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A New Stochastic Asset and Contingent Claim Valuation Framework with Augmented Schrödinger-Sturm-Liouville Eigenprice Conversion

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A New Stochastic Asset and Contingent Claim Valuation Framework with Augmented Schrödinger-Sturm-Liouville Eigenprice Conversion Coventry University DOCTOR OF PHILOSOPHY A new stochastic asset and contingent claim valuation framework with augmented Schrödinger-Sturm-Liouville EigenPrice conversion Euler, Adrian Award date: 2019 Awarding institution: Coventry University Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of this thesis for personal non-commercial research or study • This thesis cannot be reproduced or quoted extensively from without first obtaining permission from the copyright holder(s) • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 24. Sep. 2021 A New Stochastic Asset and Contingent Claim Valuation Framework with Augmented Schrödinger-Sturm-Liouville Eigen- Price Conversion By Adrian Euler July 10, 2019 A thesis submitted in partial fulfilment of the University’s requirements for the Degree of Doctor of Philosophy EXECUTIVE SUMMARY This study considers a new abstract and probabilistic stochastic finance asset price and contingent claim valuation model with an augmented Schrödinger PDE representation and Sturm-Liouville solutions, leading to additional requirements and outlooks of the probability density function and measurable price quantization effects with increased degrees of freedom. It is an analytical and valuation framework that explores existing pricing problems and models from a common point of high abstraction using a dimensionality reduction approach, under real-time trading assumptions. The context of the new model is a realistic market, made-up of a network of financial intermediaries and products whose prices are stochastic, measurable through transformable random variables, and a network of investors (individuals and firms) with rational and measurable preferences and expectations, seeking to maximize the expected utility of their final wealth in a multi-period time horizon. This study models pricing of assets and contingent claim at any time node and considers a zero-dimension reduction around each node in order to identify additional probability and price-change behavior effects, subsequently yielding new testable techniques of pricing assets and contingent claims. 2 AKNOWLEDGMENTS Firstly, I would like to express my sincere gratitude to my director of studies Professor Panagiotis Andrikopoulos for the continuous support of my PhD study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better supervisor and mentor for my PhD study. My sincere thanks also goes to the technical support team at National Algorithmic Group Ltd., especially Dr. John Muddle and his team for providing and extending the license for my use of NAGs and their technical guidance with installation of NAGs. Last but not the least, I would like to thank my family: my wife Dafine and my daughters Lowena and Lanasa for supporting me spiritually throughout the writing of this thesis and life in general. 3 TABLE OF CONTENTS CHAPTER 1: INTRODUCTION 7 CHAPTER 2: LITERATURE REVIEW 32 2.1 PRELIMINARIES 33 2.2 APPLIED STOCHASTIC FINANCE CONCEPTS 50 2.3 RELEVANT QUANTUM FINANCE MODELS: Q-MARKETS AND PRICE DISTRIBUTIONS 73 2.4 LITERATURE REVIEW CLOSING REMARKS 106 CHAPTER 3: RESEARCH METHODOLOGY 110 3.1 RESEARCH PHILOSOPHY 111 3.2 RESEARCH APPROACH 117 3.3 METHODOLOGICAL FOUNDATIONS 122 CHAPTER 4: AN ABSTRACT STOCHASTIC ASSET PRICING AND CONTIGENT CLAIM VALUATION FRAMEWORK WITH SCHRÖDINGER PDE AUGMENTATION 131 4.1 INTUITION AND RATIONALE 132 4.2. THE MODEL 146 4.3 PROPOSAL PROBABILITY DENSITY AND ASSET PRICING PDES 147 4.3.1 CASE 1.0 : TIME - INDEPENDANT GSE 151 4.3.2 CASE 2.0 - CAPITAL ASSET PRICING PDE 152 4.3.2.1 SCENARIO 1.0 154 4.3.2.2 SCENARIO 2.0 157 4.3.2.3 SCENARIO 3.0 159 4.3.2.4. SCENARIO 4.0 – PART I 160 4.3.2.5. SCENARIO 4.0 – PART II 162 4.3.3 CASE 3.0 - GENERALISED OPERATOR 164 4.3.4 CASE 4.0 – TIME INDEPENDENT EIGEN-STATE FORMULATION 168 CHAPTER 5: ASSET PRICE RAPPROCHEMENT: SPLIT PDF IDENTITIES AND STURM- LIOUVILLE QUANTUM FITTING 170 5.1 INTUITION AND RATIONALE 172 4 5.2. PROPOSED FINANCIAL DERIVATIVES’ PARTIAL DIFFERENTIAL EQUATIONS (PDES) 179 5.2.1 CASE 5.0 – TIME-DEPENDENT ASSET PRICING PDE 179 5.2.1.1. SCENARIO 1.0 – PART I 180 5.2.1.2. SCENARIO 1.0 – PART II 183 5.3. PRELIMINARY PROPOSAL FOR THE POTENTIAL ‘IDENTITY’ FUNCTION 185 5.3.1 CASE 6.0 –ASSET PRICING PDE WITH A ‘HARMONIC’’ POTENTIAL 187 5.3.1.1. SCENARIO 1.0 189 5.3.2 CASE 7.0 – SUPERSET DERIVATIVES’ PRICING PDE 191 5.3.2.1. SCENARIO 1.0 191 5.4.DENSITY OF STATES AND EIGEN-PRICE SYSTEM 193 5.5. INFORMATION DISSIPATION, REFLECTION AND RELAY FROM 0D PRICE SYSTEM 196 5.6. REBRANDING THE MASTER EXPRESSION 197 CHAPTER 6: COMPUTATION OF ASSET PRICES IN FORWARD TIME WITH EIGEN- VALUE CONVERSION 201 6.1 DISCUSSIONS AND NUMERICAL ILLUSTRATIONS 206 6.1.1 GEOMETRIC BROWNIAN MOTION SIMULATION OF STOCK PRICE 206 6.1.2 MONTE-CARLO PROBABILITY DENSITY MIXING 208 6.1.3 AUGMENTED MARKET-MOTIVE POTENTIAL FUNCTION AND INFORMATION TUNNELING 212 6.1.4 SUITABLE POTENTIAL FUNCTIONS 214 6.1.5 NUMERICAL ANALYSIS 220 6.1.6 INFORMATION TUNNELLING AND PRICE TRANSITION 231 6.1.7 INTENSITY OF INFORMATON REFLECTION ON PRICE (IIRP) 239 6.2 STURM-LIOUVILLE LOGARITHMIC PRICE EIGEN-VALUE SYSTEM 243 CHAPTER 7: COMPARATIVE VALUATION OF FINANCIAL OPTIONS WITH EIGEN- VALUE CONVERSION AND CLASSICAL MODELS 250 7.1 FINANCIAL OPTIONS WITH EIGEN-VALUE CONVERSION 251 7.2 FINANCIAL OPTION VALUATION WITH SQUARE-WELL, GAUSSIAN, COSH, AND ARCTAN ZERO-OBJECT MODELS 263 7.3 FINANCIAL OPTION VALUATION WITH BLACK-SCHOLES, CRR BINOMIAL, RG MODELS 274 7.3.1 THE BLACK-SCHOLES MODEL 274 7.3.2 THE CRR BINOMIAL MODEL 277 7.3.3 LOGNORMAL BLACK-SCHOLES MODEL 283 7.3.4 THE RECIPROCAL-GAMMA MODEL 284 7.4 EMPIRICAL TESTING AND ANALYSIS USING HISTORICAL VOLATILITY, CLASSICAL VS QUANTUM SW 288 7.5 EMPIRICAL TESTING AND ANALYSIS WITH LOCAL VOLATILITY AND PRICE- SURFACE QUANTUM SW FITTING 293 5 7.5.1 REAL-TIME OPTION PRICING, TESTING, AND ANALYSIS 293 7.5.2 REVISED OPTION PRICING MODELS 298 7.5.3 SAMPLE HYPOTHESIS TESTING 321 CHAPTER 8: CONCLUSIONS 332 BIBLIOGRAPHY 336 APPENDIX I: BROWNIAN MOTION SIMULATION USING EULER DISCRETIZATION 379 APPENDIX II: MATHEMATICAL DERIVATIONS 387 6 1.0 INTRODUCTION This research study introduces a new stochastic asset pricing model that can be used to value conventional financial assets, including financial derivatives. The new model utilises a generalised Schrödinger-Sturm-Liouville Eigen-price function and algorithm1. It stipulates a plausible probability density - price function duality phenomenon, incorporating relevant add-on price quantization effects. It allows for inclusion of increased degrees of freedom and subsequently increased accurancy in price forecasting . This is done by including effects of information dissipation and quantum ‘tunnelling’2. The new asset pricing model leads to a whole new set of pricing analytics that can be empirically applicable across asset classes such as equities, debt instruments, financial derivatives, etc. The abstracted pricing framework is formed through a process of breaking-apart, re- arranging, and re-assmbling previously known asset-pricing models, using a dimensionality-reduction approach, under real-time trading assumptions. It is accomplished by relaxing specific assumptions often used in obtaining and validating existing pricing models. Instead, I theorise using a general master expresison that reflects 1 A GSE-Sturm-Liouville algorithm implementing the modelled PDE in C++ with NAGS and allows for computation of Eigen price levels within an open-form quantum system. This is further elaborated in Chapters five and six. 2 With the term “tunnelling” we refer to the % of information known to-day that travels to the next time point through quantum market penetration. 7 (i) price developments on time-series basis, and (ii) spatial probability distribution effects, carefully formulated around the probability density - price function duality. This implies an equilibrium of effects. It it possible to explain this from an efficient market hypothesis angle (Fama, 1991; Malkiel, 2003). The information, expected to be reflected in the market price, is dissipated by the quantum structure fitted at a market point. Information is fragmented and price quantized within the structure. I hypothesize that quantum and market prices develop simultaneously. Therefore by studying information and price effects in the quantum structure, one might sufficiently predict the market price. This is in-line with the broader conceptualisation around Markov’s property (Markov, 1954:1971; Seneta, 1996; Gilks et al., 1996). In order to empirically test the new pricing techniques, I consider real-life market conditions, where a financial market is made up of an integrated system of financial intermediaries and financial assets whose prices follow a stochastic process measurable through transformable3 random variables. The financial market in this study is comprised of a network of institutional and retail investors with rational and measurable preferences and expectations, in-line with the Arrow-Pratt risk aversive attribution theory (Arrow, 1971:1988; Pratt, 1964; Gollier and Schlesinger, 1996). These investors seek to maximise the expected utility of their final wealth in a multi- 3 Changing a variable from a Cartesian system representation to spherical, but also replicative expansion of one single variable to two or more within the intended space-dimensions.
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