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This book summarizes recent advances in applying saddlepoint approximation methods to financial engineering. It addresses pricing exotic financial derivatives and calculating risk contributions to Value-at-Risk and Expected Shortfall in credit portfolios under various default correlation models. These standard problems involve the computation of tail probabilities and tail expectations of the corresponding underlying state variables. The text offers in a single source most of the saddlepoint approximation results in financial engineering, with different sets of ready-to-use approximation formulas. Much of this material may otherwise only be found in original research publications. The exposition and style are made rigorous by providing formal proofs of most of the results. Starting with a presentation of the derivation of a variety of saddlepoint approximation formulas in different contexts, this book will help new researchers to learn the fine technicalities of the topic. It will also be valuable to quantitative analysts in financial institutions who strive for effective valuation of prices of exotic financial derivatives and risk positions of portfolios of risky instruments. .
Financial engineering. --- Method of steepest descent (Numerical analysis) --- Mathematics. --- Economics, Mathematical. --- Quantitative Finance. --- Approximations, Saddlepoint --- Descent, Method of steepest (Numerical analysis) --- Method of saddle points (Numerical analysis) --- Saddle point method (Numerical analysis) --- Saddle points, Method of (Numerical analysis) --- Saddlepoint approximations --- Saddlepoint method (Numerical analysis) --- Steepest descent method (Numerical analysis) --- Approximation theory --- Asymptotic expansions --- Computational finance --- Engineering, Financial --- Finance --- Finance. --- Funding --- Funds --- Economics --- Currency question --- Economics, Mathematical . --- Mathematical economics --- Econometrics --- Mathematics --- Methodology
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Applied functional analysis has an extensive history. In the last century, this field has often been used in physical sciences, such as wave and heat phenomena. In recent decades, with the development of nonlinear functional analysis, this field has been used to model a variety of engineering, medical, and computer sciences. Two of the most significant issues in this area are modeling and optimization. Thus, we consider some recently published works on fixed point, variational inequalities, and optimization problems. These works could lead readers to obtain new novelties and familiarize them with some applications of this area.
vector variational-like inequalities --- vector optimization problems --- limiting (p,r)-α-(η,θ)-invexity --- Lipschitz continuity --- Fan-KKM theorem --- set-valued optimization problems --- higher-order weak adjacent epiderivatives --- higher-order mond-weir type dual --- benson proper efficiency --- fractional calculus --- ψ-fractional integrals --- fractional differential equations --- contraction --- hybrid contractions --- volterra fractional integral equations --- fixed point --- inertial-like subgradient-like extragradient method with line-search process --- pseudomonotone variational inequality problem --- asymptotically nonexpansive mapping --- strictly pseudocontractive mapping --- sequentially weak continuity --- method with line-search process --- pseudomonotone variational inequality --- strictly pseudocontractive mappings --- common fixed point --- hyperspace --- informal open sets --- informal norms --- null set --- open balls --- modified implicit iterative methods with perturbed mapping --- pseudocontractive mapping --- strongly pseudocontractive mapping --- nonexpansive mapping --- weakly continuous duality mapping --- set optimization --- set relations --- nonlinear scalarizing functional --- algebraic interior --- vector closure --- conjugate gradient method --- steepest descent method --- hybrid projection --- shrinking projection --- inertial Mann --- strongly convergence --- strict pseudo-contraction --- variational inequality problem --- inclusion problem --- signal processing
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