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

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of the payoff is also accountable for the severe spikes in sparse vega solution near the at-the-money position. 6.4 Discussion and Recommendations. In financial community, the quality of a computational method is, to a large extent, determined by its ability to show accurate approximations of risk sensitivities, such as ∆ and V. We can conclude from ∆ and V slice plots of each test case that the impact of the spikes effect is being magnified as we start operating on smaller ranges of values. What didn’t show in quality of price estimates, shows distinctly in the values for the Greeks. This fact makes immediate use of a sparse grid method in a discussed setting quite problematic. Steps need to be taken to improve the accuracy of approximation. There are several possible recipes one could follow: • Iterative Solver. When benchmarking against the results from[Blackham, 2004], the latter shows smoother convergence in both discretizations, and, moreover, surprisingly close results on earlier levels. Most likely, such behaviour is inflicted by the choice of an iterative solver. The author of [Blackham, 2004] reports to have used second-order finite differences with a multigrid iterative solver. The multigrid technique is designed to operate on a hierarchy of increasingly “coarser” meshes where, after desired accuracy has been reached, every individual mesh represents a part of the common solution. However, despite smoother convergence, the author of [Blackham, 2004] reports the presence of spikes in the solution. Thus, similar inaccuracies are to be expected in the results for the Greeks. • Order of FD. A possible way of with dealing discretization errors could be refining spatial discretization to a fourth-order finite differences, as explored in [Leentvaar and Oosterlee, 2006]. Again, such discretization poses difficulties for ”narrow” grids and solutions adjacent to boundary conditions, that appear to rely on ”ghost” outlier points. These issues can be addressed with application of extrapolation or rewriting of PDE with one-sided differences on the borders of solution domain. • Combination technique and sparse grid type. It would be worthwhile investigating the effect of the choice of an alternative sparse grid combination technique. Some of the possible alternatives to the “classical” sparse grid and combination technique are discussed in [Klimke, 2003] and [Leentvaar and Oosterlee, 2006], respectively. • Quantification Error. In practice, the quantification error of discontinuous payoff can be significantly reduced by peforming coordinate transformation. As described in [Travella and Randal, 2000], by rewriting the PDE in different terms, one can transform the grid in such a way that a larger amount of points is concentrated in the discontinuity region. Since this procedure does not degrade the accuracy of discretization, it could be an employed by the sparse grid method to battle the quantification error in case of digital option. 66
Chapter 7 Conclusion and Future Work In this thesis, we have investigated the application of sparse grid combination technique to the 2D and 3D interest rate products in the LIBOR Market Model framework. The selected test cases have been solved with second-order finite differences and BiCGSTAB iterative solver. In two dimensions, a chooser option and a digital option show good agreement of MC, full and sparse solutions in case of on-grid points for higher levels of discretization. For the chooser option, the price has converged to the value provided in [Blackham, 2004]. In addition, we have included results for different sets of strike and spot rates in case of on-grid solution points, where good convergence in both full and sparse discretizations was recorded. In three dimensions, the valuation of a bermudan swaption demonstrated convergence behaviour on a full grid. However, the obtained FD and MC values were different from the benchmark provided in [Blackham, 2004]. While convergence of a benchmark problem solution on a sparse grid was not smooth due to a large interpolation error, results for on-grid values were by far more convincing. The accuracy of approximations of the ’Greeks’ has been shown to suffer seriously from the ’spikes’, introduced into the final combination by approximations on the coarser grids in both 2D and 3D. Moreover, the delta of the digital option becomes a victim of a large quantification error. Our suggestions for the future work are the following: • refining discretization scheme of Finite Differences to the fourth order could help to obtain more accurate solutions on the coarse grids of sparse discretization. • it would be interesting to see how results of BiCGSTAB compare with those of a different iterative solver. In our view, a study of different combinations of discretization order and iterative solver could make a positive change on the quality of the ’Greeks’. • a study of the effect of grid stretching and coordinate transformation on the quality of digital ’Greeks’ with sparse grid approach. • combination technique is inherently well-fit for parallel computing. When solving problems in higher than 3D, parallel computation of solutions on individual grids will be required. 67
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SPARSE GRID METHOD IN THE LIBOR MAR
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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
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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
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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: Figures (6.21) and (6.22) show the
Page 77 and 78: Appendix A Implementation Details I
Page 79: A. Klimke. Piecewise multilinear sp
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