附註:Stochastic engines and partial differential equations -- Popular one-factor models by asset classes -- Fredholm integral equations -- Volterra integral equations -- Solving integral equations numerically -- Classical integral transforms -- Generalized integral transforms -- Method of heat potentials -- Barrier and American options -- On the first hitting time density for a reducible diffusion process -- Optimal mean-reverting trading strategies -- Barrier options in the hull-white model -- Barrier options in the time-dependent CEV and CIR models -- Barrier options in the BK and Verhulst models -- Calibrating the default boundary to a constant default intensity -- McKean-Vlasov equation with feedback through hitting a boundary -- Miscellaneous problems -- Double barrier options -- Multilayer heat equations: application to finance.
摘要:"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"--
