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

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900 800 Full Solution Sparse Solution 450 400 Full Solution Sparse Solution 700 350 600 300 Option value 500 400 Option value 250 200 300 150 200 100 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L1 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L1 (a) Option Values for L 2 =3.75% (b) Plot (a) zoomed in. 900 800 Full Solution Sparse Solution 450 400 Full Solution Sparse Solution 700 350 600 300 Option value 500 400 Option value 250 200 300 150 200 100 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L1 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L1 (c) Option Values for L 2 =7.5% (d) Plot (c) zoomed in. Figure 6.4: 2D Chooser. Sparse vs. full solution. Slices. The Figure (6.4) above shows how Level 6 sparse-grid approximated solution corresponds to the full grid counterpart as we freeze one of the forward rates. 48
6.1.2 Discussion: Convergence of Pricing. Results from Tables (6.3), (6.4) and (6.5) prove that the use of BiCGSTAB+ILU iterative solver with second order FD method manages to produce promising results in case of sparse discretization. In addition to impressive improvement in speed, the “per grid point“ convergence of sparse grid PDE solution is by far better than that of a full grid. This feature of sparse grid technique proves to be crucial when approximating functions in higher dimensional spaces. In order to gain a better insight on accuracy of the solution we have plotted the slices of solution surface, where full and sparse approximations were compared against each other. Figure (6.4) presents two of such slices, corresponding to L 2 = 7.5% and L 2 = 3.75% fixings. On a broader range both plots show relatively strong agreement of both full and sparse approximations. Yet, as we zoom closer, the plot(d) exhibits inaccuracies that form a distinguishable “wave” pattern, and less so plot(b). The origin of this pattern is better illustrated by Figure (6.5), that provides an insight into the absolute error distribution, characteristic of a sparse grid solution on level 8. It shows that the largest differences between solutions appear along the center of the domain and in the region of discontinuities of initial boundary conditions. Error Profile. Level 8 25 20 15 10 -10 -505 -15 Error 25 20 15 10 5 0 -5 -10 -15 0.14 0.12 0.1 0.08 0.06 Forward Rate L2 0.04 0.02 0.14 0.12 0.1 0.08 0.06 0.04 Forward Rate L1 0.02 Figure 6.5: 2D Chooser. Absolute error surface on level 8. Since all grids of sparse discretization are treated independently by our iterative solver, a discretization error is to be introduced while solving subproblems on ”narrower” and borderline grids of sparse space (as in Figure (5.2)), where spatial step h i in one dimension is significantly larger than in the other. In the process of combination, some points remain less-influenced or unaffected at all by addition and subtraction of “sparser” grids, and therefore introducing “spikes” into solution. For instance, the volatile center of the solution is such due to the points contributed by the “narrowest” 49
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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
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: Full Solution. Level 6 Option Price
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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