Pricing Models for Bermudan-style Interest Rate Derivatives_PIETERSZ

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Pricing Models for Bermudan-style Interest Rate Derivatives_PIETERSZ

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Acknowledgements vii
Notation xix
Outline xxiii
1 Introduction 1
1.1 Arbitrage-free pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Use of models in practice . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Interest rate markets and options . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Linear products: Deposits, bonds, and swaps . . . . . . . . . . . . . 7
1.2.2 Interest rate options: Caps, floors, and swaptions . . . . . . . . . . 8
1.3 Interest rate derivatives pricing models . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Short rate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Market models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Markov-functional models . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 American option pricing with Monte Carlo simulation . . . . . . . . . . . . 17
2 Risk-managing Bermudan swaptions in a LIBOR model 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Recalibration approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Swap vega and the swap market model . . . . . . . . . . . . . . . . . . . . 27
2.5 Alternative method for calculating swap vega . . . . . . . . . . . . . . . . 29
2.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Comparison with the swap market model . . . . . . . . . . . . . . . . . . . 30
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.A Appendix: Negative vega two-stock Bermudan options . . . . . . . . . . . 34
x CONTENTS
3 Rank reduction of correlation matrices by majorization 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 Modified PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Majorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 Geometric programming . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.4 Alternating projections without normal correction . . . . . . . . . . 45
3.2.5 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.6 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.7 Alternating projections with normal correction (d = n) . . . . . . . 47
3.3 Majorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 The algorithm and convergence analysis . . . . . . . . . . . . . . . . . . . 50
3.4.1 Global convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.2 Local rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5.1 Numerical comparison with other methods . . . . . . . . . . . . . . 54
3.5.2 Non-constant weights . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.3 The order effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5.4 Majorization equipped with the power method . . . . . . . . . . . . 62
3.5.5 Using an estimate for the largest eigenvalue . . . . . . . . . . . . . 62
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.A Appendix: Proof of Equation (3.11) . . . . . . . . . . . . . . . . . . . . . . 64
4 Rank reduction of correlation matrices by geometric programming 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Weighted norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Solution methodology with geometric optimisation . . . . . . . . . . . . . . 71
4.2.1 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Topological structure . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.3 A dense part of Mn;d equipped with a differentiable structure . . . . 74
4.2.4 The Cholesky manifold . . . . . . . . . . . . . . . . . . . . . . . . . 75
......

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