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This volume contains 6 chapters which cover several modern topics
of quantitative finance and reflect the most significant trends
currently shaping this field. The chapters discuss in detail and
make original contributions to stochastic/fractional volatility
models and their asymptotic solutions (Chapter 1); equity trading,
optimal portfolios and related problems (Chapters 2, 5, 6); machine
learning and NLP (Chapters 2, 3); and economic scenario generation
(Chapter 4), and are written by the leading experts in the field.
This book will be useful for both researchers and practitioners.
This book describes several techniques, first invented in physics
for solving problems of heat and mass transfer, and applies them to
various problems of mathematical finance defined in domains with
moving boundaries. These problems include: (a) semi-closed form
pricing of options in the one-factor models with time-dependent
barriers (Bachelier, Hull-White, CIR, CEV); (b) analyzing an
interconnected banking system in the structural credit risk model
with default contagion; (c) finding first hitting time density for
a reducible diffusion process; (d) describing the exercise boundary
of American options; (e) calculating default boundary for the
structured default problem; (f) deriving a semi-closed form
solution for optimal mean-reverting trading strategies; to mention
but some.The main methods used in this book are generalized
integral transforms and heat potentials. To find a semi-closed form
solution, we need to solve a linear or nonlinear Volterra equation
of the second kind and then represent the option price as a
one-dimensional integral. Our analysis shows that these methods are
computationally more efficient than the corresponding
finite-difference methods for the backward or forward Kolmogorov
PDEs (partial differential equations) while providing better
accuracy and stability.We extend a large number of known results by
either providing solutions on complementary or extended domains
where the solution is not known yet or modifying these techniques
and applying them to new types of equations, such as the Bessel
process. The book contains several novel results broadly applicable
in physics, mathematics, and engineering.
The concept of local volatility as well as the local volatility
model are one of the classical topics of mathematical finance.
Although the existing literature is wide, there still exist various
problems that have not drawn sufficient attention so far, for
example: a) construction of analytical solutions of the Dupire
equation for an arbitrary shape of the local volatility function;
b) construction of parametric or non-parametric regression of the
local volatility surface suitable for fast calibration; c)
no-arbitrage interpolation and extrapolation of the local and
implied volatility surfaces; d) extension of the local volatility
concept beyond the Black-Scholes model, etc. Also, recent
progresses in deep learning and artificial neural networks as
applied to financial engineering have made it reasonable to look
again at various classical problems of mathematical finance
including that of building a no-arbitrage local/implied volatility
surface and calibrating it to the option market data.This book was
written with the purpose of presenting new results previously
developed in a series of papers and explaining them consistently,
starting from the general concept of Dupire, Derman and Kani and
then concentrating on various extensions proposed by the author and
his co-authors. This volume collects all the results in one place,
and provides some typical examples of the problems that can be
efficiently solved using the proposed methods. This also results in
a faster calibration of the local and implied volatility surfaces
as compared to standard approaches.The methods and solutions
presented in this volume are new and recently published, and are
accompanied by various additional comments and considerations.
Since from the mathematical point of view, the level of details is
closer to the applied rather than to the abstract or pure
theoretical mathematics, the book could also be recommended to
graduate students with majors in computational or quantitative
finance, financial engineering or even applied mathematics. In
particular, the author used to teach some topics of this book as a
part of his special course on computational finance at the Tandon
School of Engineering, New York University.
This monograph presents a novel numerical approach to solving
partial integro-differential equations arising in asset pricing
models with jumps, which greatly exceeds the efficiency of existing
approaches. The method, based on pseudo-differential operators and
several original contributions to the theory of finite-difference
schemes, is new as applied to the Levy processes in finance, and is
herein presented for the first time in a single volume. The results
within, developed in a series of research papers, are collected and
arranged together with the necessary background material from Levy
processes, the modern theory of finite-difference schemes, the
theory of M-matrices and EM-matrices, etc., thus forming a
self-contained work that gives the reader a smooth introduction to
the subject. For readers with no knowledge of finance, a short
explanation of the main financial terms and notions used in the
book is given in the glossary. The latter part of the book
demonstrates the efficacy of the method by solving some typical
problems encountered in computational finance, including structural
default models with jumps, and local stochastic volatility models
with stochastic interest rates and jumps. The author also adds
extra complexity to the traditional statements of these problems by
taking into account jumps in each stochastic component while all
jumps are fully correlated, and shows how this setting can be
efficiently addressed within the framework of the new method.
Written for non-mathematicians, this book will appeal to financial
engineers and analysts, econophysicists, and researchers in applied
numerical analysis. It can also be used as an advance course on
modern finite-difference methods or computational finance.
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