Download Computational Methods for Quantitative Finance: Finite by Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph PDF

By Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Many mathematical assumptions on which classical by-product pricing equipment are dependent have come below scrutiny in recent times. the current quantity deals an advent to deterministic algorithms for the short and actual pricing of spinoff contracts in glossy finance. This unified, non-Monte-Carlo computational pricing method is able to dealing with particularly common sessions of stochastic industry types with jumps, together with, specifically, all at the moment used Lévy and stochastic volatility versions. It permits us e.g. to quantify version threat in computed costs on undeniable vanilla, in addition to on a variety of varieties of unique contracts. The algorithms are constructed in classical Black-Scholes markets, after which prolonged to industry types in keeping with multiscale stochastic volatility, to Lévy, additive and likely sessions of Feller techniques. This e-book is meant for graduate scholars and researchers, in addition to for practitioners within the fields of quantitative finance and utilized and computational arithmetic with a pretty good historical past in arithmetic, statistics or economics.​

Table of Contents

Cover

Computational tools for Quantitative Finance - Finite aspect tools for spinoff Pricing

ISBN 9783642354007 ISBN 9783642354014

Preface

Contents

Part I simple ideas and Models

Notions of Mathematical Finance
1.1 monetary Modelling
1.2 Stochastic Processes
1.3 additional Reading
components of Numerical equipment for PDEs
2.1 functionality Spaces
2.2 Partial Differential Equations
2.3 Numerical tools for the warmth Equation
o 2.3.1 Finite distinction Method
o 2.3.2 Convergence of the Finite distinction Method
o 2.3.3 Finite aspect Method
2.4 additional Reading
Finite point tools for Parabolic Problems
3.1 Sobolev Spaces
3.2 Variational Parabolic Framework
3.3 Discretization
3.4 Implementation of the Matrix Form
o 3.4.1 Elemental types and Assembly
o 3.4.2 preliminary Data
3.5 balance of the .-Scheme
3.6 errors Estimates
o 3.6.1 Finite aspect Interpolation
o 3.6.2 Convergence of the Finite point Method
3.7 additional Reading
ecu concepts in BS Markets
4.1 Black-Scholes Equation
4.2 Variational Formulation
4.3 Localization
4.4 Discretization
o 4.4.1 Finite distinction Discretization
o 4.4.2 Finite aspect Discretization
o 4.4.3 Non-smooth preliminary Data
4.5 Extensions of the Black-Scholes Model
o 4.5.1 CEV Model
o 4.5.2 neighborhood Volatility Models
4.6 extra Reading
American Options
5.1 optimum preventing Problem
5.2 Variational Formulation
5.3 Discretization
o 5.3.1 Finite distinction Discretization
o 5.3.2 Finite aspect Discretization
5.4 Numerical resolution of Linear Complementarity Problems
o 5.4.1 Projected Successive Overrelaxation Method
o 5.4.2 Primal-Dual lively Set Algorithm
5.5 additional Reading
unique Options
6.1 Barrier Options
6.2 Asian Options
6.3 Compound Options
6.4 Swing Options
6.5 additional Reading
rate of interest Models
7.1 Pricing Equation
7.2 rate of interest Derivatives
7.3 extra Reading
Multi-asset Options
8.1 Pricing Equation
8.2 Variational Formulation
8.3 Localization
8.4 Discretization
o 8.4.1 Finite distinction Discretization
o 8.4.2 Finite aspect Discretization
8.5 additional Reading
Stochastic Volatility Models
9.1 marketplace Models
o 9.1.1 Heston Model
o 9.1.2 Multi-scale Model
9.2 Pricing Equation
9.3 Variational Formulation
9.4 Localization
9.5 Discretization
o 9.5.1 Finite distinction Discretization
o 9.5.2 Finite point Discretization
9.6 American Options
9.7 extra Reading
L�vy Models
10.1 L�vy Processes
10.2 L�vy Models
o 10.2.1 Jump-Diffusion Models
o 10.2.2 natural bounce Models
o 10.2.3 Admissible marketplace Models
10.3 Pricing Equation
10.4 Variational Formulation
10.5 Localization
10.6 Discretization
o 10.6.1 Finite distinction Discretization
o 10.6.2 Finite aspect Discretization
10.7 American techniques below Exponential L�vy Models
10.8 additional Reading
Sensitivities and Greeks
11.1 alternative Pricing
11.2 Sensitivity Analysis
o 11.2.1 Sensitivity with appreciate to version Parameters
o 11.2.2 Sensitivity with admire to answer Arguments
11.3 Numerical Examples
o 11.3.1 One-Dimensional Models
o 11.3.2 Multivariate Models
11.4 additional Reading
Wavelet Methods
12.1 Spline Wavelets
o 12.1.1 Wavelet Transformation
o 12.1.2 Norm Equivalences
12.2 Wavelet Discretization
o 12.2.1 area Discretization
o 12.2.2 Matrix Compression
o 12.2.3 Multilevel Preconditioning
12.3 Discontinuous Galerkin Time Discretization
o 12.3.1 Derivation of the Linear Systems
o 12.3.2 resolution Algorithm
12.4 extra Reading

Part II complicated concepts and Models

Multidimensional Diffusion Models
13.1 Sparse Tensor Product Finite point Spaces
13.2 Sparse Wavelet Discretization
13.3 totally Discrete Scheme
13.4 Diffusion Models
o 13.4.1 Aggregated Black-Scholes Models
o 13.4.2 Stochastic Volatility Models
13.5 Numerical Examples
o 13.5.1 Full-Rank d-Dimensional Black-Scholes Model
o 13.5.2 Low-Rank d-Dimensional Black-Scholes
13.6 additional Reading
Multidimensional L�vy Models
14.1 L�vy Processes
14.2 L�vy Copulas
14.3 L�vy Models
o 14.3.1 Subordinated Brownian Motion
o 14.3.2 L�vy Copula Models
o 14.3.3 Admissible Models
14.4 Pricing Equation
14.5 Variational Formulation
14.6 Wavelet Discretization
o 14.6.1 Wavelet Compression
o 14.6.2 totally Discrete Scheme
14.7 software: effect of Approximations of Small Jumps
o 14.7.1 Gaussian Approximation
o 14.7.2 Basket Options
o 14.7.3 Barrier Options
14.8 extra Reading
Stochastic Volatility versions with Jumps
15.1 marketplace Models
o 15.1.1 Bates Models
o 15.1.2 BNS Model
15.2 Pricing Equations
15.3 Variational Formulation
15.4 Wavelet Discretization
15.5 additional Reading
Multidimensional Feller Processes
16.1 Pseudodifferential Operators
16.2 Variable Order Sobolev Spaces
16.3 Subordination
16.4 Admissible industry Models
16.5 Variational Formulation
o 16.5.1 quarter Condition
o 16.5.2 Well-Posedness
16.6 Numerical Examples
16.7 extra Reading

Elliptic Variational Inequalities

Parabolic Variational Inequalities

Index

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Example text

24). Then, the following error estimates hold: N+1 (u − IN u)(n) 2 L2 (G) ≤C 2( −n) hi u( ) 2 , L2 (Ki ) n = 0, 1, = 1, 2. e. hi = h, (u − IN u)(n) L2 (G) ≤ Ch −n u( ) L2 (G) , n = 0, 1, = 1, 2. 36) 1 Proof Consider G = (0, 1) and u ∈ H 2 (G). Then, u − c ∈ H 1 (G) for c = 0 u . 5)), u −c L2 (G) ≤C u L2 (G) . 37) 42 3 Finite Element Methods for Parabolic Problems we have (IN u)(1) = u(0) + c = u(1) and u − IN u ∈ H01 (G) ∩ H 2 (G). 4), u − IN u L2 (G) ≤ C u − (IN u) L2 (G) =C u −c L2 (G) ≤ C2 u L2 (G) .

M − 1 and i = 1, . . 12) where the constant C(u) > 0 depends on the exact solution u and its derivatives. For the convergence of the FDM, we are interested in estimating the error between the finite difference solution um i and the exact solution u(t, x) at the grid 18 2 Elements of Numerical Methods for PDEs point (tm , xi ). e. εim := u(tm , xi ) − um i , 0 ≤ i ≤ N + 1, 0≤m≤M. 15) where ηm := (k −1 I + θ G)−1 E m , and where Aθ := (k −1 I + θG)−1 (−k −1 I + (1 − θ )G), is called an amplification matrix.

10). For this section, we assume the uniform mesh width h in space and constant time steps k = T /M. We define 1 v a := a(v, v) 2 . 28) In the analysis, we will use for f ∈ VN∗ the following notation: f ∗ := sup vN ∈VN (f, vN ) . 29) We will also need λA defined by 2 vN vN λA := sup vN ∈VN 2 ∗ . In the case 12 ≤ θ ≤ 1, the θ -scheme is stable for any time step k > 0, whereas in the case 0 ≤ θ < 12 the time step k must be sufficiently small. 1 In the case 0 ≤ θ < 12 , assume σ := k(1 − 2θ )λA < 2.

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