Affine Yield Models
15.3.2.1 The Ho–Lee Model in the HJM Framework
15.3.2.2 The Vasiček Model in the HJM Framework
15.4 Multifactor Affine Term Structure Models
15.4.1 Gaussian Multifactor Models
15.4.2 Equivalent Classes of Affine Models
15.5 Pricing Derivatives under Forward Measures
15.5.1 Forward Measures
15.5.2 Pricing Stock Options under Stochastic Interest Rates
15.5.3 Pricing Options on Zero-Coupon Bonds
15.6 LIBOR Model
15.6.1 LIBOR Rates
15.6.2 Brace–Gatarek–Musiela Model of LIBOR Rates
15.6.3 Pricing Caplets, Caps, and Swaps
15.7 Exercises
Chapter 16 Alternative Models of Asset Price Dynamics
16.1 Stochastic Volatility Diffusion Models
16.1.1 Local Volatility Models
16.1.2 Constant Elasticity of Variance Model
16.1.2.1 Definition and Basic Properties
16.1.2.2 Transition Probability Law
16.1.2.3 Pricing European Options
16.1.3 The Heston Model
16.2 Models with Jumps
16.2.1 The Poisson Process
16.2.2 Jump-Diffusion Models with a Compound Poisson Component
16.2.3 The Variance Gamma Model
16.3 Exercises
Figure 16.1
Figure 16.1
Figure 16.2
Figure 16.3
Part IV Computational Techniques
Chapter 17 Introduction to Monte Carlo and Simulation Methods
17.1 Introduction
17.1.1 The “Hit-or-Miss” Method
17.1.2 The Law of Large Numbers
17.1.3 Approximation Error and Confidence Interval
17.1.4 Parallel Monte Carlo Methods
17.1.5 One Monte Carlo Application: Numerical Integration
17.2 Generation of Uniformly Distributed Random Numbers
17.2.1 Uniform Probability Distributions
17.2.2 Linear Congruential Generator
17.3 Generation of Nonuniformly Distributed Random Numbers
17.3.1 Transformations of Random Variables
17.3.2 Inversion Method
17.3.2.1 Inverse Distribution Function
17.3.2.2 The Chop-Down Search Method
17.3.2.3 The Binomial Search Method
17.3.3 Composition Methods
17.3.3.1 Mixture of PDFs
17.3.3.2 Randomized Gamma Distributions
17.3.3.3 The Alias Method by Walker
17.3.4 Acceptance-Rejection Methods
17.3.5 Multivariate Sampling
17.3.5.1 Sampling by Conditioning
17.3.5.2 The Box–Müller method
17.3.5.3 Simulation of Multivariate Normals
17.4 Simulation of Random Processes
17.4.1 Simulation of Brownian Processes
17.4.1.1 Sequential Sampling
17.4.1.2 Bridge Sampling
17.4.2 Simulation of Gaussian Processes
17.4.3 Diffusion Processes: Exact Simulation Methods
17.4.3.1 The Stochastic Calculus Approach
17.4.3.2 The PDF Approach
17.4.4 Diffusion Processes: Approximation Schemes
17.4.4.1 Types of Convergence
17.4.4.2 The Euler Scheme
17.4.4.3 Extrapolation
17.4.4.4 Error Analysis
17.4.5 Simulation of Processes with Jumps
17.4.5.1 Poisson Processes
17.4.5.2 Subordinated Processes
17.5 Variance Reduction Methods
17.5.1 Numerical Integration by a Direct Monte Carlo Method
17.5.2 Importance Sampling Method
17.5.3 Change of Probability Measure
17.5.4 Control Variate Method
17.5.5 Antithetic Variate
17.5.6 Conditional Sampling
17.5.7 Stratified Sampling
17.6 Exercises
References
Figure 17.2
Figure 17.2
Figure 17.3
Figure 17.4
Figure 17.5
Figure 17.6
Table 17.1
Table 17.1
Table 17.7
Chapter 18 Numerical Applications to Derivative Pricing
18.1 Overview of Deterministic Numerical Methods
18.1.1 Quadrature Formulae
18.1.1.1 Newton–Cotes Quadrature Formulae
18.1.1.2 Gaussian Quadrature Formulae
18.1.1.3 Composite Quadrature Formulae
18.1.1.4 Extrapolation and Romberg Integration
18.1.2 Finite-Difference Methods
18.1.2.1 Finite-Difference Approximations for ODEs
18.1.2.2 Second-Order Linear PDEs
18.1.2.3 Finite-Difference Approximations for the Heat Equation
The Explicit Method
The Implicit Method
The Crank–Nicolson Method
18.1.2.4 Stability Analysis
Stability Analysis of the Explicit Method
Stability Analysis of the Implicit Method
Stability Analysis of the Crank–Nicolson Method
18.2 Pricing European Options
18.2.1 Pricing European Options by Quadrature Rules
18.2.2 Pricing European Options by the Monte Carlo Method
18.2.3 Pricing European Options by Tree Methods
18.2.3.1 Binomial Model
18.2.3.2 Multinomial Models
18.2.4 Pricing European Options by PDEs
18.2.4.1 Pricing by the Heat Equation
18.2.4.2 Pricing by the Black–Scholes PDE
The Explicit Method
The Implicit Method
The Crank–Nicolson Method
18.2.5 Calibration of Asset Price Models to Empirical Data
18.2.5.1 Least Squares Method
18.2.5.2 Maximum Likelihood Estimation
18.3 Pricing Early-Exercise and Path-Dependent Options
18.3.1 Pricing American and Bermudan Options
18.3.1.1 Pricing American Options by Tree Methods
18.3.1.2 Pricing Bermudan Options by the Monte Carlo Method
18.3.2 Pricing Asian Options
18.3.2.1 Pricing Discrete-Time Asian Options by the Monte Carlo Method
Figure 18.2
Figure 18.2
Figure 18.7
Figure 18.9
Figure 18.11
Figure 18.12
Figure 18.14
Figure 18.16
Figure 18.17
Table 18.1
Table 18.1
Table 18.3
Table 18.4
Table 18.5
Table 18.6
Table 18.8
Table 18.10
Table 18.13
Table 18.15
Table 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 Processes
Introduction to Mathematics of Finance
Mathematics of Finance (Discrete-Time)
Mathematics of Finance (Continuous-Time)
Computational Methods
Financial Economics
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