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

More documents

Recommendations

Info

$Learning LATEX by Doing$

0.035 0.03 L1=3.75% L2=3.75% Shift in L3 = 10bp Full Vega L6 Sparse Vega L7 Sparse Vega L9 0.035 0.03 L1=3.75% L2=3.75% Shift in L3 = 0.1bp Full Vega L6 Sparse Vega L7 Sparse Vega L9 0.025 0.025 Vega value 0.02 0.015 0.01 Vega value 0.02 0.015 0.01 0.005 0.005 0 0 -0.005 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 -0.005 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 (a) L 2 = 3.75% Step in σ 3 = 10bp (b) L 2 = 3.75% Step in σ 3 = 0.1bp 0.04 0.03 L1=5.62% L2=5.62% Shift in L3 = 10bp Full Vega L6 Sparse Vega L7 Sparse Vega L9 0.04 0.03 L1=5.62% L2=5.62% Shift in L3 = 0.1bp Full Vega L6 Sparse Vega L7 Sparse Vega L9 0.02 0.02 Vega value 0.01 Vega value 0.01 0 0 -0.01 -0.01 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L3 (c) L 2 = 5.62% Step in σ 3 = 10bp (d) L 2 = 5.62% Step in σ 3 = 0.1bp Figure 6.15: 3D Bermudan Swaption. L 3 -Vega. Slices corresponding to a fixing in L 1 = 5.62% Remarks. One can make several observations when looking at the plots above. First, sparse solution for the greeks has proved to be stable with respect to the size of the shift in the underlying and volatility step. Second, despite the clear presence of well-defined ’wiggles’, sparse grid approximation is capable of outlining the shape of the delta and vega slopes. 60
6.3 Discontinuous Payoff. 2D Digital option. The third and final task case that we have considered is the case of a discontinuous payoff, characterictic of so-called digital options. Figures (6.16) and (6.5) show the digital payoff that describes our specific product: { V Digital 0.02 L1 > K OR L (T ) = 2 > K 0 otherwise (6.5) With full grid of level 6 and K=5.5%, such payoff can be demonstrated visually: Discontinuous Payoff 0.02 0.015 0.01 0.005 0 0.16 0.14 0.12 0.1 0.08 Forward Rate L2 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L1 0.02 0.015 0.01 0.005 0 Figure 6.16: 2D Digital Option. Terminal payoff discretized on a L6 full grid. The stripe of different color in the corner hints that in such setting the strike rate K is an off-grid point. Indeed, in case of level 6 discretization, K falls roughly between values 0.05391 and 0.05625. 6.3.1 Valuation Results. Level of Discretization 7 7.5 8 8.5 9 9.5 10 21.2 MC Simulation Value+Std.Error Full-Grid FD value 21 20.8 Option price 20.6 20.4 20.2 20 19.8 19.6 500000 1e+006 1.5e+006 2e+006 2.5e+006 3e+006 Number of MC simulations Figure 6.17: 2D Digital Option. Convergence of Monte Carlo and full grid FD. 61
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 and 66: Level L 2 =1.87% L 2 =1.875% L 2 =5
Page 67: Figures (6.14) and (6.15) show the
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?