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

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6.3.2 Greeks. Delta and Vega. Delta Figures (6.19) and (6.20) show the delta profile characteristic of a 2D digital option and how full approximation compares with sparse grid representation. 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0.02 0.04 0.06 0.08 0.1 Forward Rate L1 0.12 0.14 Forward Rate L2 0.02 0.040.060.080.10.120.14 Figure 6.19: 2D Digital Option. Full grid Delta on L7. L1=4.68% L1=4.68% 0.8 0.7 Full Delta L7 Sparse Delta L7 Sparse Delta L9 0.8 0.7 Full Delta L7 Sparse Delta L7 Sparse Delta L9 0.6 0.6 0.5 0.5 Delta value 0.4 0.3 Delta value 0.4 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L2 Forward Rate L2 (a) L 1 = 4.68%. Shift in L 1 = 10bp (b) L 1 = 4.68%. Shift in L 1 = 1bp L1=5.62% L1=5.62% 0.6 0.5 Full Delta L7 Sparse Delta L7 Sparse Delta L9 0.6 0.5 Full Vega L7 Sparse Vega L7 Sparse Vega L9 0.4 0.4 Delta value 0.3 0.2 Vega value 0.3 0.2 0.1 0.1 0 0 -0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 Forward Rate L2 (c) L 1 = 5.62%. Shift in L 1 = 10bp (d) L 1 = 5.62%. Shift in L 1 = 1bp Figure 6.20: 2D Digital Option. Sparse Delta L7 and L9 slices. 64
Figures (6.21) and (6.22) show the vega profile characteristic of a 2D digital option and how full approximation compares with sparse grid representation. 0.015 0.01 0.005 -0.005 0 -0.01 -0.015 -0.02 -0.025 0 0.02 0.04 0.06 0.08 0.1 0.12 Forward Rate L1 0.14 0.16 0.14 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 0.02 0.04 0.06 0.08 0.10.12 Forward Rate L2 Figure 6.21: 2D Digital Option. Full grid vega on L7. L1=4.68% L1=4.68% 0.014 0.012 Full Vega L7 Sparse Vega L7 Sparse Vega L9 0.014 0.012 Full Vega L7 Sparse Vega L7 Sparse Vega L9 0.01 0.01 0.008 0.008 Vega value 0.006 0.004 Vega value 0.006 0.004 0.002 0.002 0 0 -0.002 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -0.002 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L2 Forward Rate L2 (a) L 1 = 4.68%. Shift in σ 1 = 10bp (b) L 1 = 4.68%. Shift in σ 1 = 0.1bp L1=5.62% L1=5.62% 0.005 0 Full Vega L7 Sparse Vega L7 Sparse Vega L9 0.005 0 Full Vega L7 Sparse Vega L7 Sparse Vega L9 -0.005 -0.005 Vega value -0.01 Vega value -0.01 -0.015 -0.015 -0.02 -0.02 -0.025 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -0.025 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L2 Forward Rate L2 (c) L 1 = 5.62%. Shift in σ 1 = 10bp (d) L 1 = 5.62%. Shift in σ 1 = 0.1bp Figure 6.22: 2D Digital Option. Sparse Vega L7 and L9 slices. Remarks. As anticipated, sparse grid delta approximation appeared to be very demanding to the size of the shift. It becomes obvious from Figure (6.20), plot(b), that many of the less accurate solution points become irresponsive to the shift as large as 1%. When the step size is reduced to one-tenth of a percent, the differential of the value with respect to step in the underlying becomes zero. Such behavior of the delta solution is, again, the magnified effect of quantification error mentioned earlier. Discontinuity 65
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SPARSE GRID METHOD IN THE LIBOR MAR
Page 3 and 4:
Acknowledgements This project would
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6.3.1 Valuation Results. . . . . .
Page 7 and 8:
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
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Risk-Neutral Pricing Another import
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S 0 u d u S 0 S u 2 0 Su 3 0 S du 2
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or log(S T ) d −→ log(S 0 ) +
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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: Level L 1 =3.75% L 1 =4.68% L 1 =5.
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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