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

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List of Tables 6.1 Setup of LMM. Input data for LMM . . . . . . . . . . . . . . . . . . 44 6.2 Setup of LMM. LIBOR rates and caplet volatilies . . . . . . . . . . . 44 6.3 2D Chooser. Convergence values for full and sparse grid solutions. . . 45 6.4 2D Chooser. At-the-money and in-the-money option values. . . . . . 46 6.5 2D Chooser. Deep out-of-the-money and at-the-money with low strike. 46 6.6 3D Bermudan Swaption. Spot Rates as Table (6.2) . . . . . . . . . . 56 6.7 3D Bermudan Swaption. Full and sparse on-grid solution values for L 1 =3.75%. 500000 MC simulation trials. . . . . . . . . . . . . . . . 56 6.8 3D Bermudan Swaption. Full and sparse on-grid solution values for L 1 =5.62%. 500000 MC simulation trials. . . . . . . . . . . . . . . . 57 6.9 2D Digital. Full and sparse grid convergence of on-grid points. 500000 MC simulation trials. . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.10 2D Digital Continued. Full and sparse grid convergence of on-grid points. 500000 MC simulation trials. . . . . . . . . . . . . . . . . . . 63 vi
Chapter 1 Introduction The modern field of Derivative Finance is a very broad and highly scientific subject that spans several areas of knowledge. First of all, it naturally requires profound understanding of the dynamics of financial markets and financial products traded in the real world. Secondly, over the past several decades it has grown into an accomplished branch of applied mathematics that heavily employs probability theory and the theory of continuous-time stochastic processes. Thirdly, the problems of derivative pricing, growing in complexity along with the demands of the market players, increasingly rely on efficient numerical approximation and simulation methods. Last but not least, high performance and parallel computing environments have become an indispensable tool for the process of valuation and risk-analysis of complex multi-asset financial instruments. The main focus of this thesis is on the family of interest-rate derivatives, a very important class of products used to hedge exposure to interest-rate risk. Since the demand for such insurance is generated by large financial institutions (e.g. pension funds, insurance companies), banks and market makers selling interest-rate derivatives have to deal with the risks of serious losses. The necessity of controlling these risks, and thus, finding the correct value of a derivative instrument, formulates the motivation for the theory behind Derivative Finance. The second chapter of the thesis is meant to provide a brief introduction to the main concepts of option pricing in general, as well as mathematical theory needed while operating in the economy of interest rates. Both in academia and industry, a great deal of research is devoted to the development of realistic models for interest rate dynamics 1 and frameworks for pricing and hedging options. Among those discussed in the literature, one can distinguish two fundamental approaches: spot rate models and market rate models. The former approach takes a fictional instantaneous spot rate (alive for an infinitesimal time-period dt) as a basis for modelling of the term structure 2 . The rest of the rates are obtained by integrating with respect to a spot interest rate. From a mathematical standpoint, formulation and pricing products in terms of this abstraction is quite convenient. However, calibra- 1 of course, these models are being designed to simulate and not to forecast the future developments in the interest rate behaviour. 2 the set of consequent interest rates with increasing maturity 1
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: 6.18 2D Digital Option. Full and sp
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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