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

More documents

Recommendations

Info

$Learning LATEX by Doing$

5.3 A Combination technique So far, we’ve had a d-linear basis function as the unit of approximation on a discrete function space. The sparse grid combination technique, introduced in [Griebel et al., 1992], employes a different technique leading to the function representation on a sparse grid. Similar to Richardson extrapolation, it is based on the principle of multi-variate extrapolation and relies on the fact that leading-order discretization errors cancel out as the result of linear combination of grids. The combination technique uses a node-based discretization, as opposed to function-based one. The grids are combined as follows 1 : • 2D case : • 3D case : • dD case: û c n,n = ∑ u i,j − ∑ u i,j (5.10) i+j=n+1 i+j=n û c n,n = ∑ ∑ u i,j − 2 u i,j + ∑ u i,j (5.11) i+j=n+2 i+j=n+1 i+j=n ∑d−1 ( ) d − 1 ∑ û c n,n = (−1) k k k=0 |l| 1=n−k u |l| (5.12) First, PDE is solved on the individual grids by means of a numerical method (e.g., finite difference method, as in Chapter 4). Second, solutions are linearly combined to produce a interpolated approximation to u. Essencially, the combined approximation of a solution to the PDE u c l is not the same as the u ∗ l . However, it can be shown [Griebel et al., 1992] via asymptotic error expansion for 2D, that the estimation is of the same order of accuracy: ∥ ∥ u − û c ∥∞ n,n ≤ Kh 2 l (1 + 5 4 log 2(h −1 l )) = O(h −1 n log(h −1 n )) (5.13) where K is some constant and h l = 2 −l . Consider referring to [Blackham, 2004] for a d-dimensional error analysis. Recently, advances have been made in the subject of combination techniques for sparse grids. Novel combination approaches have been introduced in [Leentvaar and Oosterlee, 2006] or [Reisinger, 2004]. 1 here we usePnotation for a direct sum operator to emphasize the node-based nature of discretization, used in combination technique, instead of the commonLdirect sum notation 42
Chapter 6 Numerical Results. Valuation and the Greeks In Chapter 1 we have outlined the list of objectives for the practical part of this thesis. According to the first objective, our intention is to revisit valuation results of [Blackham, 2004]. Therefore, we choose to price a chooser option and a bermudan swaption products in the setting specified in [Blackham, 2004]. Even though real-world LMM derivatives are contingent on a larger number of forward rates, it has been decided to constrain the scope of work by looking into two and three rate cases. As a starting point for each case, the convergence behaviour of the price is studied by means of Monte Carlo simulation. The following step involves implementation of a finite difference solver to obtain a PDE solution on a full mesh of points, while earlier Monte Carlo results serve as a useful benchmark. Convergence of a sparse grid solution, obtained with the combination technique of Chapter 5, is then compared with a full grid approximation. As required by our original motivation, further tests include solving for different strike and spot rates and taking a closer look at the ’Greeks’ (∆ and V). Finally, we address the case of a discontinuous payoff and present results for valuation and the ’Greeks’ of a 2D digital option. Setup of Pricing Framework Before any valuation can take place, one needs to calibrate the pricer code to the real market data. The same calibration dataset is used as in [Blackham, 2004]. Movements in the underlying rates are restricted to the domain 1 of [0.0, 0.15] d . Volatility term structure is assumed flat, meaning the value of an instantaneous volatility function remains constant throughout the tenor period. Oneyear LIBOR forward rates and corresponding caplet volatilities are provided in (6.2). The amount of cash that underlies gains or losses in interest rate movements (a notional amount) is chosen to be 10000. The correlation matrix is given by formula (3.32). We assume strong correlation between the neighbouring rates, which can be achieved by setting β = 0.1. The following two tables give a summary of our setup: 1 unless stated otherwise 43
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: 00 03 02 03 01 03 02 03 00 30 30 20
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

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?