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

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6.2.1 Convergence Study Table (6.6) shows steady convergence of full grid values. On the other hand, convergence of a sparse grid solution is not very smooth. However, it soon becomes clear that such convergence behaviour should be attributed to a large interpolation error. Level Full Exec. Time Sparse Exec. Time Full point Sparse point solution (in sec.) solution (in sec.) count count 4 31.639 11.94 4913 593 5 29.802 121.11 39.691 1.51 35937 1505 6 29.212 1107.16 38.494 5.27 274625 3713 7 33.341 17.48 2146689 8961 8 36.259 54.39 16974593 21249 9 32.532 158.14 135005697 49665 Table 6.6: 3D Bermudan Swaption. Spot Rates as Table (6.2) Figure (6.11) shows the discrepancy between the calculated price and the benchmark provided in [Blackham, 2004]. 500000 simulation runs of MC showed the value of 29.064 with standard error 0.133. As suggested in [Andersen, 1999], numeraire-rebased swap value has been chosen as the explanatory variable for regression. Conditional expectations were fitted with 4th-degree Legendre polynomial basis function. Tables (6.7) and (6.8) below show good convergence of on-grid points for a set of different at-the-money and out-of-the-money forward rates. Option price 38 37 36 35 34 33 32 31 30 29 Level of Discretization 3 3.5 4 4.5 5 5.5 6 MC Simulation Value+Std.Error Full-Grid FD value Full-Grid J.Blackham value 28 100000 150000 200000 250000 300000 350000 400000 450000 500000 Number of MC simulations Figure 6.11: 3D Bermudan Swaption. Benchmarking against [Blackham, 2004] and Monte Carlo. Level L 2 =5.62% L 2 =1.875% L 2 =3.75% L 3 =3.75% L 3 =3.75% L 3 =2.81% Full Sparse Full Sparse Full Sparse 4 51.335 8.189 4.684 5 51.828 50.361 8.243 10.985 4.337 6 51.897 51.694 8.232 9.472 4.246 9.546 7 52.605 8.772 4.823 8 51.499 8.272 4.348 9 51.711 8.107 4.245 MC Price Std.error Price Std.error Price Std.error 52.2593 0.2053 8.3734 0.0601 4.294 0.0493 Table 6.7: 3D Bermudan Swaption. Full and sparse on-grid solution values for L 1 =3.75%. 500000 MC simulation trials. 56
Level L 2 =1.87% L 2 =1.875% L 2 =5.62% L 3 =7.5% L 3 =3.75% L 3 =5.62% Full Sparse Full Sparse Full Sparse 4 196.542 10.945 176.422 5 196.766 204.006 10.903 178.124 6 196.628 207.901 10.875 13.080 178.474 7 194.337 11.263 165.244 8 196.054 11.067 178.232 9 196.378 10.796 177.958 MC Price Std.error Price Std.error Price Std.error 196.514 0.337 11.062 0.075 177.208 0.383 Table 6.8: 3D Bermudan Swaption. Full and sparse on-grid solution values for L 1 =5.62%. 500000 MC simulation trials. 400 350 Full Solution L6 Sparse Solution L6 Sparse Solution L8 L1=3.75% L2=3.75% 200 L1=3.75% L2=3.75% Full Solution L6 Sparse Solution L6 Sparse Solution L8 300 150 Option value 250 200 150 Option value 100 100 50 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 (a) L 1 and L 2 out-of-the-money (b) Plot (a) Zoomed. 400 350 Full Solution L6 Sparse Solution L6 Sparse Solution L8 L1=5.62% L2=5.62% 200 L1=5.62% L2=5.62% Full Solution L6 Sparse Solution L6 Sparse Solution L8 300 150 Option value 250 200 150 Option value 100 100 50 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 (c) L 1 and L 2 in-the-money (d) Plot (b) Zoomed. Figure 6.12: 3D Bermudan Swaption. Solution slices corresponding to fixings in L 1 and L 2 . 57
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SPARSE GRID METHOD IN THE LIBOR MAR
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Acknowledgements This project would
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6.3.1 Valuation Results. . . . . .
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6.18 2D Digital Option. Full and sp
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Chapter 1 Introduction The modern f
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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: L1=5.62% Sparse Solution Swaption P
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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