The Variance Gamma Process and Option Pricing
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Numerical Solution of Jump-Diffusion LIBOR Market Models
Finance Stochast. 7, 1–27 (2003) c Springer-Verlag 2003 Numerical solution of jump-diffusion LIBOR market models Paul Glasserman1, Nicolas Merener2 1 403 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027, USA (e-mail: [email protected]) 2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA (e-mail: [email protected]) Abstract. This paper develops, analyzes, and tests computational procedures for the numerical solution of LIBOR market models with jumps. We consider, in par- ticular, a class of models in which jumps are driven by marked point processes with intensities that depend on the LIBOR rates themselves. While this formulation of- fers some attractive modeling features, it presents a challenge for computational work. As a first step, we therefore show how to reformulate a term structure model driven by marked point processes with suitably bounded state-dependent intensities into one driven by a Poisson random measure. This facilitates the development of discretization schemes because the Poisson random measure can be simulated with- out discretization error. Jumps in LIBOR rates are then thinned from the Poisson random measure using state-dependent thinning probabilities. Because of discon- tinuities inherent to the thinning process, this procedure falls outside the scope of existing convergence results; we provide some theoretical support for our method through a result establishing first and second order convergence of schemes that accommodates thinning but imposes stronger conditions on other problem data. The bias and computational efficiency of various schemes are compared through numerical experiments. Key words: Interest rate models, Monte Carlo simulation, market models, marked point processes JEL Classification: G13, E43 Mathematics Subject Classification (1991): 60G55, 60J75, 65C05, 90A09 The authors thank Professor Steve Kou for helpful comments and discussions. -
MERTON JUMP-DIFFUSION MODEL VERSUS the BLACK and SCHOLES APPROACH for the LOG-RETURNS and VOLATILITY SMILE FITTING Nicola Gugole
International Journal of Pure and Applied Mathematics Volume 109 No. 3 2016, 719-736 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v109i3.19 ijpam.eu MERTON JUMP-DIFFUSION MODEL VERSUS THE BLACK AND SCHOLES APPROACH FOR THE LOG-RETURNS AND VOLATILITY SMILE FITTING Nicola Gugole Department of Computer Science University of Verona Strada le Grazie, 15-37134, Verona, ITALY Abstract: In the present paper we perform a comparison between the standard Black and Scholes model and the Merton jump-diffusion one, from the point of view of the study of the leptokurtic feature of log-returns and also concerning the volatility smile fitting. Provided results are obtained by calibrating on market data and by mean of numerical simulations which clearly show how the jump-diffusion model outperforms the classical geometric Brownian motion approach. AMS Subject Classification: 60H15, 60H35, 91B60, 91G20, 91G60 Key Words: Black and Scholes model, Merton model, stochastic differential equations, log-returns, volatility smile 1. Introduction In the early 1970’s the world of option pricing experienced a great contribu- tion given by the work of Fischer Black and Myron Scholes. They developed a new mathematical model to treat certain financial quantities publishing re- lated results in the article The Pricing of Options and Corporate Liabilities, see [1]. The latter work became soon a reference point in the financial scenario. Received: August 3, 2016 c 2016 Academic Publications, Ltd. Revised: September 16, 2016 url: www.acadpubl.eu Published: September 30, 2016 720 N. Gugole Nowadays, many traders still use the Black and Scholes (BS) model to price as well as to hedge various types of contingent claims. -
Sensitivity Estimation of Sabr Model Via Derivative of Random Variables
Proceedings of the 2011 Winter Simulation Conference S. Jain, R. R. Creasey, J. Himmelspach, K. P. White, and M. Fu, eds. SENSITIVITY ESTIMATION OF SABR MODEL VIA DERIVATIVE OF RANDOM VARIABLES NAN CHEN YANCHU LIU The Chinese University of Hong Kong 709A William Mong Engineering Building Shatin, N. T., HONG KONG ABSTRACT We derive Monte Carlo simulation estimators to compute option price sensitivities under the SABR stochastic volatility model. As a companion to the exact simulation method developed by Cai, Chen and Song (2011), this paper uses the sensitivity of “vol of vol” as a showcase to demonstrate how to use the pathwise method to obtain unbiased estimators to the price sensitivities under SABR. By appropriately conditioning on the path generated by the volatility, the evolution of the forward price can be represented as noncentral chi-square random variables with stochastic parameters. Combined with the technique of derivative of random variables, we can obtain fast and accurate unbiased estimators for the sensitivities. 1 INTRODUCTION The ubiquitous existence of the volatility smile and skew poses a great challenge to the practice of risk management in fixed and foreign exchange trading desks. In foreign currency option markets, the implied volatility is often relatively lower for at-the-money options and becomes gradually higher as the strike price moves either into the money or out of the money. Traders refer to this stylized pattern as the volatility smile. In the equity and interest rate markets, a typical aspect of the implied volatility, which is also known as the volatility skew, is that it decreases as the strike price increases. -
Option Pricing with Jump Diffusion Models
UNIVERSITY OF PIRAEUS DEPARTMENT OF BANKING AND FINANCIAL MANAGEMENT M. Sc in FINANCIAL ANALYSIS FOR EXECUTIVES Option pricing with jump diffusion models MASTER DISSERTATION BY: SIDERI KALLIOPI: MXAN 1134 SUPERVISOR: SKIADOPOULOS GEORGE EXAMINATION COMMITTEE: CHRISTINA CHRISTOU PITTIS NIKITAS PIRAEUS, FEBRUARY 2013 2 TABLE OF CONTENTS ς ABSTRACT .............................................................................................................................. 3 SECTION 1 INTRODUCTION ................................................................................................ 4 ώ SECTION 2 LITERATURE REVIEW: .................................................................................... 9 ι SECTION 3 DATASET .......................................................................................................... 14 SECTION 4 .............................................................................................................................α 16 4.1 PRICING AND HEDGING IN INCOMPLETE MARKETS ....................................... 16 4.2 CHANGE OF MEASURE ............................................................................................ρ 17 SECTION 5 MERTON’S MODEL .........................................................................................ι 21 5.1 THE FORMULA ........................................................................................................... 21 5.2 PDE APPROACH BY HEDGING ................................................................ε ............... 24 5.3 EFFECT -
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). -
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. -
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. -
Introduction to Lévy Processes
Introduction to L´evyprocesses Graduate lecture 22 January 2004 Matthias Winkel Departmental lecturer (Institute of Actuaries and Aon lecturer in Statistics) 1. Random walks and continuous-time limits 2. Examples 3. Classification and construction of L´evy processes 4. Examples 5. Poisson point processes and simulation 1 1. Random walks and continuous-time limits 4 Definition 1 Let Yk, k ≥ 1, be i.i.d. Then n X 0 Sn = Yk, n ∈ N, k=1 is called a random walk. -4 0 8 16 Random walks have stationary and independent increments Yk = Sk − Sk−1, k ≥ 1. Stationarity means the Yk have identical distribution. Definition 2 A right-continuous process Xt, t ∈ R+, with stationary independent increments is called L´evy process. 2 Page 1 What are Sn, n ≥ 0, and Xt, t ≥ 0? Stochastic processes; mathematical objects, well-defined, with many nice properties that can be studied. If you don’t like this, think of a model for a stock price evolving with time. There are also many other applications. If you worry about negative values, think of log’s of prices. What does Definition 2 mean? Increments , = 1 , are independent and Xtk − Xtk−1 k , . , n , = 1 for all 0 = . Xtk − Xtk−1 ∼ Xtk−tk−1 k , . , n t0 < . < tn Right-continuity refers to the sample paths (realisations). 3 Can we obtain L´evyprocesses from random walks? What happens e.g. if we let the time unit tend to zero, i.e. take a more and more remote look at our random walk? If we focus at a fixed time, 1 say, and speed up the process so as to make n steps per time unit, we know what happens, the answer is given by the Central Limit Theorem: 2 Theorem 1 (Lindeberg-L´evy) If σ = V ar(Y1) < ∞, then Sn − (Sn) √E → Z ∼ N(0, σ2) in distribution, as n → ∞. -
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 -
Nonparametric Bayesian Latent Factor Models for Network Reconstruction
NONPARAMETRIC BAYESIAN LATENT FACTOR MODELS FORNETWORKRECONSTRUCTION Dem Fachbereich Elektrotechnik und Informationstechnik der TECHNISCHE UNIVERSITÄT DARMSTADT zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) vorgelegte Dissertation von SIKUN YANG, M.PHIL. Referent: Prof. Dr. techn. Heinz Köppl Korreferent: Prof. Dr. Kristian Kersting Darmstadt 2019 The work of Sikun Yang was supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement 668858 at the Technische Universsität Darmstadt, Germany. Yang, Sikun : Nonparametric Bayesian Latent Factor Models for Network Reconstruction, Darmstadt, Technische Universität Darmstadt Jahr der Veröffentlichung der Dissertation auf TUprints: 2019 URN: urn:nbn:de:tuda-tuprints-96957 Tag der mündlichen Prüfung: 11.12.2019 Veröffentlicht unter CC BY-SA 4.0 International https://creativecommons.org/licenses/by-sa/4.0/ ABSTRACT This thesis is concerned with the statistical learning of probabilistic models for graph-structured data. It addresses both the theoretical aspects of network modelling–like the learning of appropriate representations for networks–and the practical difficulties in developing the algorithms to perform inference for the proposed models. The first part of the thesis addresses the problem of discrete-time dynamic network modeling. The objective is to learn the common structure and the underlying interaction dynamics among the entities involved in the observed temporal network. Two probabilistic modeling frameworks are developed. First, a Bayesian nonparametric framework is proposed to capture the static latent community structure and the evolving node-community memberships over time. More specifi- cally, the hierarchical gamma process is utilized to capture the underlying intra-community and inter-community interactions. The appropriate number of latent communities can be automatically estimated via the inherent shrinkage mechanism of the hierarchical gamma process prior. -
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.