DOCSLIB.ORG

Option Pricing with Selfsimilar Additive Processes Mack L

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Option Pricing with Selfsimilar Additive Processes Mack L Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2006 Option Pricing with Selfsimilar Additive Processes Mack L. (Mack Laws) Galloway Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES OPTION PRICING WITH SELFSIMILAR ADDITIVE PROCESSES By MACK L. GALLOWAY A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Fall Semester, 2006 The members of the Committee approve the Dissertation of Mack L. Galloway defended on August 7, 2006. Craig Nolder Professor Directing Dissertation Fred Huffer Outside Committee Member Paul Beaumont Committee Member Bettye Anne Case Committee Member John R. Quine Committee Member The Office of Graduate Studies has verified and approved the above named committee members. ii ACKNOWLEDGEMENTS I thank the following friends and faculty for their help and support: Craig Nolder, Warren Beaves, Bettye Anne Case, Jennifer Mann, Fred Huffer, Paul Beaumont, Alec Kercheval, David Kopriva, John Quine, Max Gunzberger, James Doran, and Giles Levy. I also thank my parents, Harriett and Louie Galloway, III for their love and support throughout my life. iii TABLE OF CONTENTS List of Tables ...................................... vi List of Figures ..................................... vii Abstract ........................................ ix 1. Introduction .................................... 1 2. Review of Stochastic Processes .......................... 5 2.1 Additive Processes .............................. 5 2.2 L´evy Stochastic Volatility Models ...................... 9 3. Selfsimilar Additive Processes ........................... 11 3.1 Selfdecomposability and Selfsimilarity ................... 11 3.2 Strictly Stable L´evy Processes ........................ 12 3.3 Sato Processes ................................ 13 4. Subordinated, Additive Processes ......................... 17 4.1 Supporting Theorems and Definition .................... 17 4.2 The Generalized Subordination Theorem .................. 20 5. Construction of Additive Models ......................... 23 5.1 Derivation of Characteristic Functions ................... 23 5.2 Expressions for the Central Moments .................... 28 5.3 Properties of the Increments: ........................ 30 6. Option Pricing and Calibration .......................... 32 6.1 Definition of the Price Process ........................ 32 6.2 Numerical Pricing Method for European Options ............. 35 6.3 Calibration .................................. 37 7. Empirical Study .................................. 39 7.1 Models of Interest .............................. 39 7.2 Data Specifications .............................. 39 7.3 Algorithmic Specifications .......................... 40 7.4 Results .................................... 41 iv 8. Conclusion ..................................... 51 8.1 Future Research ............................... 51 9. Plots ......................................... 53 A. Truncation and Sampling Error: Lee’s Theorems ................ 67 A.1 Bounds on the Discounted Characteristic Function ............ 68 B. Plots of Difference between Calculated Pricing Error and Upper Bound on Theoretical Pricing Error ............................. 73 C. Additive Models: Discounted Characteristic Function Bounds for the European Call Option ..................................... 78 C.1 Variance Gamma Family ........................... 78 C.2 Normal Inverse Gaussian Family ...................... 80 D. VG-OU-Γ Discounted Characteristic Function Bounds for the European Call Option ........................................ 83 E. NIG-OU-Γ Discounted Characteristic Function Bounds for the European Call Option ........................................ 91 REFERENCES ..................................... 101 BIOGRAPHICAL SKETCH ............................. 103 v LIST OF TABLES 7.1 Classification of Calibrated Models ....................... 39 7.2 Upper Bound on the Time-averaged Theoretical Average Pricing Error (Jan. - Dec. 2005) .................................... 45 7.3 Upper Bound on the Time-averaged Theoretical Root Mean Square Pricing Error (Jan. - Dec. 2005) ............................. 45 7.4 Intra-family Comparison: Twelve Quote Date Time-averaged Ratio of the Upper Bounds on the Theoretical Average Pricing Errors (Jan. - Dec. 2005) 46 7.5 Intra-family Comparison: Twelve Quote Date Time-averaged Ratio of the Upper Bounds on the Theoretical Root Mean Square Pricing Errors (Jan. - Dec. 2005) ..................................... 46 7.6 Intra-class Comparison: Twelve Quote Date Time-averaged Ratio of the Upper Bounds on the Theoretical Average Pricing Errors (Jan. - Dec. 2005) 47 7.7 Intra-class Comparison: Twelve Quote Date Time-averaged Ratio of the Upper Bounds on the Theoretical Root Mean Square Pricing Errors (Jan. - Dec. 2005) .................................... 47 7.8 Variance Gamma Models: Twelve Quote Date Time-average of the Risk- Neutral Parameters (Jan. - Dec. 2005): “( )” denotes the sample standard deviation of the mean. .............................. 49 7.9 Normal Inverse Gaussian Models: Twelve Quote Date Time-average of the Risk-Neutral Parameters (Jan. - Dec. 2005): “( )” denotes the sample standard deviation of the mean. ........................ 49 7.10 Normal H-Inverse Gaussian Model: Three Quote Date Time-average of the Risk-Neutral Parameters (Sep. - Nov. 2005): “( )” denotes the sample standard deviation of the mean. ........................ 50 vi LIST OF FIGURES 9.1 Upper Bounds on Theoretical Average Pricing Error [in-sample, out-of-the- money options] .................................. 54 9.2 Upper Bounds on Theoretical Average Pricing Error [out-of-sample, out-of- the-money options] ................................ 55 9.3 Upper Bounds on Theoretical Root Mean Square Error [in-sample, out-of-the- money options] .................................. 56 9.4 Upper Bounds on Theoretical Root Mean Square Error [out-of-sample, out- of-the-money options] ............................... 57 9.5 Market and Model Option Prices vs. Moneyness (K/S) for Variance Gamma family : April 2005 quote date, out-of-the-money options ........... 58 9.6 Market and Model Option Prices vs. Moneyness (K/S) for Variance Gamma family : August 2005 quote date, out-of-the-money options .......... 59 9.7 Market and Model Option Prices vs. Moneyness (K/S) for Variance Gamma family : December 2005 quote date, out-of-the-money options ........ 60 9.8 Market and Model Option Prices vs. Moneyness (K/S) for Normal Inverse Gaussian family : April 2005 quote date, out-of-the-money options ...... 61 9.9 Market and Model Option Prices vs. Moneyness (K/S) for Normal Inverse Gaussian family : August 2005 quote date, out-of-the-money options .... 62 9.10 Market and Model Option Prices vs. Moneyness (K/S) for Normal Inverse Gaussian family : December 2005 quote date, out-of-the-money options ... 63 9.11 Risk-neutral Mean, Variance, Skewness, Kurtosis for April 2005 quote date [VG models - top row][NIG models - bottom row] ............... 64 9.12 Risk-neutral Mean, Variance, Skewness, Kurtosis for August 2005 quote date [VG models - top row][NIG models - bottom row] ............... 65 9.13 Risk-neutral Mean, Variance, Skewness, Kurtosis for December 2005 quote date [VG models - top row][NIG models - bottom row] ............ 66 vii B.1 In-sample Average Pricing Error Differences: (Upper Bound of Theoretical Value - Calculated Value using out-of-the-money options) ........... 74 B.2 Out-of-sample Average Pricing Error Differences: (Upper Bound of Theoret- ical Value - Calculated Value using out-of-the-money options) ........ 75 B.3 In-sample Root Mean Square Error Differences: (Upper Bound of Theoretical Value - Calculated Value using out-of-the-money options) .......... 76 B.4 Out-of-sample Root Mean Square Error Differences: (Upper Bound of Theo- retical Value - Calculated Value using out-of-the-money options) ...... 77 viii ABSTRACT The use of time-inhomogeneous additive models in option pricing has gained attention in recent years due to their potential to adequately price options across both strike and maturity with relatively few parameters. In this thesis two such classes of models based on the selfsimilar additive processes of Sato are developed. One class of models consists of the risk-neutral exponentials of a selfsimilar additive process, while the other consists of the risk-neutral exponentials of a Brownian motion time-changed by an independent, increasing, selfsimilar additive process. Examples from each class are constructed in which the time one distributions are Variance Gamma or Normal Inverse Gaussian distributed. Pricing errors are assessed for the case of Standard and Poor’s 500 index options from the year 2005. Both sets of time-inhomogeneous additive models show dramatic improvement in pricing error over their associated L´evy processes. Furthermore, with regard to the average of the pricing errors over the quote dates studied, the selfsimilar Normal Inverse Gaussian model yields a mean pricing error significantly less than that implied by the bid-ask spreads of the options, and also significantly less than that given by its associated, less parsimonious, L´evy stochastic volatility model. ix CHAPTER 1 Introduction The purpose of this thesis is to study the ability of certain non-L´evy, additive processes
Load more

Recommended publications
  • A University of Sussex Phd Thesis Available Online Via Sussex
    A University of Sussex PhD thesis Available online via Sussex Research Online: http://sro.sussex.ac.uk/ This thesis is protected by copyright which belongs to the author. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Please visit Sussex Research Online for more information and further details NON-STATIONARY PROCESSES AND THEIR APPLICATION TO FINANCIAL HIGH-FREQUENCY DATA Mailan Trinh A thesis submitted for the degree of Doctor of Philosophy University of Sussex March 2018 UNIVERSITY OF SUSSEX MAILAN TRINH A THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NON-STATIONARY PROCESSES AND THEIR APPLICATION TO FINANCIAL HIGH-FREQUENCY DATA SUMMARY The thesis is devoted to non-stationary point process models as generalizations of the standard homogeneous Poisson process. The work can be divided in two parts. In the first part, we introduce a fractional non-homogeneous Poisson process (FNPP) by applying a random time change to the standard Poisson process. We character- ize the FNPP by deriving its non-local governing equation. We further compute moments and covariance of the process and discuss the distribution of the arrival times. Moreover, we give both finite-dimensional and functional limit theorems for the FNPP and the corresponding fractional non-homogeneous compound Poisson process. The limit theorems are derived by using martingale methods, regular vari- ation properties and Anscombe's theorem.
    [Show full text]
  • Lévy Finance *[0.5Cm] Models and Results
    Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Stochastic Calculus for L´evy Processes L´evy-Process Driven Financial Market Models Jump-Diffusion Models Merton-Model Kou-Model General L´evy Models Variance-Gamma model CGMY model GH models Variance-mean mixtures European Style Options Equivalent Martingale Measure Jump-Diffusion Models Variance-Gamma Model NIG Model Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Stochastic Integral for L´evyProcesses Let (Xt ) be a L´evy process with L´evy-Khintchine triplet (α, σ, ν(dx)). By the L´evy-It´odecomposition we know X = X (1) + X (2) + X (3), where the X (i) are independent L´evyprocesses. X (1) is a Brownian motion with drift, X (2) is a compound Poisson process with jump (3) distributed concentrated on R/(−1, 1) and X is a square-integrable martingale (which can be viewed as a limit of compensated compound Poisson processes with small jumps). We know how to define the stochastic integral with respect to any of these processes! Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Canonical Decomposition From the L´evy-It´odecomposition we deduce the canonical decomposition (useful for applying the general semi-martingale theory) Z t Z X (t) = αt + σW (t) + x µX − νX (ds, dx), 0 R where Z t Z X xµX (ds, dx) = ∆X (s) 0 R 0<s≤t and Z t Z Z t Z Z X X E xµ (ds, dx) = xν (ds, dx) = t xν(dx).
    [Show full text]
  • Pricing, Risk and Volatility in Subordinated Market Models
    risks Article Pricing, Risk and Volatility in Subordinated Market Models Jean-Philippe Aguilar 1,* , Justin Lars Kirkby 2 and Jan Korbel 3,4,5,6 1 Covéa Finance, Quantitative Research Team, 8-12 rue Boissy d’Anglas, FR75008 Paris, France 2 School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30318, USA; [email protected] 3 Section for the Science of Complex Systems, Center for Medical Statistics, Informatics, and Intelligent Systems (CeMSIIS), Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria; [email protected] 4 Complexity Science Hub Vienna, Josefstädterstrasse 39, 1080 Vienna, Austria 5 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, 11519 Prague, Czech Republic 6 The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod Vodárenskou Vˇeží4, 182 00 Prague 8, Czech Republic * Correspondence: jean-philippe.aguilar@covea-finance.fr Received: 26 October 2020; Accepted: 13 November 2020; Published: 17 November 2020 Abstract: We consider several market models, where time is subordinated to a stochastic process. These models are based on various time changes in the Lévy processes driving asset returns, or on fractional extensions of the diffusion equation; they were introduced to capture complex phenomena such as volatility clustering or long memory. After recalling recent results on option pricing in subordinated market models, we establish several analytical formulas for market sensitivities and portfolio performance in this class of models, and discuss some useful approximations when options are not far from the money. We also provide some tools for volatility modelling and delta hedging, as well as comparisons with numerical Fourier techniques.
    [Show full text]
  • STOCHASTIC COMPARISONS and AGING PROPERTIES of an EXTENDED GAMMA PROCESS Zeina Al Masry, Sophie Mercier, Ghislain Verdier
    STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS Zeina Al Masry, Sophie Mercier, Ghislain Verdier To cite this version: Zeina Al Masry, Sophie Mercier, Ghislain Verdier. STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS. Journal of Applied Probability, Cambridge University press, 2021, 58 (1), pp.140-163. 10.1017/jpr.2020.74. hal-02894591 HAL Id: hal-02894591 https://hal.archives-ouvertes.fr/hal-02894591 Submitted on 9 Jul 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Applied Probability Trust (4 July 2020) STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS ZEINA AL MASRY,∗ FEMTO-ST, Univ. Bourgogne Franche-Comt´e,CNRS, ENSMM SOPHIE MERCIER & GHISLAIN VERDIER,∗∗ Universite de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, Pau, France Abstract Extended gamma processes have been seen to be a flexible extension of standard gamma processes in the recent reliability literature, for cumulative deterioration modeling purpose. The probabilistic properties of the standard gamma process have been well explored since the 1970's, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and aging properties for the corresponding level-crossing times are nowadays well understood.
    [Show full text]
  • Lecture Notes
    Lecture Notes Lars Peter Hansen October 8, 2007 2 Contents 1 Approximation Results 5 1.1 One way to Build a Stochastic Process . 5 1.2 Stationary Stochastic Process . 6 1.3 Invariant Events and the Law of Large Numbers . 6 1.4 Another Way to Build a Stochastic Process . 8 1.4.1 Stationarity . 8 1.4.2 Limiting behavior . 9 1.4.3 Ergodicity . 10 1.5 Building Nonstationary Processes . 10 1.6 Martingale Approximation . 11 3 4 CONTENTS Chapter 1 Approximation Results 1.1 One way to Build a Stochastic Process • Consider a probability space (Ω, F, P r) where Ω is a set of sample points, F is an event collection (sigma algebra) and P r assigns proba- bilities to events . • Introduce a function S :Ω → Ω such that for any event Λ, S−1(Λ) = {ω ∈ Ω: S(ω) ∈ Λ} is an event. • Introduce a (Borel measurable) measurement function x :Ω → Rn. x is a random vector. • Construct a stochastic process {Xt : t = 1, 2, ...} via the formula: t Xt(ω) = X[S (ω)] or t Xt = X ◦ S . Example 1.1.1. Let Ω be a collection of infinite sequences of real numbers. Specifically, ω = (r0, r1, ...), S(ω) = (r1, r2, ...) and x(ω) = r0. Then Xt(ω) = rt. 5 6 CHAPTER 1. APPROXIMATION RESULTS 1.2 Stationary Stochastic Process Definition 1.2.1. The transformation S is measure-preserving if: P r(Λ) = P r{S−1(Λ)} for all Λ ∈ F. Proposition 1.2.2. When S is measure-preserving, the process {Xt : t = 1, 2, ...} has identical distributions for every t.
    [Show full text]
  • Arbitrage Free Approximations to Candidate Volatility Surface Quotations
    Journal of Risk and Financial Management Article Arbitrage Free Approximations to Candidate Volatility Surface Quotations Dilip B. Madan 1,* and Wim Schoutens 2,* 1 Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA 2 Department of Mathematics, KU Leuven, 3000 Leuven, Belgium * Correspondence: [email protected] (D.B.M.); [email protected] (W.S.) Received: 19 February 2019; Accepted: 10 April 2019; Published: 21 April 2019 Abstract: It is argued that the growth in the breadth of option strikes traded after the financial crisis of 2008 poses difficulties for the use of Fourier inversion methodologies in volatility surface calibration. Continuous time Markov chain approximations are proposed as an alternative. They are shown to be adequate, competitive, and stable though slow for the moment. Further research can be devoted to speed enhancements. The Markov chain approximation is general and not constrained to processes with independent increments. Calibrations are illustrated for data on 2695 options across 28 maturities for SPY as at 8 February 2018. Keywords: bilateral gamma; fast Fourier transform; sato process; matrix exponentials JEL Classification: G10; G12; G13 1. Introduction A variety of arbitrage free models for approximating quoted option prices have been available for some time. The quotations are typically in terms of (Black and Scholes 1973; Merton 1973) implied volatilities for the traded strikes and maturities. Consequently, the set of such quotations is now popularly referred to as the volatility surface. Some of the models are nonparametric with the Dupire(1994) local volatility model being a popular example. The local volatility model is specified in terms of a deterministic function s(K, T) that expresses the local volatility for the stock if the stock were to be at level K at time T.
    [Show full text]
  • Informs 2007 Proceedings
    informs14th ® Applied Probability Conference July 9–11, 2007 Program Monday July 9, 2007 Track 1 Track 2 Track 3 Track 4 Track 5 Track 6 Track 7 Track 8 Track 9 Room CZ 4 CZ 5 CZ 10 CZ 11 CZ 12 CZ 13 CZ 14 CZ 15 CZ 16 9:00am - 9:15am Opening (Room: Blauwe Zaal) 9:15am - 10:15am Plenary - Peter Glynn (Room: Blauwe Zaal) MA Financial Random Fields Rare Event Asymptotic Scheduling Call Centers 1 MDP 1 Retrial Inventory 1 10:45am - 12:15pm Engineering 1 Simulation 1 Analysis 1 Queues Kou Kaj Dupuis Bassamboo / Borst / Koole Feinberg Artalejo Van Houtum Randhawa Wierman Keppo Scheffler Blanchet Lin Gupta Taylor Bispo Machihara Buyukkaramikli DeGuest Ruiz-Medina Glasserman Tezcan Ayesta Jongbloed Van der Laan Nobel Qiu Peng Kaj Juneja Gurvich Wierman Henderson Haijema Shin Timmer Weber Mahmoodi Dupuis Randhawa Winands Koole Feinberg Artalejo Van Houtum 12:45pm - 1.45pm Tutorial Philippe Robert MB Financial Percolation and Simulation 1 Stability of Stoch. Communication Many-server Games 1 Fluid Queues Search 2:00pm - 3:30pm Engineering 2 Related Topics Networks Systems 1 Models 1 Models Schoutens / Van den Berg Henderson Ramanan Choi Armony Economou Adan Klafter Valdivieso Werker Newman Chick Gamarnik Bae Tezcan Economou Dieker Benichou Koch Newman Haas Reiman Kim Jennings Amir Nazarathy Oshanin Scherer Meester Blanchet Williams Park Ward Dube Margolius Eliazar Valdivieso Kurtz Henderson Zachary Roubos Armony Economou Adan Metzler MC Exit Times Interacting Stoch. Prog. Stoch. Netw. & Flow-Level Markov Control Queueing Inventory 2 4:00pm - 5:30pm
    [Show full text]
  • Portfolio Optimization and Option Pricing Under Defaultable Lévy
    Universit`adegli Studi di Padova Dipartimento di Matematica Scuola di Dottorato in Scienze Matematiche Indirizzo Matematica Computazionale XXVI◦ ciclo Portfolio optimization and option pricing under defaultable L´evy driven models Direttore della scuola: Ch.mo Prof. Paolo Dai Pra Coordinatore dell’indirizzo: Ch.ma Prof.ssa. Michela Redivo Zaglia Supervisore: Ch.mo Prof. Tiziano Vargiolu Universit`adegli Studi di Padova Corelatori: Ch.mo Prof. Agostino Capponi Ch.mo Prof. Andrea Pascucci Johns Hopkins University Universit`adegli Studi di Bologna Dottorando: Stefano Pagliarani To Carolina and to my family “An old man, who had spent his life looking for a winning formula (martingale), spent the last days of his life putting it into practice, and his last pennies to see it fail. The martingale is as elusive as the soul.” Alexandre Dumas Mille et un fantˆomes, 1849 Acknowledgements First and foremost, I would like to thank my supervisor Prof. Tiziano Vargiolu and my co-advisors Prof. Agostino Capponi and Prof. Andrea Pascucci, for coauthoring with me all the material appearing in this thesis. In particular, thank you to Tiziano Vargiolu for supporting me scientifically and finan- cially throughout my research and all the activities related to it. Thank you to Agostino Capponi for inviting me twice to work with him at Purdue University, and for his hospital- ity and kindness during my double stay in West Lafayette. Thank you to Andrea Pascucci for wisely leading me throughout my research during the last four years, and for giving me the chance to collaborate with him side by side, thus sharing with me his knowledge and his experience.
    [Show full text]
  • Some Mathematical Aspects of Market Impact Modeling by Alexander Schied and Alla Slynko
    EMS Series of Congress Reports EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowro´nski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes Jochen Blath Peter Imkeller Sylvie Rœlly Editors Editors: Jochen Blath Peter Imkeller Sylvie Rœlly Institut für Mathematik Institut für Mathematik Institut für Mathematik der Technische Universität Berlin Humboldt-Universität zu Berlin Universität Potsdam Straße des 17. Juni 136 Unter den Linden 6 Am Neuen Palais, 10 10623 Berlin 10099 Berlin 14469 Potsdam Germany Germany Germany [email protected] [email protected] [email protected] 2010 Mathematics Subject Classification: Primary: 60-06, Secondary 60Gxx, 60Jxx Key words: Stochastic processes, stochastic finance, stochastic analysis,statistical physics, stochastic differential equations ISBN 978-3-03719-072-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks.
    [Show full text]
  • A Representation Free Quantum Stochastic Calculus
    JOURNAL OF FUNCTIONAL ANALYSIS 104, 149-197 (1992) A Representation Free Quantum Stochastic Calculus L. ACCARDI Centro Matemarico V. Volterra, Diparfimento di Matematica, Universitci di Roma II, Rome, Italy F. FACNOLA Dipartimento di Matematica, Vniversild di Trento, Povo, Italy AND J. QUAEGEBEUR Department of Mathemarics, Universiteit Leuven, Louvain, Belgium Communicated by L. Gross Received January 2; revised March 1, 1991 We develop a representation free stochastic calculus based on three inequalities (semimartingale inequality, scalar forward derivative inequality, scalar conditional variance inequality). We prove that our scheme includes all the previously developed stochastic calculi and some new examples. The abstract theory is applied to prove a Boson Levy martingale representation theorem in bounded form and a general existence, uniqueness, and unitarity theorem for quantum stochastic differential equations. 0 1992 Academic Press. Inc. 0. INTRODUCTION Stochastic calculus is a powerful tool in classical probability theory. Recently various kinds of stochastic calculi have been introduced in quan- tum probability [18, 12, 10,211. The common features of these calculi are: - One starts from a representation of the canonical commutation (or anticommutation) relations (CCR or CAR) over a space of the form L2(R+, dr; X) (where 3? is a given Hilbert space). - One introduces a family of operator valued measures on R,, related to this representation, e.g., expressed in terms of creation or annihilation or number or field operators. 149 0022-1236192 $3.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved. 150 QUANTUM STOCHASTIC CALCULUS -- One shows that it is possible to develop a theory of stochastic integration with respect to these operator valued measures sufficiently rich to allow one to solve some nontrivial stochastic differential equations.
    [Show full text]
  • Normalized Random Measures Driven by Increasing Additive Processes
    The Annals of Statistics 2004, Vol. 32, No. 6, 2343–2360 DOI: 10.1214/009053604000000625 c Institute of Mathematical Statistics, 2004 NORMALIZED RANDOM MEASURES DRIVEN BY INCREASING ADDITIVE PROCESSES By Luis E. Nieto-Barajas1, Igor Prunster¨ 2 and Stephen G. Walker3 ITAM-M´exico, Universit`adegli Studi di Pavia and University of Kent This paper introduces and studies a new class of nonparamet- ric prior distributions. Random probability distribution functions are constructed via normalization of random measures driven by increas- ing additive processes. In particular, we present results for the dis- tribution of means under both prior and posterior conditions and, via the use of strategic latent variables, undertake a full Bayesian analysis. Our class of priors includes the well-known and widely used mixture of a Dirichlet process. 1. Introduction. This paper considers the problem of constructing a stochastic process, defined on the real line, which has sample paths be- having almost surely (a.s.) as a probability distribution function (d.f.). The law governing the process acts as a prior in Bayesian nonparametric prob- lems. One popular idea is to take the random probability d.f. as a normal- ized increasing process F (t)= Z(t)/Z¯, where Z¯ = limt→∞ Z(t) < +∞ (a.s.). For example, the Dirichlet process [Ferguson (1973)] arises when Z is a suitably reparameterized gamma process. We consider the case of normal- ized random d.f.s driven by an increasing additive process (IAP) L, that is, Z(t)= k(t,x) dL(x) and provide regularity conditions on k and L to ensure F is a random probability d.f.
    [Show full text]
  • Multilevel Particle Filters for L\'Evy-Driven Stochastic Differential
    Multilevel Particle Filters for L´evy-driven stochastic differential equations BY AJAY JASRA 1, KODY J. H. LAW 2 & PRINCE PEPRAH OSEI 1 1 Department of Statistics & Applied Probability, National University of Singapore, Singapore, 117546, SG. E-Mail: [email protected], [email protected] 2 School of Mathematics, University of Manchester, UK, AND Computer Science and Mathematics Division, Oak Ridge National Laboratory Oak Ridge, TN, 37831, USA. E-Mail: [email protected] Abstract We develop algorithms for computing expectations with respect to the laws of models associated to stochastic differential equations (SDEs) driven by pure L´evyprocesses. We consider filtering such processes and well as pricing of path dependent options. We propose a multilevel particle filter (MLPF) to address the computational issues involved in solving these continuum problems. We show via numerical simulations and theoretical results that under suitable assumptions regarding the discretization of the underlying driving L´evyproccess, our proposed method achieves optimal convergence rates: the cost to obtain MSE O(2) scales like O(−2) for our method, as compared with the standard particle filter O(−3). Keywords: L´evy-driven SDE; L´evyprocesses; Particle Filters; Multilevel Particle Filters; Barrier options. 1 Introduction arXiv:1804.04444v2 [stat.CO] 11 Jul 2018 L´evyprocesses have become very useful recently in several scientific disciplines. A non-exhaustive list includes physics, in the study of turbulence and quantum field theory; economics, for con- tinuous time-series models; insurance mathematics, for computation of insurance and risk, and mathematical finance, for pricing path dependent options. Earlier application of L´evyprocesses in modeling financial instruments dates back in [23] where a variance gamma process is used to model market returns.
    [Show full text]