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

Contents
Preface
Acknowledgment
Using this Book
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, It ˆo integral
		1.3.1 Elementary processes
		1.3.2 Ito 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 Ito process
		2.1.2 Ito’s lemma
		2.1.3 Distributions of S(t) and logS(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 Ito’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 Levy processes
		5.3.1 Finite activity exponential L´evy processes
		5.3.2 PIDE and the L´evy triplet
		5.3.3 Equivalent martingale measure
	5.4 Infinite activity exponential Levy 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 Ito’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 num´eraire 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 Sch¨ obel-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 Schobel-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