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

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$Learning LATEX by Doing$

• reduction in the dimension of the PDE could be attempted by fast-drift approximation, as in [Pietersz et al., 2005]. At the end we conclude that sparse grid combination technique is a promising method to help revive Finite Difference approach in the interest-rate option pricing field in higher dimensions. 68
Appendix A Implementation Details In this appendix, we present a brief list of software tools we employed throughout the project: 1. C++, as the language of choice, allowed us to take advantage of the power of templates, operator overloading and generic STL containers and algorithms. 2. Subgrids have been implemented on top of Blitz++ generic array library, described in [Veldhuizen]. 3. LMM PDE solver made use of the BiCGSTAB routine and supplimentary data structures of the free-source iterative methods library, described in [Dongarra et al.]. 4. As in case of full grid, off-grid points of sparse grid solution need to be found with interpolation. The current implementation of sparse grid solver doesn’t provide a routine to interpolating non-uniform meshes, so this procedure has been outsourced to Matlab. Interpolation of off-grid values has been peformed as a post-processing step with griddata and griddata3 routines. 5. GnuPlot software came very handy in drawing pretty graphs and color-mapped surfaces. 6. The text of the thesis has been typeset in L A TEX. 69
Page 1 and 2:
SPARSE GRID METHOD IN THE LIBOR MAR
Page 3 and 4:
Acknowledgements This project would
Page 5 and 6:
6.3.1 Valuation Results. . . . . .
Page 7 and 8:
6.18 2D Digital Option. Full and sp
Page 9 and 10:
Chapter 1 Introduction The modern f
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Chapter 2 Asset Pricing Overview Th
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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: Chapter 7 Conclusion and Future Wor
Page 79: A. Klimke. Piecewise multilinear sp
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