Computational Methods for Option Pricing
This book is a must for becoming better acquainted with the modern tools of numerical analysis for several significant computational problems arising in finance. Important aspects of finance modeling are reviewed, involving partial differential equations and numerical algorithms for the fast and accurate pricing of financial derivatives and the calibration of parameters. The best numerical algorithms are fully explored and discussed, from their mathematical analysis up to their implementation in C++ with efficient numerical libraries. This is one of the few books that thoroughly covers the following topics: mathematical results and efficient algorithms for pricing American options; modern algorithms with adaptive mesh refinement for European and American options; regularity and error estimates are derived and give strong support to the mesh adaptivity, an essential tool for speeding up the numerical implementations; calibration of volatility with European and American options; the use of automatic differentiation of computer codes for computing greeks.
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adjoint algorithm American options Asian option assume Assumption 2.1 automatic differentiation barrier options Black–Scholes equation Black–Scholes model boundary condition calibration Chapter computed const daouble const int converges Crank–Nicolson scheme ddouble ddouble operator ddouble& define denote error indicators Euler implicit scheme European option Figure finite difference finite element method friend ddouble gradient grid iloc inequality int i−0 int j=0 int operator jloc Lemma LÚvy process Lipschitz continuous local volatility M-matrix matrix maturity maximum principle mesh nodes nonnegative norm obtain parabolic parameters partial differential equation payoff function piecewise plot positive constant Proof Proposition prove put option resp S_nodes S_steps satisfies smooth solving space step stochastic Theorem underlying asset vanilla call vanilla put variable variational inequality vector void