Generalized Integral Transforms in Mathematical Finance

✍ Scribed by Andrey Itkin, Alexander Lipton, Dmitry Muravey

• Publisher: World Scientific
• Year: 2021
• Tongue: English
• Leaves: 508
• Category: Library

✦ Table of Contents

Contents
Foreword
Preface
About the Authors
Acknowledgments
List of Figures
PART 1. One-Factor Financial Models and Problems
1. Stochastic Engines and Partial Differential Equations
1.1 Introduction
1.2 Stochastic processes
1.3 Markov processes
1.4 Diffusions
1.5 Wiener processes
1.6 SDE and mappings
1.7 Linear SDEs
1.7.1 Arithmetic Brownian motion
1.7.2 Geometric Brownian motion
1.7.3 Mixed Brownian motion
1.7.4 The Ornstein-Uhlenbeck process
2. Popular One-Factor Models by Asset Classes
2.1 Introduction
2.2 Equities, foreign exchange, and commodities
2.2.1 The Black-Scholes-Merton model
2.2.2 The Bachelier model
2.2.3 The CEV model
2.2.4 The local volatility model
2.3 Fixed Income
2.3.1 The Hull-White model
2.3.2 The Cox-Ingersoll-Ross model
2.3.3 The Black-Karasinski model
2.3.4 The modified Black-Karasinski (Verhulst) model
2.4 Credit
2.5 Analytical solution of financial PDEs
2.5.1 General considerations
2.5.2 Reducibility to the Wiener process
2.5.3 Reducibility to the Bessel process
2.5.4 Cherkasov's transformation
PART 2. Integral Equations
3. Fredholm Integral Equations
3.1 Basic definitions
3.2 Fredholm equations with degenerate kernels
3.3 Fredholm equations with non-degenerate kernels
3.4 The Fredholm determinant: Resolvent
3.5 Fredholm equations of the first kind
3.6 Nonlinear Fredholm equations
4. Volterra Integral Equations
4.1 Picard iterations and the resolvent formalism
4.2 Convolution kernels
4.2.1 Solution by using the Laplace transform
4.3 Equations of the second kind with weakly singular kernels
4.4 Systems of linear Volterra integral equations
4.5 Nonlinear Volterra equations
5. Solving Integral Equations Numerically
5.1 Approximation of the integral by quadratures
5.1.1 Block-by-block method based on quadratic interpolation
5.1.2 Higher order quadratures
5.2 Approximation by an Abel integral equation
5.3 The differential transform method
5.4 Other methods
5.5 Solving Fredholm equations of the first kind
PART 3. Integral Transforms
6. Classical Integral Transforms
6.1 The Fourier transform
6.2 The Laplace transform
6.3 The Hankel transform
6.4 The Index transform
6.5 The Mellin transform
7. Generalized Integral Transforms
7.1 PDE with moving boundaries
7.2 Sturm–Liouville expansion
7.3 Construction of the GIT
7.3.1 Another illustration of the GIT method
7.4 General case
8. Method of Heat Potentials
8.1 Statement of the problem
8.1.1 The limiting value of φ(t)
8.1.2 The limiting value of ψ(t)
8.1.3 Other useful results
8.2 Solution via a Volterra integral equation
8.2.1 Other domains
8.3 A generalized method of heat potentials
8.4 Connection to fractional differential equations
PART 4. Equities, FX, and Commodities
9. Barrier and American Options
9.1 Problem for pricing barrier options
9.1.1 Transformation to the heat equation
9.1.2 Solution of the barrier pricing problem
9.1.3 The inverse transform
9.1.3.1 Residual machinery
9.1.3.2 The final solution
9.2 Pricing American options
9.2.1 Solution for the American Call option price
9.3 Numerical example
9.4 Discussion
9.5 Extension to the Black-Scholes model
10. On the First Hitting Time Density for a Reducible Diffusion Process
10.1 Problem formulation and initial transformations
10.2 Main results
10.3 Representative examples
10.3.1 Linear boundary
10.3.2 Time-dependent geometric Brownian motion
10.3.3 Time-dependent OU process
10.4 Alternative transformation for the OU process
10.5 Numerical examples
10.5.1 Wiener process with linear boundary
10.5.2 Standard OU process with constant boundary
10.5.3 Comparison with other methods
10.6 Final notes
11. Optimal Mean-Reverting Trading Strategies
11.1 Formulation of the problem
11.1.1 Formulation in terms of HPs
11.2 The method of heat potentials
11.3 Numerical method
11.3.1 Computation of the Sharpe ratio
11.3.2 Comparison with Monte Carlo simulations
11.3.3 Optimization of the Sharpe ratio
11.4 Traditional approaches
11.4.1 Expectation and variance of the trade's duration
11.4.2 Renewal theory approach
11.4.3 Perpetual value function
11.4.4 Linear transaction costs
PART 5. Fixed Income
12. Barrier Options in the Hull-White Model
12.1 Down-and-Out barrier option
12.1.1 Transformation to the heat equation
12.1.2 Solution of the barrier pricing problem
12.1.3 Second solution of the barrier pricing problem
12.2 Numerical example
12.3 Final notes
13. Barrier Options in the Time-Dependent CEV and CIR Models
13.1 The CEV model
13.2 The CIR model
13.2.1 Down-and-Out barrier option
13.3 The method of Bessel potentials
13.3.1 Domain y(τ) ≤ z < ∞
13.3.2 Domain 0 < z < y(τ)
13.3.3 Double barrier options
13.4 The method of generalized integral transform
13.4.1 Domain 0 < z < y(τ)
13.4.1.1 The inverse transform
13.4.1.2 Some approximations
13.4.1.3 Connection to the first passage time problem
13.4.2 Domain z > y(τ)
13.5 Numerical experiments
13.5.1 Comparison with the BP method
13.5.2 Comparison with the GIT method
13.6 Discussion
14. Barrier Options in the BK and Verhulst Models
14.1 Integral equation for the ZCB price in the BK model
14.2 Integral equation for the ZCB price in the Verhulst model
14.3 A closed-form solution for the ZCB price in the Verhulst model
14.3.1 Residuals and branching points
14.3.2 Calculation of residuals
14.3.3 Calculation of the integrals at different branches
14.3.4 A closed-form solution for the ZCB price
14.4 Numerical examples
14.4.1 Solving Eq. (14.9) by using RDTM
PART 6. Credit and Miscellaneous Problems
15. Calibrating the Default Boundary to a Constant Default Intensity
15.1 Governing system of integral equations and its solutions
16. McKean-Vlasov Equation with Feedback Through Hitting a Boundary
16.1 Mean-field limit for large banking system
16.2 Transition density and known regularity results
16.3 The method of heat potentials
16.3.1 Semi-closed formula for the transition density
16.3.2 Computation of loss rate over the boundary
16.3.3 Direct computation of loss rate
16.4 Solution of the McKean-Vlasov equation
16.4.1 Particular cases
16.4.2 Perturbation solution
16.4.3 Numerical solution
16.5 Numerical tests and results
16.5.1 Parameter studies
17. Miscellaneous Problems
17.1 The supercooled Stefan problem
17.1.1 Solution by the HP method
17.1.2 Solution by the GIT method
17.2 Time-dependent coefficient of heat exchange
17.3 The integrate-and-fire neuron excitation model
17.3.1 The stationary problem
17.3.2 The unsteady problem
PART 7. Multilayer Problems
18. Double Barrier Options
18.1 Statement of the problem
18.2 Solution by the GIT method
18.2.1 The inverse transform
18.2.2 Connection to the Jacobi theta function
18.2.3 Determining Ψ(τ) and Φ(τ)
18.2.4 The Poisson summation formula and alternative representations
18.2.5 A system of Volterra equations for Ψ(τ) and Φ(τ)
18.3 Solution by the HP method
18.3.1 A system of Volterra equations
18.4 Numerical example
18.5 Final notes
Appendices
18.A Simplification of Eq. (18.29)
18.B Transformation of Eqs. (18.30) to (18.39)
18.B.1 The limiting values x → y(τ) and x → z(τ) in Eq. (18.80)
19. Multilayer Heat Equations: Application to Finance
Introduction
19.1 Solving the ML heat equation via the HP method
19.2 Solving the ML heat equation via the GIT method
19.2.1 Background
19.2.2 Solution of the heat equation
19.2.3 Solution of Eq. (19.10) when σ is piecewise constant
19.3 Application to finance
19.3.1 One-factor short-rate models
19.3.1.1 Pricing zero-coupon bonds and barrier options for the Black-Karasinski and similar models
19.3.1.2 The modified BK (Verhulst) model
19.3.1.3 Pricing barrier options in the Verhulst model
19.3.2 Local volatility and Dupire's equation
19.4 Solution of the Volterra equations
19.4.1 The heat equation in a strip
19.4.2 The Laplace transform
19.5 Numerical experiments
19.5.1 Constant volatility σi
19.5.2 Piecewise constant volatility σi
19.6 Discussion
Appendices
19.A Transformation of a non-divergent heat equation to a divergent form
19.B Multilayer method for time-inhomogeneous coefficients and the domain
19.C Coefficients of Eq. (19.94)
Bibliography
Index