sparse grid method in the libor market model. option valuation and the

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$Learning LATEX by Doing$

List of Figures 2.1 From Single Period to Multi-Period . . . . . . . . . . . . . . . . . . 7 3.1 Tenor Structure of Interest Rates . . . . . . . . . . . . . . . . . . . . 18 4.1 Monte Carlo. Delta of the option with a digital feature. . . . . . . . . 32 4.2 2D Solution mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Band coefficient matrix. . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.1 Standard Basis functions S 3 . . . . . . . . . . . . . . . . . . . . . . 38 5.2 Hierarchical representation. . . . . . . . . . . . . . . . . . . . . . . . 39 5.3 Mapping of hierarchical representation S ∗ 3 onto a full grid. . . . . . . 40 5.4 Mapping of hierarchical representation Ŝ∗ 3 onto a sparse grid. . . . . . 41 6.1 2D Chooser Option. MC vs. FD Convergence. . . . . . . . . . . . . . 45 6.2 2D Chooser. Convergence of Full and Sparse Grid. . . . . . . . . . . 46 6.3 2D Chooser. Solution Surface for level 6. . . . . . . . . . . . . . . . 47 6.4 2D Chooser. Sparse vs. full solution. Slices. . . . . . . . . . . . . . . 48 6.5 2D Chooser. Absolute error surface on level 8. . . . . . . . . . . . . . 49 6.6 2D Chooser. Sparse delta surface on level 7. . . . . . . . . . . . . . . 50 6.7 2D Chooser. Full vs. sparse delta profile on level 7 . . . . . . . . . . 51 6.8 2D Chooser. Sparse vega surface on level 8. . . . . . . . . . . . . . . 52 6.9 2D Chooser. Full vs. sparse vega slices on level 8. . . . . . . . . . . . 53 6.10 3D Bermudan Swaption. Full and Sparse solution surfaces. . . . . . . 55 6.11 3D Bermudan Swaption. Benchmarking against [Blackham, 2004] and Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.12 3D Bermudan Swaption. Solution slices corresponding to fixings in L 1 and L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.13 3D Bermudan Swaption. Delta and Vega Surfaces for L 1 =5.62%. Full grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.14 3D Bermudan Swaption. L 3 -Delta. Slices corresponding to a fixing in L 1 = 5% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.15 3D Bermudan Swaption. L 3 -Vega. Slices corresponding to a fixing in L 1 = 5.62% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.16 2D Digital Option. Terminal payoff discretized on a L6 full grid. . . . 61 6.17 2D Digital Option. Convergence of Monte Carlo and full grid FD. . . 61 iv
6.18 2D Digital Option. Full and sparse grid solution surfaces. Level 7. . . 62 6.19 2D Digital Option. Full grid Delta on L7. . . . . . . . . . . . . . . . 64 6.20 2D Digital Option. Sparse Delta L7 and L9 slices. . . . . . . . . . . . 64 6.21 2D Digital Option. Full grid vega on L7. . . . . . . . . . . . . . . . . 65 6.22 2D Digital Option. Sparse Vega L7 and L9 slices. . . . . . . . . . . . 65 v
Page 1 and 2: SPARSE GRID METHOD IN THE LIBOR MAR
Page 3 and 4: Acknowledgements This project would
Page 5: 6.3.1 Valuation Results. . . . . .
Page 9 and 10: Chapter 1 Introduction The modern f
Page 11 and 12: Chapter 2 Asset Pricing Overview Th
Page 13 and 14: Risk-Neutral Pricing Another import
Page 15 and 16: S 0 u d u S 0 S u 2 0 Su 3 0 S du 2
Page 17 and 18: or log(S T ) d −→ log(S 0 ) +
Page 19 and 20: There is an important difference be
Page 21 and 22: Delta. In order to set up a riskfre
Page 23 and 24: Theorem 2.3.2. Let W P (t) = (W1 P
Page 25 and 26: Chapter 3 LIBOR Market Model Having
Page 27 and 28: The right-hand side of the equality
Page 29 and 30: At this point of the text all neces
Page 31 and 32: pays off at expiry. Conveniently, t
Page 33 and 34: 1. Tenor structure of N interest ra
Page 35 and 36: and weight factor parts from the co
Page 37 and 38: we can substitute into the equation
Page 39 and 40: with ɛ i being independent standar
Page 41 and 42: a good compromise on the size of ɛ
Page 43 and 44: Three different θ values represent
Page 45 and 46: Chapter 5 Sparse Grids In the previ
Page 47 and 48: T 0 , 0 T 0 , 1 T 0 , 2 T 0 , 3 |i|
Page 49 and 50: 00 03 02 03 01 03 02 03 00 30 30 20
Page 51 and 52: Chapter 6 Numerical Results. Valuat
Page 53 and 54: 6.1.1 Convergence of MC, Full and S
Page 55 and 56: Full Solution. Level 6 Option Price
Page 57 and 58:
6.1.2 Discussion: Convergence of Pr
Page 59 and 60:
1 Full Delta Value Sparse Delta Val
Page 61 and 62:
0.03 0.025 Full vega Value Sparse v
Page 63 and 64:
L1=5.62% Sparse Solution Swaption P
Page 65 and 66:
Level L 2 =1.87% L 2 =1.875% L 2 =5
Page 67 and 68:
Figures (6.14) and (6.15) show the
Page 69 and 70:
6.3 Discontinuous Payoff. 2D Digita
Page 71 and 72:
Level L 1 =3.75% L 1 =4.68% L 1 =5.
Page 73 and 74:
Figures (6.21) and (6.22) show the
Page 75 and 76:
Chapter 7 Conclusion and Future Wor
Page 77 and 78:
Appendix A Implementation Details I
Page 79:
A. Klimke. Piecewise multilinear sp
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