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thesis by Gutachter
Foundations 3
1.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Research survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 From American Options to Variational Inequalities 13
2.1 Strategy for Dealing with American Options . . . . . . . . . . . 14
2.2 Formulation as Free Boundary-Value Problem . . . . . . . . . . 15
2.3 Formulation as Linear Complementarity Problem . . . . . . . . 18
2.4 Transformation of the Black-Scholes-Equation . . . . . . . . . . 19
2.5 Classes of Second Order PDEs . . . . . . . . . . . . . . . . . . . 22
2.6 Weak Derivatives and Sobolev Spaces . . . . . . . . . . . . . . . 22
2.7 Formulation as Variational Inequality . . . . . . . . . . . . . . . 25
3 The Method of Finite Differences 29
3.1 Discretization of the Domain . . . . . . . . . . . . . . . . . . . . 29
3.2 Types of Finite Differences . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 The Explicit Scheme . . . . . . . . . . . . . . . . . . . . 31
3.2.2 The Implicit Scheme . . . . . . . . . . . . . . . . . . . . 32
3.2.3 The Crank-Nicholson Scheme . . . . . . . . . . . . . . . 33
3.3 Existence of a Uniform Solution . . . . . . . . . . . . . . . . . . 35
3.4 Stability of Finite Differences . . . . . . . . . . . . . . . . . . . 35
3.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Finite Differences Applied to American Options . . . . . . . . . 39
4 The Method of Finite Elements 41
4.1 Subdivision of the Domain . . . . . . . . . . . . . . . . . . . . . 41
4.2 The Ritz-Galerkin Approach . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Stability and Convergence of Finite Elements . . . . . . 44
4.2.2 Conform Finite Elements and Interpolation Accuracy . . 45
ii
4.3 Finite Elements applied to American Options . . . . . . . . . . 47
4.4 Types of Finite Elements . . . . . . . . . . . . . . . . . . . . . . 49
4.4.1 Hat Functions . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4.2 Cubic B-Splines . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Grid Improvements . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Implementation of the Methods 62
5.1 Iterative Solution of a Linear System of Equations . . . . . . . . 62
5.2 The Projected SOR Method . . . . . . . . . . . . . . . . . . . . 65
5.3 An Algorithm to Compute American Options . . . . . . . . . . 67
6 Numerical Observations 72
6.1 Example 1: American Put Option . . . . . . . . . . . . . . . . . 73
6.2 Example 2: American Call Option . . . . . . . . . . . . . . . . . 88
6.3 Test Summary and Conclusions . . . . . . . . . . . . . . . . . . 90
7 Options on Multiple Assets 91
7.1 American Option on Two Assets . . . . . . . . . . . . . . . . . . 92
7.2 2D Finite Element Discretization . . . . . . . . . . . . . . . . . 95
7.3 Implementation of the 2D Algorithm . . . . . . . . . . . . . . . 99
7.3.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . 100
7.3.2 Assembling of the Matrices . . . . . . . . . . . . . . . . . 102
7.4 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . 107
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1. �� Introduction, DE, Galerkin
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Georgios Foufas and Mats G. Larson
The main objective of this paper is to develop an adaptive nite element method for computation of the values and dierent sensitivity measures of ordinary European options, barrier options, and lookback options. The options are priced using the Black-Scholes PDE-model, and the resulting PDE:s are of parabolic type in one spatial dimension with different boundary conditions and jump conditions at monitoring dates. The adaptive nite element method is based on a posteriori estimates of the error in desired quantities, which we derive using duality techniques. The suggested adaptive nite element method is stable and gives fast and accurate results.

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Chensong Zhang
The evaluation of the price of an option is of considerable importance in
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Gunter Winkler
Thomas Apel
Uwe Wystup
Quoted:

Introduction Due to the smile observed in options markets numerous authors have suggested different models such as generalized Levy processes, fractional Brownian motion, entropy based models [4], jump diffusions and stochastic volatility models. For vanilla options (put and call options) the dependence of the price on the volatility is monotone, whence using the Black-Scholes formula along with a volatility smile matrix is sufficient. Values of exotic options, however, do not always depend on the volatility in a monotone fashion, whence pricing consistently with the smile requires a more sophisticated model. Therefore, it is important to find efficient ways to calculate exotic option values in exotic models.

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