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

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6.2.2 Greeks. Delta and Vega. In this section we provide some results and comments on the results for the delta and vega of the 3D Bermudan Swaption. Figure (6.13) shows the corresponding surface plots in case of a full grid approximation. L1=5% Full Delta L6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.16 0.14 0.12 0.1 0.08 Forward Rate L3 0.06 0.04 0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L2 (a) Delta L1=5.62% Full Vega Solution 0.04 0.03 0.02 0.01 -0.01 0 -0.02 -0.03 0.14 0.12 0.1 0.08 0.06 Forward Rate L3 0.04 0.02 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 Forward Rate L2 (b) Vega Figure 6.13: 3D Bermudan Swaption. Delta and Vega Surfaces for L 1 =5.62%. Full grid. 58
Figures (6.14) and (6.15) show the slices of a sparse grid delta and vega profiles, respectively, plotted against a full grid solution. Basis point (bp) is equal to 0.01%. 1.4 L1=5% L2=1.25% 1.4 L1=5% L2=1.25% 1.2 1.2 1 1 Delta value 0.8 0.6 Delta value 0.8 0.6 0.4 0.4 0.2 0.2 Full Delta L6 Sparse Delta L8. dL3 = 7bp Sparse Delta L8. dL3 =2bp 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 0 Full Delta L6 Sparse Delta L10. dL = 2bp -0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 (a) Full L6 vs. Sparse L8. L 2 = 1.25% (b) Full L6 vs. Sparse L10. L 2 = 1.25% 1.4 L1=L2=5% 1.4 L1=5% L2=5% 1.2 1.2 1 1 Delta value 0.8 0.6 Delta value 0.8 0.6 0.4 0.4 0.2 Full Delta L6 Sparse Delta L6. dL3 = 30bp Sparse Delta L6. dL3 = 7bp 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 0.2 Full Delta L6 Sparse Delta L8. dL8 = 7bp 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 (c) Full L6 vs. Sparse L6. L 2 = 5% (d) Full L6 vs. Sparse L8. L 2 = 5% 1.4 L1=5% L2=5.62% 1.4 L1=5% L2=5.62% 1.2 1.2 1 1 Delta value 0.8 0.6 Delta value 0.8 0.6 0.4 0.4 0.2 Full Delta L6 Sparse Delta L8. dL3 = 7bp Sparse Delta L8. dL3 = 2bp 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 0.2 Full Delta L6 Sparse Delta L10. dL10 = 2bp 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 (e) Full L6 vs. Sparse L8. L 2 = 5.62% (f) Full L6 vs. Sparse L10. L 2 = 5.62% Figure 6.14: 3D Bermudan Swaption. L 3 -Delta. Slices corresponding to a fixing in L 1 = 5% 59
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 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: Level L 2 =1.87% L 2 =1.875% L 2 =5
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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