Physics and Finance

292 Pages·2021·6.317 MB·English

by  Volker Ziemann

Undergraduate Lecture Notes in Physics
Volker Ziemann
Physics and
Finance
Undergraduate Lecture Notes in Physics
Series Editors
Neil Ashby, University of Colorado, Boulder, CO, USA
WilliamBrantley,DepartmentofPhysics,FurmanUniversity,Greenville,SC,USA
MatthewDeady,PhysicsProgram,BardCollege,Annandale-on-Hudson,NY,USA
Michael Fowler, Department of Physics, University of Virginia, Charlottesville,
VA, USA
Morten Hjorth-Jensen, Department of Physics, University of Oslo, Oslo, Norway
Michael Inglis, Department of Physical Sciences, SUNY Suffolk County
Community College, Selden, NY, USA
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Volker Ziemann
Physics and Finance
123
Volker Ziemann
Department ofPhysics andAstronomy
Uppsala University
Uppsala, Sweden
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-3-030-63642-5 ISBN978-3-030-63643-2 (eBook)
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Preface
One fine day, my son, who was studying economics at the time, brought along
books on theuse of stochastic differential equations in finance. Following a dialog
alongthelinesof“Oh,Iknowthat,itsaFokker-Planckequation”—“Nodad,that’s
Black-Scholes,”Igotcurious.Afterall,theremightbesomefunineconomicsand
finance,besidesthemoney.So,IborrowedHull’sbookaboutthebasicsoffinancial
economics, because I wanted to understand the basic concepts and the lingo. Just
looking through the book, I recognized those differential equations that look so
similar to a diffusion equation with a drift term. So, I set out to understand what
finance has to do with diffusion.
Ilaterpresentedsomelecturesaboutmyexplorationstoafewinterestedstudents
and colleagues, which was very stimulating and caused me to explore the subject
further.Thatwashowthelaterchapterscameabout.Theyalldealwithsomeaspect
of random processes and have some overlap between physics, finance, and other
neighboring disciplines.
At that point, I prepared a 5 ECTS (European transfer credits) lecture series for
masters students at Uppsala University and expanded the manuscript to serve as
lecturenotesforthiscourse,whichranforthefirsttimeinthespringof2019,with
about15interestedstudents.Thefeedbackafterthecoursewasratherpositivesuch
thatIgavethecourseagaininthespringof2020.Thistimewith24students,who
providedmuchmorefeedbackandcriticism,whichcausedmetorevisepartsofthe
manuscript to bring it to its present form.
Obviously,manypeoplehelpedtoimprovethemanuscript.First,Ihavetothank
my son Ingvar. He stimulated my interest in finance and also critically read parts
of themanuscript. Likewise, Iam indebted to mycolleaguesand thestudents who
participated in the early lectures and in the course later. Many of them gave
valuable criticism and feedback on the growing manuscript. I want to single out a
few students, who were particularly diligent: Joe and Martin from the course in
2019; Friedrich, Sebastian, and Elias from 2020. They helped me weed out many
ambiguities and errors. They are, however, not to blame for any remaining bugs,
v
vi Preface
those aremy responsibility alone. I also need to thank our director of studies, Lisa
Freyhult, for her support at the faculty to include this course in the curriculum.
Finally, I want to thank my family for their patience with me when I was “a bit”
overfocused on the manuscript.
Uppsala, Sweden Volker Ziemann
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Concepts of Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Stocks and Other Tradeable Goods . . . . . . . . . . . . . . . . . . . . 5
2.2 Hedging and Shorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Discounting and Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6 Efficient Market Hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Theoretical Markets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Market Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Portfolio Theory and CAPM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Variational Calculus and Lagrange Multipliers . . . . . . . . . . . . 16
3.2 Portfolio with Risky Assets Only. . . . . . . . . . . . . . . . . . . . . . 18
3.3 Portfolio with a Risk-Free Asset . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Capital Market Line and Sharpe Ratio . . . . . . . . . . . . . . . . . . 22
3.5 Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Wiener Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Diffusion Processes and Green’s Functions. . . . . . . . . . . . . . . 33
4.4 Stochastic Integrals and Ito’s Lemma . . . . . . . . . . . . . . . . . . . 36
4.5 Master and Fokker-Planck Equations . . . . . . . . . . . . . . . . . . . 37
vii
viii Contents
4.6 A First Look at Option Pricing . . . . . . . . . . . . . . . . . . . . . . . 40
4.7 Digression on Expectation Values . . . . . . . . . . . . . . . . . . . . . 44
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Black-Scholes Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1 Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 The Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Risk-Neutrality and Martingales. . . . . . . . . . . . . . . . . . . . . . . 52
5.4 Dynamic Hedging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.5 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 The Greeks and Risk Management. . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1 The Greeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Volatility Smile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.4 Tailoring Risk to One’s Desire . . . . . . . . . . . . . . . . . . . . . . . 64
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 Regression Models and Hypothesis Testing. . . . . . . . . . . . . . . . . . . 69
7.1 Regression and Linear Fitting . . . . . . . . . . . . . . . . . . . . . . . . 70
7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.3 Goodness-of-Fit R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.4 v2-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 Student’s t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.6 Hypothesis Testing and p-Values . . . . . . . . . . . . . . . . . . . . . . 84
7.7 F-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.8 Parsimony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8 Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.1 Trend and Seasonality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.2 MA, AR, and ARMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8.3 Auto-Covariance and Autocorrelation. . . . . . . . . . . . . . . . . . . 96
8.4 Partial Autocorrelation Function. . . . . . . . . . . . . . . . . . . . . . . 99
8.5 Determining the Model Coefficients . . . . . . . . . . . . . . . . . . . . 101
8.6 Box-Jenkins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.7 Forecasting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.8 Zoo of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Contents ix
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions. . . . . . . 113
9.1 Historical Bubbles and Crashes . . . . . . . . . . . . . . . . . . . . . . . 114
9.2 Bubble-Crash Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.3 Behavioral Economics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.4 Fat-Tailed Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.5 Power Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.6 Fractals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.7 Sums of Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.8 Lévy-Stable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.9 Extreme-Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.10 Finite-Time Divergence and Log-Periodic Oscillations . . . . . . 138
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10 Quantum Finance and Path Integrals . . . . . . . . . . . . . . . . . . . . . . . 145
10.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10.2 Black-Scholes Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.3 Pricing Kernel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.4 Barrier Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.5 Path Integrals in Quantum Mechanics. . . . . . . . . . . . . . . . . . . 156
10.6 Path Integrals in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.7 Monte-Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
10.8 Numerical Evaluation of Path Integrals. . . . . . . . . . . . . . . . . . 167
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
11 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11.1 Macroeconomic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.2 Control and Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.3 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11.4 Hamiltonians for Optimal Control . . . . . . . . . . . . . . . . . . . . . 182
11.5 Donkey Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
11.6 Linear Quadratic Regulators. . . . . . . . . . . . . . . . . . . . . . . . . . 187
11.7 Controlling the Robinson-Crusoe Economy . . . . . . . . . . . . . . 189
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
12 Cryptocurrencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12.1 Information, Probabilities, and Codes. . . . . . . . . . . . . . . . . . . 195
12.2 Relation to the Thermodynamic Entropy. . . . . . . . . . . . . . . . . 198
12.3 Moving Information Through Discrete Channels. . . . . . . . . . . 199
12.4 Continuous Information Channels . . . . . . . . . . . . . . . . . . . . . 204