! " #$% & '( ' ) *% + , - %+ , ./ % ! / %+ /%+ ./ ' 0 "1%2 / '/ // " %2 '/ - - % + / 3 % / / %+ / % / - %2 '/ 4 35 %+ % ! " ! " # " # 677 %% 7 8 9 6 6 66 *#*$# 2.!:#;:#<*:::*; 2.!:#;:#<*:::== #$ ' <: SOME EXTENSIONS OF FRACTIONAL ORNSTEIN-UHLENBECK MODEL José Igor Morlanes Some Extensions of Fractional Ornstein-Uhlenbeck Model Arbitrage and Other Applications José Igor Morlanes ©José Igor Morlanes, Stockholm University 2017 ISBN print 978-91-7649-994-8 ISBN PDF 978-91-7649-995-5 Printed in Sweden by Universitetsservice US-AB, 2017 Distributor: Department of Statistics, Stockholm University List of Papers The following papers, referred to in the text by their Roman numerals, are included in this thesis. PAPER I: MORLANES, J.I, RASILA, A.,SOTTINEN, T. (2009) Empir- ical Evidence on Arbitrage by Changing the Stock Exchange. Advances and Applications in Statistics. Vol.12, Issue 2, pages 223-233. PAPER II: GASBARRA, D., MORLANES, J.I, VALKEILA, E. (2011) Ini- tial Enlargement in a Markov Chain Market Model. Stochastics and Dynamics. Vol. 11, Nos. 2& 3, pages 389-413. PAPER III: AZMOODEH, E., MORLANES, J.I (2015) Drift Parameter Esti- mation for fractional Ornstein-Uhlenbeck process of the Second Kind. Statistics: A Journal of Theoretical and Applied Statis- tics. Vol. 49, Issue 1, pages 1-18. PAPER IV: MORLANES, J.I., ANDREEV, A. (2017) On Simulation of Frac- tional Ornstein-Uhlenbeck of the Second Kind by Circulant Em- bedding Method. Research Report. No. 2017:1, SU, pages 1-11. PAPER V: ANDREEV, A., MORLANES, J.I. (2017) Simulations-based Study of Covariance structure for Fractional Ornstein-Uhlenbeck process of the Second Kind.. Preprint. Reprints were made with permission from the publishers. Motivation There may be many paths that one can follow to become a well-rounded sci- entific researcher. The path I decided to follow involves three main steps: first to master the theoretical concepts and methods in the field of research; second, to develop tools that can be applied to theoretical or empirical problems; and third, to use the accumulated knowledge in the first two steps to solve problems with real data in an area of interest. This doctoral dissertation is an effort to accomplish the first two steps. The family of fractional Brownian motion and Ornstein-Uhlenbeck process have been the focus. As I have been moving toward my goal, I have always kept third step in mind too. This dissertation has thus been frame-worked in the area of finance and economics. As a result, this thesis consists of five scientific articles. In the first three articles, I settle my theoretical background. In the last two, I master my programming skills and gain a deeper understanding of numerical methods. Contents List of Papers iii Motivation v 1 Introduction 1 2 Computational Aspects of Stochastic Differential Equations 3 2.1 Generalities of Stochastic Processes .............. 3 2.1.1 Martingales ....................... 4 2.2 Brownian Motion ........................ 4 2.2.1 Brownian motion as the limit of a random walk .... 5 2.2.2 Karhunen-Loeve expansion of Brownian motion . 6 2.2.3 Brownian Bridge .................... 7 2.3 Itô and Stratonovich Diffusion Processes ............ 8 2.3.1 Numerical Stochastic Integral ............. 9 2.3.2 Switching from a Itô to a Stratonovich Differential Equa- tion ........................... 10 2.3.3 Itô Lemma ....................... 11 2.3.4 The Lamperti Transform ................ 11 2.3.5 Families of Stochastic Processes ............ 11 2.4 Simulating Diffusion Processes ................. 13 2.4.1 Euler-Maruyama Approximation ............ 13 2.4.2 Milstein Scheme .................... 14 2.4.3 Error and Accuracy ................... 14 3 Ornstein-Uhlenbeck Process 17 3.1 Numerical Implementation of UO ............... 18 3.1.1 Simulating SDE .................... 18 3.1.2 Integral Solution .................... 19 3.1.3 Analytical Solution ................... 19 4 Fractional Brownian Motion 21 4.1 Definition and Properties of fBm ................ 21 4.1.1 Correlation and Long-Range Dependence of Time Se- ries Data ........................ 22 4.1.2 Numerical Integral respect to fBm ........... 23 4.1.3 Fractional Itô Lemma .................. 24 4.2 Fractional Ornstein-Uhlenbeck SDE .............. 25 4.3 Numerical Simulation ...................... 25 4.3.1 Circulant Embedding Method in a Nutshell ...... 26 5 Fundamental Theory Background 29 5.1 Malliavin Calculus ....................... 29 5.1.1 Malliavin derivative .................. 29 5.1.2 Divergente Operator and Skorohod integral ...... 33 5.1.3 Wick–Itô Integral .................... 34 5.2 Kernels ............................. 35 5.3 Regular conditional expectation ................ 37 5.3.1 Construction of regular conditional probability .... 40 5.3.2 Likelihood ratio function ................ 41 5.4 Maximum likelihood ratio process and inference ....... 43 Bibliography 45 6 Summary of Papers 49 7 Swedish Summary 53 1. Introduction A key contribution of this thesis is that it gives a deeper insight of some aspects of mathematical finance from a statistical and probabilistic point of view. This thesis offers the possibility of improving the modelling of financial data and of gaining an insight into modelling of information and arbitrage strategies which may be used to write financial instruments that take into account a potential insider. My journey began by learning the concepts of several important topics of Stochastic Calculus and Malliavin Calculus. I next developed probability and statistical tools to model random dynamic systems. These tools and models can be applied to any research field such as physics, engineering and telecom- munications. Although I have background in these fields, my true enthusiasm has been to seek applications in financial and economical problems. I have thus also learnt the main concepts of Continuous Stochastic Finance and Economet- rics. The main topic of this dissertation is the so-called fractional Ornstein- Uhlenbeck process of the second kind ( fOU2). The motivation for studying this process is that it exhibits a short-range dependence for all values of the self-similar parameter. This combined with a long-range dependence process is a more flexible dynamic model to capture the covariance structure of sam- pled data such as traffic networking data, interest rates of treasury bonds or stochastic volatility of financial derivatives. The fOU2 process is introduced in the scholarly paper by Kaarakka and Salminen [11]. The authors prove many mathematical properties of the process e.g. the process is locally Hölder continuous and stationary. They also consider the kernel representation of its covariance. The purpose of the articles III-V in this dissertation is to extend the re- search on the fOU2 process into the field of simulation and statistical infer- ence. These papers represent the first two research steps mentioned in the motivation. An estimator for the drift part of the process is constructed and simulation procedures to synthesize its sample trajectories are explored. The knowledge in these articles can then be used in a third step involving modelling and calibration of the process with real data. Article III describes a least-squared estimator and study its asymptotic properties. A logical next step is to examine the robustness of the estima- 1 tor. In order to achieve this, article IV-V present different algorithmic proce- dures to create the sample paths of the fOU2 process. The procedures syn- thesize the exact covariance structure and also consider the fOU2 paths with the most widely used marginal distributions by means of the circulant embed- ding method (CEM). The CEM is exact and easy to extend to non-Gaussian distributions with several dimensions. Articles I and II are devoted to financial derivatives and trading of non- public information. They are motivated by the need toreduce the likelihood of arbitrage opportunities for traders with access to insider information. Article I shows the possibility of making money without risk when a publicly- traded company switches marketplace, for example, from NASDAQ to the New York Stock Exchange. A trader with this information thus has an ar- bitrage opportunity. This is because the switching of marketplaces creates a shift in the price of the company derivative products. This shift is studied em- pirically in the paper and extended in a chapter of this dissertation by means of advanced Econometric techniques such the Logistic Smooth Transition Au- toregressive model and the Artificial Neural Network model. In article II an insider trader places orders in a marketplace modeled by a Markov chain. We investigate the possibility or not of arbitrage opportunities for an insider in a high frequency framework. The Markov chain process mod- els high frequency trading by tiny jumps between transactions. Some of the jumps may be accessible or predictable, but in the original filtration all jumps are inaccessible. Although the jumps change to accessible or predictable, the insider does not necessarily have arbitrage opportunities. This helps one to understand how the insider may profit even in the situation of being forced to close his position before expiration time. Along the research