Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes

Cover
Halftitle
Title
Copyright
Dedication
Preface
Acknowledgment
Using this Book
Contents
1 Basics about Stochastic Processes
    1.1 Stochastic variables
        1.1.1 Density function, expectation, variance
        1.1.2 Characteristic function
        1.1.3 Cumulants and moments
    1.2 Stochastic processes, martingale property
        1.2.1 Wiener process
        1.2.2 Martingales
        1.2.3 Iterated expectations (Tower property)
    1.3 Stochastic integration, Itô integral
        1.3.1 Elementary processes
        1.3.2 Itô isometry
        1.3.3 Martingale representation theorem
    1.4 Exercise set
2 Introduction to Financial Asset Dynamics
    2.1 Geometric Brownian motion asset price process
        2.1.1 Itô process
        2.1.2 Itô’s lemma
        2.1.3 Distributions of S(t) and log S(t)
    2.2 First generalizations
        2.2.1 Proportional dividend model
        2.2.2 Volatility variation
        2.2.3 Time-dependent volatility
    2.3 Martingales and asset prices
        2.3.1 P-measure prices
        2.3.2 Q-measure prices
        2.3.3 Parameter estimation under real-world measure P
    2.4 Exercise set
3 The Black-Scholes Option Pricing Equation
    3.1 Option contract definitions
        3.1.1 Option basics
        3.1.2 Derivation of the partial differential equation
        3.1.3 Martingale approach and option pricing
    3.2 The Feynman-Kac theorem and the Black-Scholes model
        3.2.1 Closed-form option prices
        3.2.2 Green’s functions and characteristic functions
        3.2.3 Volatility variations
    3.3 Delta hedging under the Black-Scholes model
    3.4 Exercise set
4 Local Volatility Models
    4.1 Black-Scholes implied volatility
        4.1.1 The concept of implied volatility
        4.1.2 Implied volatility; implications
        4.1.3 Discussion on alternative asset price models
    4.2 Option prices and densities
        4.2.1 Market implied volatility smile and the payoff
        4.2.2 Variance swaps
    4.3 Non-parametric local volatility models
        4.3.1 Implied volatility representation of local volatility 
        4.3.2 Arbitrage-free conditions for option prices
        4.3.3 Advanced implied volatility interpolation
        4.3.4 Simulation of local volatility model
    4.4 Exercise set
5 Jump Processes
    5.1 Jump diffusion processes
        5.1.1 Itô’s lemma and jumps
        5.1.2 PIDE derivation for jump diffusion process
        5.1.3 Special cases for the jump distribution
    5.2 Feynman-Kac theorem for jump diffusion process
        5.2.1 Analytic option prices
        5.2.2 Characteristic function for Merton’s model
        5.2.3 Dynamic hedging of jumps with the Black-Scholes model
    5.3 Exponential Lévy processes
        5.3.1 Finite activity exponential Lévy processes
        5.3.2 PIDE and the Lévy triplet
        5.3.3 Equivalent martingale measure
    5.4 Infinite activity exponential Lévy processes
        5.4.1 Variance Gamma process
        5.4.2 CGMY process
        5.4.3 Normal inverse Gaussian process
    5.5 Discussion on jumps in asset dynamics
    5.6 Exercise set
6 The COS Method for European Option Valuation
    6.1 Introduction into numerical option valuation
        6.1.1 Integrals and Fourier cosine series
        6.1.2 Density approximation via Fourier cosine expansion
    6.2 Pricing European options by the COS method
        6.2.1 Payoff coefficients
        6.2.2 The option Greeks
        6.2.3 Error analysis COS method
        6.2.4 Choice of integration range
    6.3 Numerical COS method results
        6.3.1 Geometric Brownian Motion 
        6.3.2 CGMY and VG processes
        6.3.3 Discussion about option pricing
    6.4 Exercise set
7 Multidimensionality, Change of Measure, Affine Processes
    7.1 Preliminaries for multi-D SDE systems
        7.1.1 The Cholesky decomposition
        7.1.2 Multi-D asset price processes
        7.1.3 Itô’s lemma for vector processes
        7.1.4 Multi-dimensional Feynman-Kac theorem
    7.2 Changing measures and the Girsanov theorem
        7.2.1 The Radon-Nikodym derivative
        7.2.2 Change of numéraire examples
        7.2.3 From P to Q in the Black-Scholes model
    7.3 Affine processes
        7.3.1 Affine diffusion processes
        7.3.2 Affine jump diffusion processes
        7.3.3 Affine jump diffusion process and PIDE
    7.4 Exercise set
8 Stochastic Volatility Models
    8.1 Introduction into stochastic volatility models
        8.1.1 The Schöbel-Zhu stochastic volatility model
        8.1.2 The CIR process for the variance
    8.2 The Heston stochastic volatility model
        8.2.1 The Heston option pricing partial differential equation
        8.2.2 Parameter study for implied volatility skew and smile
        8.2.3 Heston model calibration
    8.3 The Heston SV discounted characteristic function
        8.3.1 Stochastic volatility as an affine diffusion process
        8.3.2 Derivation of Heston SV characteristic function
    8.4 Numerical solution of Heston PDE
        8.4.1 The COS method for the Heston model
        8.4.2 The Heston model with piecewise constant parameters
        8.4.3 The Bates model
    8.5 Exercise set
9 Monte Carlo Simulation
    9.1 Monte Carlo basics
        9.1.1 Monte Carlo integration
        9.1.2 Path simulation of stochastic differential equations
    9.2 Stochastic Euler and Milstein schemes
        9.2.1 Euler scheme
        9.2.2 Milstein scheme: detailed derivation
    9.3 Simulation of the CIR process
        9.3.1 Challenges with standard discretization schemes
        9.3.2 Taylor-based simulation of the CIR process
        9.3.3 Exact simulation of the CIR model
        9.3.4 The Quadratic Exponential scheme
    9.4 Monte Carlo scheme for the Heston model
        9.4.1 Example of conditional sampling and integrated variance
        9.4.2 The integrated CIR process and conditional sampling
        9.4.3 Almost exact simulation of the Heston model
        9.4.4 Improvements of Monte Carlo simulation
    9.5 Computation of Monte Carlo Greeks
        9.5.1 Finite differences
        9.5.2 Pathwise sensitivities
        9.5.3 Likelihood ratio method
    9.6 Exercise set
10 Forward Start Options; Stochastic Local Volatility Model
    10.1 Forward start options
        10.1.1 Introduction into forward start options
        10.1.2 Pricing under the Black-Scholes model
        10.1.3 Pricing under the Heston model
        10.1.4 Local versus stochastic volatility model
    10.2 Introduction into stochastic-local volatility model
        10.2.1 Specifying the local volatility
        10.2.2 Monte Carlo approximation of SLV expectation
        10.2.3 Monte Carlo AES scheme for SLV model
    10.3 Exercise set
11 Short-Rate Models
    11.1 Introduction to interest rates
        11.1.1 Bond securities, notional
        11.1.2 Fixed-rate bond
    11.2 Interest rates in the Heath-Jarrow-Morton framework
        11.2.1 The HJM framework
        11.2.2 Short-rate dynamics under the HJM framework
        11.2.3 The Hull-White dynamics in the HJM framework
    11.3 The Hull-White model
        11.3.1 The solution of the Hull-White SDE
        11.3.2 The HW model characteristic function
        11.3.3 The CIR model under the HJM framework
    11.4 The HJM model under the T-forward measure
        11.4.1 The Hull-White dynamics under the T-forward measure
        11.4.2 Options on zero-coupon bonds under Hull-White model
    11.5 Exercise set
12 Interest Rate Derivatives and Valuation Adjustments
    12.1 Basic interest rate derivatives and the Libor rate
        12.1.1 Libor rate
        12.1.2 Forward rate agreement
        12.1.3 Floating rate note
        12.1.4 Swaps
        12.1.5 How to construct a yield curve
    12.2 More interest rate derivatives
        12.2.1 Caps and floors
        12.2.2 European swaptions
    12.3 Credit Valuation Adjustment and Risk Management
        12.3.1 Unilateral Credit Value Adjustment
        12.3.2 Approximations in the calculation of CVA
        12.3.3 Bilateral Credit Value Adjustment (BCVA)
        12.3.4 Exposure reduction by netting
    12.4 Exercise set
13 Hybrid Asset Models, Credit Valuation Adjustment
    13.1 Introduction to affine hybrid asset models
        13.1.1 Black-Scholes Hull-White (BSHW) model
        13.1.2 BSHW model and change of measure
        13.1.3 Schöbel-Zhu Hull-White (SZHW) model
        13.1.4 Hybrid derivative product
    13.2 Hybrid Heston model
        13.2.1 Details of Heston Hull-White hybrid model
        13.2.2 Approximation for Heston hybrid models
        13.2.3 Monte Carlo simulation of hybrid Heston SDEs
        13.2.4 Numerical experiment, HHW versus SZHW model
    13.3 CVA exposure profiles and hybrid models
        13.3.1 CVA and exposure
        13.3.2 European and Bermudan options example
    13.4 Exercise set
14 Advanced Interest Rate Models and Generalizations
    14.1 Libor market model
        14.1.1 General Libor market model specifications
        14.1.2 Libor market model under the HJM framework
    14.2 Lognormal Libor market model
        14.2.1 Change of measure in the LMM
        14.2.2 The LMM under the terminal measure
        14.2.3 The LMM under the spot measure
        14.2.4 Convexity correction
    14.3 Parametric local volatility models
        14.3.1 Background, motivation
        14.3.2 Constant Elasticity of Variance model (CEV)
        14.3.3 Displaced diffusion model
        14.3.4 Stochastic volatility LMM
    14.4 Risk management: The impact of a financial crisis
        14.4.1 Valuation in a negative interest rates environment
        14.4.2 Multiple curves and the Libor rate
        14.4.3 Valuation in a multiple curves setting
    14.5 Exercise set
15 Cross-Currency Models
    15.1 Introduction into the FX world and trading
        15.1.1 FX markets
        15.1.2 Forward FX contract
        15.1.3 Pricing of FX options, the Black-Scholes case
    15.2 Multi-currency FX model with short-rate interest rates
        15.2.1 The model with correlated, Gaussian interest rates
        15.2.2 Pricing of FX options
        15.2.3 Numerical experiment for the FX-HHW model
        15.2.4 CVA for FX swaps
    15.3 Multi-currency FX model with interest rate smile
        15.3.1 Linearization and forward characteristic function
        15.3.2 Numerical experiments with the FX-HLMM model
    15.4 Exercise set
References
Index