Stochastic Processes and Applications Mongolia 2015
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Poisson Processes Stochastic Processes
Poisson Processes Stochastic Processes UC3M Feb. 2012 Exponential random variables A random variable T has exponential distribution with rate λ > 0 if its probability density function can been written as −λt f (t) = λe 1(0;+1)(t) We summarize the above by T ∼ exp(λ): The cumulative distribution function of a exponential random variable is −λt F (t) = P(T ≤ t) = 1 − e 1(0;+1)(t) And the tail, expectation and variance are P(T > t) = e−λt ; E[T ] = λ−1; and Var(T ) = E[T ] = λ−2 The exponential random variable has the lack of memory property P(T > t + sjT > t) = P(T > s) Exponencial races In what follows, T1;:::; Tn are independent r.v., with Ti ∼ exp(λi ). P1: min(T1;:::; Tn) ∼ exp(λ1 + ··· + λn) . P2 λ1 P(T1 < T2) = λ1 + λ2 P3: λi P(Ti = min(T1;:::; Tn)) = λ1 + ··· + λn P4: If λi = λ and Sn = T1 + ··· + Tn ∼ Γ(n; λ). That is, Sn has probability density function (λs)n−1 f (s) = λe−λs 1 (s) Sn (n − 1)! (0;+1) The Poisson Process as a renewal process Let T1; T2;::: be a sequence of i.i.d. nonnegative r.v. (interarrival times). Define the arrival times Sn = T1 + ··· + Tn if n ≥ 1 and S0 = 0: The process N(t) = maxfn : Sn ≤ tg; is called Renewal Process. If the common distribution of the times is the exponential distribution with rate λ then process is called Poisson Process of with rate λ. Lemma. N(t) ∼ Poisson(λt) and N(t + s) − N(s); t ≥ 0; is a Poisson process independent of N(s); t ≥ 0 The Poisson Process as a L´evy Process A stochastic process fX (t); t ≥ 0g is a L´evyProcess if it verifies the following properties: 1. -
Stochastic Processes and Applications Mongolia 2015
NATIONAL UNIVERSITY OF MONGOLIA Stochastic Processes and Applications Mongolia 2015 27th July - 7th August 2015 National University of Mongolia Ulan Bator Mongolia National University of Mongolia 1 1 Basic information Venue: The meeting will take place at the National University of Mongolia. The map below shows the campus of the University which is located in the North-Eastern block relative to the Government Palace and Chinggis Khaan Square (see the red circle with arrow indicating main entrance in map below) in the very heart of down-town Ulan Bator. • All lectures, contributed talks and tutorials will be held in the Room 320 at 3rd floor, Main building, NUM. • Registration and Opening Ceremony will be held in the Academic Hall (Round Hall) at 2nd floor of the Main building. • The welcome reception will be held at the 2nd floor of the Broadway restaurant pub which is just on the West side of Chinggis Khaan Square (see the blue circle in map below). NATIONAL UNIVERSITY OF MONGOLIA 2 National University of Mongolia 1 Facilities: The main venue is equipped with an electronic beamer, a blackboard and some movable white- boards. There will also be magic whiteboards which can be used on any vertical surface. White-board pens and chalk will be provided. Breaks: Refreshments will be served between talks (see timetable below) at the conference venue. Lunches: Arrangements for lunches will be announced at the start of the meeting. Accommodation: Various places are being used for accommodation. The main accommodation are indi- cated on the map below relative to the National University (red circle): The Puma Imperial Hotel (purple circle), H9 Hotel (black circle), Ulanbaatar Hotel (blue circle), Student Dormitories (green circle) Mentoring: A mentoring scheme will be running which sees more experienced academics linked with small groups of junior researchers. -
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). -
Application of Malliavin Calculus and Wiener Chaos to Option Pricing Theory
Application of Malliavin Calculus and Wiener Chaos to Option Pricing Theory by Eric Ben-Hamou The London School of Economics and Political Science Thesis submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy at the University of London UMI Number: U615447 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U615447 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 weseS F 7855 % * A bstract This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al. -
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. -
Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications
Purdue University Purdue e-Pubs Open Access Dissertations Theses and Dissertations January 2015 Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications. Rolando Dangnanan Navarro Purdue University Follow this and additional works at: https://docs.lib.purdue.edu/open_access_dissertations Recommended Citation Navarro, Rolando Dangnanan, "Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications." (2015). Open Access Dissertations. 1422. https://docs.lib.purdue.edu/open_access_dissertations/1422 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Graduate School Form 30 Updated 1/15/2015 PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Rolando D. Navarro, Jr. Entitled Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications For the degree of Doctor of Philosophy Is approved by the final examining committee: Frederi Viens Chair Jonathon Peterson Michael Levine Jose Figueroa-Lopez To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material. Approved by Major Professor(s): Frederi Viens Approved by: Jun Xie 11/24/2015 Head of the Departmental Graduate Program Date MALLIAVIN CALCULUS IN THE CANONICAL LEVY´ PROCESS: WHITE NOISE THEORY AND FINANCIAL APPLICATIONS A Dissertation Submitted to the Faculty of Purdue University by Rolando D. Navarro, Jr. -
Weak Approximations for Arithmetic Means of Geometric Brownian Motions and Applications to Basket Options Romain Bompis
Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options Romain Bompis To cite this version: Romain Bompis. Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options. 2017. hal-01502886 HAL Id: hal-01502886 https://hal.archives-ouvertes.fr/hal-01502886 Preprint submitted on 6 Apr 2017 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. Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options ∗ Romain Bompis y Abstract. In this work we derive new analytical weak approximations for arithmetic means of geometric Brownian mo- tions using a scalar log-normal Proxy with an averaged volatility. The key features of the approach are to keep the martingale property for the approximations and to provide new integration by parts formulas for geometric Brownian motions. Besides, we also provide tight error bounds using Malliavin calculus, estimates depending on a suitable dispersion measure for the volatilities and on the maturity. As applications we give new price and im- plied volatility approximation formulas for basket call options. The numerical tests reveal the excellent accuracy of our results and comparison with the other known formulas of the literature show a valuable improvement. -
Extinction, Survival and Duality to P-Jump Processes
Markov branching processes with disasters: extinction, survival and duality to p-jump processes by Felix Hermann and Peter Pfaffelhuber Albert-Ludwigs University Freiburg Abstract A p-jump process is a piecewise deterministic Markov process with jumps by a factor of p. We prove a limit theorem for such processes on the unit interval. Via duality with re- spect to probability generating functions, we deduce limiting results for the survival proba- bilities of time-homogeneous branching processes with arbitrary offspring distributions, un- derlying binomial disasters. Extending this method, we obtain corresponding results for time-inhomogeneous birth-death processes underlying time-dependent binomial disasters and continuous state branching processes with p-jumps. 1 Introduction Consider a population evolving according to a branching process ′. In addition to reproduction events, global events called disasters occur at some random times Z(independent of ′) that kill off every individual alive with probability 1 p (0, 1), independently of each other.Z The resulting process of population sizes will be called− a branching∈ process subject to binomial disasters with survival probability p. ProvidedZ no information regarding fitness of the individuals in terms of resistance against disasters, this binomial approach appears intuitive, since the survival events of single individuals are iid. Applications span from natural disasters as floods and droughts to effects of radiation treatment or chemotherapy on cancer cells as well as antibiotics on popula- tions of bacteria. Also, Bernoulli sampling comes into mind as in the Lenski Experiment (cf. arXiv:1808.00073v2 [math.PR] 7 Jan 2019 Casanova et al., 2016). In the general setting of Bellman-Harris processes with non-lattice lifetime-distribution subject to binomial disasters, Kaplan et al. -
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. -
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 -
Joint Pricing and Inventory Control for a Stochastic Inventory System With
Joint pricing and inventory control for a stochastic inventory system with Brownian motion demand Dacheng Yao Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China; [email protected] Abstract In this paper, we consider an infinite horizon, continuous-review, stochastic inventory system in which cumulative customers' demand is price-dependent and is modeled as a Brownian motion. Excess demand is backlogged. The revenue is earned by selling products and the costs are incurred by holding/shortage and ordering, the latter consists of a fixed cost and a proportional cost. Our objective is to simultaneously determine a pricing strategy and an inventory control strategy to maximize the expected long-run average profit. Specifically, the pricing strategy provides the price pt for any time t ≥ 0 and the inventory control strategy characterizes when and how much we need to order. We show that an (s∗;S∗; p∗) policy is ∗ ∗ optimal and obtain the equations of optimal policy parameters, where p = fpt : t ≥ 0g. ∗ Furthermore, we find that at each time t, the optimal price pt depends on the current inventory level z, and it is increasing in [s∗; z∗] and is decreasing in [z∗; 1), where z∗ is a negative level. Keywords: Stochastic inventory model, pricing, Brownian motion demand, (s; S; p) policy, impulse control, drift rate control. arXiv:1608.03033v2 [math.OC] 11 Jul 2017 1 Introduction Exogenous selling price is always assumed in the classic production/inventory models; see e.g., Scarf(1960) and the research thereafter. In practice, however, dynamic pricing is one important tool for revenue management by balancing customers' demand and inventory level. -
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.