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

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Naturally, the characteristic feature of ’spikes’ of a sparse combination solution is to be expected in the higher dimensional problems, where the contribution of coarse grids is bigger. The following section summarizes results for 3D Bermudan Swaption product. 6.2 Three Dimensional Case. 3D Bermudan Swaption As discussed in Section (3.3), a Bermudan swaption is an interest rate derivative with the ’early-exercise’ feature. It allows the holder to enter into a swap contract on a set of prescribed dates and its general payoff is given by (3.27). Since we are computing the price of a swaption in the framework of a LIBOR Market Model, the terminal bond price D(t, T N+1 ) will play a role of a numeraire asset and Q will be its associated martingale measure. In case of 3 underlying rates, one can engage in a swap at the following set of fixing dates (T 0 ,T 1 ,T 2 ,T 3 ). Denote Vt BSw (T t , . . . , T N : T N+1 ) to be a Bermudan Swaption value at time t with expiry time T N+1 and set of fixing dates (T t , . . . , T N ). Then, its value at T 0 follows: V BSw 0 (T 0 , T 1 , T 2 , T 3 : T 4 ) = max ( H 0 (T 0 ), E 0 (T 0 ) ) (6.3) with a hold value H 0 (T 0 ) recursively defined from T 3 : [ ( max H 2 (T 2 ) = D(2, 4) E Q H3 (T 3 ), E 3 (T 3 ) ) ] D(3, 4) ∣ F 2 = E Q[ ] (L 3 − K) + |F 2 [ ( max H 1 (T 1 ) = D(1, 4) E Q H2 (T 2 ), E 2 (T 2 ) ) ] D(2, 4) ∣ F 1 [ ( max H 0 (T 0 ) = D(0, 4) E Q H1 (T 1 ), E 1 (T 1 ) ) ] D(1, 4) ∣ F 0 where the first relation holds since: E 3 (T 3 ) = D(3, 4) (L 3 − K) + and H 3 (T 3 ) = 0 (6.4) Figure (6.10) shows solution surfaces for 3D Bermudan Swaption with L 1 =5.62% fixing: 54
L1=5.62% Sparse Solution Swaption Price 1800 1600 1400 1200 1000 800 600 400 200 0 1800 1600 1400 1200 1000 800 600 0 200 400 0.14 0.12 0.1 0.08 Forward Rate L3 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L2 (a) Full Solution Surface for L 1 =5.62% L1=5.62% Sparse Solution Swaption Price 1800 1600 1400 1200 1000 800 600 400 200 0 1800 1600 1400 1200 1000 800 600 400 200 0 0.14 0.12 0.1 0.08 Forward Rate L3 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Forward Rate L2 (b) Sparse Solution Surface for L 1 =5.62% Figure 6.10: 3D Bermudan Swaption. Full and Sparse solution surfaces. 55
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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. . . . . .
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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: 0.03 0.025 Full vega Value Sparse v
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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