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

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$Learning LATEX by Doing$

Contents 1 Introduction 1 2 Asset Pricing Overview 3 2.1 Arbitrage and Risk-Neutral World . . . . . . . . . . . . . . . . . . . 3 2.2 From Discrete Time to Continuous Time . . . . . . . . . . . . . . . . 6 2.3 Pricing with Martingales and Measures . . . . . . . . . . . . . . . . 13 3 LIBOR Market Model 17 3.1 Forward LIBOR Rates . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 LIBOR Market Model . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Interest Rate Derivatives. Caplets, Caps and Swaptions. . . . . . . . . 22 3.4 Calibration of LIBOR Market Model . . . . . . . . . . . . . . . . . . 24 3.4.1 Calibration to caplets . . . . . . . . . . . . . . . . . . . . . . 25 3.4.2 Correlation structure of forward rates. . . . . . . . . . . . . . 25 3.5 The LIBOR Market Model PDE . . . . . . . . . . . . . . . . . . . . 28 4 Computational Methods 30 4.1 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 33 5 Sparse Grids 37 5.1 Full grid approximations. . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Sparse grid approximations. . . . . . . . . . . . . . . . . . . . . . . 41 5.3 A Combination technique . . . . . . . . . . . . . . . . . . . . . . . . 42 6 Numerical Results. Valuation and the Greeks 43 6.1 Two Dimensional Case. 2D Chooser Option . . . . . . . . . . . . . . 44 6.1.1 Convergence of MC, Full and Sparse Grid solutions. . . . . . 45 6.1.2 Discussion: Convergence of Pricing. . . . . . . . . . . . . . . 49 6.1.3 Greeks: Delta . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.1.4 Greeks: Vega . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Three Dimensional Case. 3D Bermudan Swaption . . . . . . . . . . . 54 6.2.1 Convergence Study . . . . . . . . . . . . . . . . . . . . . . . 56 6.2.2 Greeks. Delta and Vega. . . . . . . . . . . . . . . . . . . . . 58 6.3 Discontinuous Payoff. 2D Digital option. . . . . . . . . . . . . . . . 61 ii
6.3.1 Valuation Results. . . . . . . . . . . . . . . . . . . . . . . . 61 6.3.2 Greeks. Delta and Vega. . . . . . . . . . . . . . . . . . . . . 64 6.4 Discussion and Recommendations. . . . . . . . . . . . . . . . . . . . 66 7 Conclusion and Future Work 67 A Implementation Details 69 iii
Page 1 and 2: SPARSE GRID METHOD IN THE LIBOR MAR
Page 3: Acknowledgements This project would
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 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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