Multiscale calculus with applications in quantitative finance [Elektronische Ressource] / Stefan Dirnstorfer

Informations

• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2006
• Nombre de lectures: 24
• Langue: English

Extrait

Institut fu¨r Informatik
der Technischen Universit¨at Mu¨nchen
Lehrstuhl fu¨r Informatik mit Schwerpunkt wissenschaftliches Rechnen
Multiscale calculus with applications in quantitative
finance
Stefan Dirnstorfer
Vollst¨andiger Abdruck der von der Fakult¨at fu¨r Informatik der Technischen
Universi¨at Mu¨nchenzurErlangungdesakademischen Grades eines Doktors der
Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. Ru¨diger Westermann
Pru¨fer de Dissertation: 1. Univ.-Prof. Dr. Hans Joachim Bungartz
2. Univ.-Prof. Dr. Rudi Zagst
DieDissertationwurdeam28.09.2005beiderTechnischenUniversit¨atMu¨nchen
eingereicht und durch die Fakult¨at fu¨r Informatik am 24.01.2006 angenomen.Acknowledgement
TheautorwantstothankhisprofessorsHansBungartz,RudiZagstandChristoph
Zenger for supervising this thesis. Special thanks to Michael Ege for his conti-
butions to this document, his criticism and the many discussions on practical
implications. The author feels much obliged to Andreas Grau for proof read-
ing the document and for the many discussions on possible applications. Arnd
Pauwels has helped the author in questions of mathematical rigor.
iiContents
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.1.1 Algorithms on the functions domain . . . . . . . . . . . . 1
0.1.2 Layered model . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 Summary of each chapter . . . . . . . . . . . . . . . . . . . . . . 4
0.3 Operator index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Multiscale Calculus 6
1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1 Multiscale functions . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 From real functions to multiscale functions . . . . . . . . 9
1.2 Computer implementation . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Java . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.4 Other languages . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Basic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 Differential . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.3 Fundamental theorem of multiscale calculus . . . . . . . . 25
1.4 Advanced calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.1 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.2 Blur operator . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.3 Continuous shift operator . . . . . . . . . . . . . . . . . . 31
1.4.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5 Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.5.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.5.2 A priori adaptivity . . . . . . . . . . . . . . . . . . . . . . 34
1.5.3 A posteriori adaptivity . . . . . . . . . . . . . . . . . . . . 35
1.5.4 Sparse grid . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Proxy operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.1 Cache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.2 Multiplexer . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.6.3 Other proxies . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Pattern in portfolios 41
2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1 The economic state . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Valuation function . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iiiContents
2.3 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.1 Discrete process . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.3 The Black&Scholes model . . . . . . . . . . . . . . . . . . 44
2.4 The Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.1 Transaction . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.2 Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Pricing via arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5.1 The process . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5.2 Arbitrage-free price . . . . . . . . . . . . . . . . . . . . . 51
2.5.3 Equivalent martingale measure . . . . . . . . . . . . . . . 51
2.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.1 Hedging activity . . . . . . . . . . . . . . . . . . . . . . . 53
2.6.2 Supply and demand . . . . . . . . . . . . . . . . . . . . . 53
2.6.3 Market impact . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 The Theta-notation for stochastic processes 56
3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1 Stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 Transition probabilities . . . . . . . . . . . . . . . . . . . 58
3.1.2 Deterministic transition . . . . . . . . . . . . . . . . . . . 58
3.2 Differential processes . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Drift operator . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Blur operator . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Stock price models . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 The Black&Scholes model . . . . . . . . . . . . . . . . . . 67
3.3.2 GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.3 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Term structure models . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1 Deterministic model . . . . . . . . . . . . . . . . . . . . . 71
3.4.2 Heath-Jarrow-Morton . . . . . . . . . . . . . . . . . . . . 72
3.4.3 Hull&White . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.1 Present value . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.2 Risk measures . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5.3 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Put into practice 80
4.1 Economic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.1 The product . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.2 The process . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Algebraic results . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.1 Simplified setting . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.2 Integral form . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.3 Logarithmic scale . . . . . . . . . . . . . . . . . . . . . . . 83
ivContents
4.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.2 Standard scale . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.3 Logarithmic scale . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 Further tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.1 Evaluation nodes . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.2 Local adaptivity . . . . . . . . . . . . . . . . . . . . . . . 88
5 Conclusion 90
vAbout this document
0.1 Introduction
This document establishes a concept for a computer implementation of a func-
tion data type on which mathematical operators can be implemented. The
functions should have the flexibility to allow for common operations such as
integration, differentiation, inversion and the solution of differential equations.
Furthermore its flexibility should extend to the algorithmic concepts adaptiv-
ity and sparse grids. Chapter 1 shows how to implement such a data type
and how the common mathematical operations work on it. The algorithms
are presented in a discrete algebra and as computer code. Chapter 2 shows
the transformation of economic investment strategies into functional operators.
Based on functions as basic data type the evaluation of typical problems in
quantitative finance is kept compact and short. Chapter 3 goes into the details
of specific models for the dynamics of economic parameters. Many well-known
processes from quantitative finance are turned into the modular and computer
implementable operator form. Thefinalchapter 4 putseverything into practice
and demonstrates the applicability of the concept for mathematical modeling
as well as for numeric implementation.
0.1.1 Algorithms on the functions domain
Sincetheveryfirstdays,computersciencewasdrivenbythesearchforsolutions
to mathematical problems. It started with basic arithmetics on the domain of
integer andfloa