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Books > Science & Mathematics > Mathematics > Applied mathematics
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Generalized Integral Transforms In Mathematical Finance (Hardcover)
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Generalized Integral Transforms In Mathematical Finance (Hardcover)
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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.
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