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

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11.8 11.6 11.4 Black-Scholes Boundary Fixed Boundary Full Grid, Level 9 Value 11.2 Option price 11 10.8 10.6 10.4 10.2 10 9.8 6 7 8 9 10 11 Discretization Level Figure 6.2: 2D Chooser. Convergence of Full and Sparse Grid. Level L 1 =L 2 =5.5%, K=5.5% K = 3.9% L 1 =L 2 =7.5%, K=5.5% Full Sparse Full Sparse Full Sparse 5 83.669 71.9602 49.325 37.3126 251.755 262.1945 6 82.947 83.0537 48.869 37.9287 251.535 258.5324 7 82.547 81.1169 48.818 41.3625 251.449 253.7469 8 82.551 79.0752 48.790 46.9370 251.438 253.1529 9 82.4895 48.0288 251.9835 10 82.5755 48.0717 251.8426 MC Price Std.error Price Std.error Price Std.error 82.399 0.01298 48.653 0.0089 251.468 0.0218 Table 6.4: 2D Chooser. At-the-money and in-the-money option values. Level L 1 =L 2 =2%, K=5.5% L 1 =L 2 =2%, K=2% Full Sparse Full Sparse 5 0.0572 0.5587 32.7686 23.4880 6 0.0269 0.0239 31.2000 28.7051 7 0.0202 0.0701 30.1360 31.3129 8 0.0193 0.1163 30.1523 28.5730 9 0.0383 28.5550 10 0.0246 29.8075 MC Price Std.error Price Std.error 0.0194 0.000153 30.0825 0.00472 Table 6.5: 2D Chooser. Deep out-of-the-money and at-the-money with low strike. In addition to the out-of-the money case from [Blackham, 2004], we were interested in the convergence of sparse solution in case of different strike prices and different values of spot forward rates. The results are presented in Tables (6.4) and (6.5), where MC prices have been calculated for 500000 simulation trials. 46
Full Solution. Level 6 Option Price 900 800 700 600 500 400 300 200 100 0 900 800 700 600 500 400 300 0 100 200 0.16 0.14 0.12 0.1 0.08 0.06 Forward Rate L2 0.04 0.02 0 0 0.120.140.16 Forward Rate L1 0.020.040.060.080.1 (a) Full Grid Surface. Sparse Grid. Level 6 Option Price 900 800 700 600 500 400 300 200 100 0 900 800 700 600 500 400 300 200 100 0 0.16 0.14 0.12 0.1 0.08 0.06 Forward Rate L2 0.04 0.02 0 0 0.120.140.16 Forward Rate L1 0.020.040.060.080.1 (b) Sparse Grid Surface. Figure 6.3: 2D Chooser. Solution Surface for level 6. 47
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
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: 6.1.1 Convergence of MC, Full and S
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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