Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 10

This book is a detailed introduction to the mathematical theory and foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance.

Major benefits of the book are:

 Step-by-step, incremental build-up of the material.

 Examples and algorithms worked out in detail. Opportunity to modify the algorithms and extend them to your own applications.

 Modern, state-of-the art numerical schemes for PDEs in finance.

 Guidelines on C++ coding (C++11 to C++20); the book is the ideal companion to the author's book Financial Instrument Pricing Using C++ (second edition, 2018).

 The book is structured so that the material can be applied to a range of existing and new application areas.

 We resolve a number of outstanding issues and improve several less-than-optimal numerical methods in finance.

We have divided the book into five parts, with each part addressing a single major issue.

Part A (Chapters 1 to 7) introduces the mathematical and numerical analysis concepts that are needed to understand the finite difference method and its application to computational finance. The main reason for writing the chapters is to make the book as self-contained as possible and to introduce and define standardised notation and results that we use in later chapters. Furthermore, the presented material can also be used as a standalone reference.

We realise that some readers will not be familiar with all of the building blocks that are needed to write finite difference schemes for the Black–Scholes PDE; for this reason, Part A was written to resolve this issue. We identify and discuss all the steps to design and implement a finite difference solver for one-factor finance PDEs. To this end, we take an incremental and single responsibility approach by focusing on one major topic in each chapter:

 Initial value problems and boundary value problems and their numerical approximation.

 Vector spaces, matrix theory and numerical linear algebra.

 My first Crank–Nicolson and Alternating Direction Explicit (ADE) methods for the one-factor Black–Scholes PDE.

Finally, Chapter 1 is devoted to major concepts (such as continuity and differentiability) in real analysis that permeate the book, and for this reason it is important to understand them.

Part A can be used by readers with no prior knowledge of partial differential equations or the finite difference method. It can be used as a mini-course or mini-project to learn the material.

Part B (Chapters 8 to 13) discusses a number of rigorous mathematical techniques relating to boundary value problems and initial boundary value problems for elliptic and parabolic partial differential equations in two-space variables. The goal is to identify and elaborate the underlying theory to unambiguously specify these problems before mapping them to a numerical solution, thereby filling some ‘mathematical gaps’ in current practice. In particular, we develop strategies to preprocess and modify a PDE before we approximate it by the finite difference method, thus removing ad hoc and heuristic tricks that are often used to arrive at (hopefully) robust numerical schemes.

The chapters in this part fill a major gap in the application of PDE/FDM to finance. In general, most of the finance literature glosses over the niceties of analysing PDEs mathematically before approximating them using the finite difference method. The new approach resolves many of issues and heuristic approaches. Some new improvements for two-factor PDEs are:

 Transform a PDE with a mixed derivative term to one in which this term has been removed (the canonical PDE).

 Why domain transformation is better than domain truncation in general.

 A rigorous set of mathematical techniques (Fichera theory, energy estimates) to discover the correct boundary conditions for finance problems.

 The deep relationship between PDEs and stochastic differential equations (SDEs). We discuss formulations and results that are important in calibration applications.

Part B prepares the way for a seamless route from PDEs to robust and understandable finite difference schemes that approximate them. It eliminates much trial-and-error experimentation. In a sense the topics in Part B serve as a reference for the more hands-on topics in later chapters. At the very least, it is important to be aware of the main results.

Part C (Chapters 14 to 17) introduces the mathematical background to the finite difference method for initial boundary value problems for parabolic partial differential equations. It encapsulates in one place all the background information that is needed to construct stable and accurate finite difference schemes for time-dependent problems. The schemes will be applied to one-factor and two-factor finance PDEs in later chapters. The advantage is that the chapters discuss finite difference schemes for generic PDEs that can then be applied to finance PDEs. We also devote two dedicated chapters to Sensitivity Analysis, and we propose at least five methods to compute the derivatives of solutions of initial boundary value problems with respect to underlying parameters. In finance, these are sometimes called option greeks.

The chapters in this part discuss both new as well as established methods to gain a deep understanding of the foundations of the finite difference method and its applications to finance, and beyond:

 Defining stability for initial value problems, consistency and the Lax Equivalence Theorem.

 Modern stability analysis for initial boundary value problems : discrete maximum principle, convergence.

 Six ways to compute sensitivities.

 A compact introduction to complex analysis: the Complex Step Method (CSM).

A good working knowledge of the topics in Part C is essential in order to proceed in an effective manner.

Part D (Chapters 18 to 22) introduces a number of modern and popular finite difference methods (approximately six) to approximate the solution of initial boundary value problems for two-factor partial differential equations. To our knowledge, this is the only book that discusses these methods as well as their comparative strengths together with their applications to option pricing and hedging.

In this part we offer a number of robust and accurate schemes for a range of PDEs:

 Soviet Splitting (Marchuk, Yanenko, Strang).

 Alternating Direction Explicit (ADE) (Saul'yev)

 Method of Lines (MOL).

 Front-fixing and variational methods for free boundary value problems.

We lay the mathematical foundations of all splitting methods by introducing semi-group theory and generalised exponential functions. In short, these schemes in combination with the PDE preprocessing techniques in Part B open up a world beyond the perennial Crank–Nicolson and Alternating Direction Implicit (ADI) methods.

A good working knowledge of the algorithms in Part D is essential. In particular, the ability to program these algorithms (in C++ for obvious reasons) is ideal. Algorithms should work on paper and in the computer.

Part E (Chapters 23 to 26) is concerned with applications of the techniques from the first twenty-two chapters. We discuss finite difference schemes for one-factor and two-factor, stochastic volatility, and interest problems. We also revisit some earlier chapters with a view to examining them in even more detail or from new perspectives. Finally, we offer tips and guidelines on extending the results in this book to other finance problems.

This part uses most of the mathematical and numerical techniques of the first twenty-two chapters, and we apply them to a range of linear and nonlinear PDEs. For each problem, we can preprocess it using the techniques in Part B before deciding which numerical schemes from Part D to use. In particular, we scope the chapters by addressing the following problems and solutions:

 Spread options with domain transformation with Yanenko, Predictor-Corrector and Marchuk two-cycle methods.

 Asian option pricing using ADE. Clarification of some issues relating to boundary conditions. Modern ADI methods.

 Interest rate models (CIR), Feller condition and energy inequalities. ADE method. Relationship with Heston model.

 Monotone schemes (viscosity solution) for two-factor problems with mixed derivatives and applications to Uncertain Volatility Models (UVM).

 One-factor and two-factor Hull–White models using ADE and MOL.