Stochastic Volatility Libor Modeling and Efficient Algorithms for Optimal Stopping Problems
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
CURRICULUM VITAE Anatoliy SWISHCHUK Department Of
CURRICULUM VITAE Anatoliy SWISHCHUK Department of Mathematics & Statistics, University of Calgary 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 Office: MS552 E-mails: [email protected] Tel: +1 (403) 220-3274 (office) home page: http://www.math.ucalgary.ca/~aswish/ Education: • Doctor of Phys. & Math. Sci. (1992, Doctorate, Institute of Mathematics, National Academy of Sciences of Ukraine (NASU), Kiev, Ukraine) • Ph.D. (1984, Institute of Mathematics, NASU, Kiev, Ukraine) • M.Sc., B.Sc. (1974-1979, Kiev State University, Faculty of Mathematics & Mechanics, Probability Theory & Mathematical Statistics Department, Kiev, Ukraine) Work Experience: • Full Professor, Department of Mathematics and Statistics, University of Calgary, Calgary, Canada (April 2012-present) • Co-Director, Mathematical and Computational Finance Laboratory, Department of Math- ematics and Statistics, University of Calgary, Calgary, Canada (October 2004-present) • Associate Professor, Department of Mathematics and Statistics, University of Calgary, Calgary, Canada (July 2006-March 2012) • Assistant Professor, Department of Mathematics and Statistics, University of Calgary, Calgary, Canada (August 2004-June 2006) • Course Director, Department of Mathematics & Statistics, York University, Toronto, ON, Canada (January 2003-June 2004) • Research Associate, Laboratory for Industrial & Applied Mathematics, Department of Mathematics & Statistics, York University, Toronto, ON, Canada (November 1, 2001-July 2004) • Professor, Probability Theory & Mathematical Statistics -
Introduction to Black's Model for Interest Rate
INTRODUCTION TO BLACK'S MODEL FOR INTEREST RATE DERIVATIVES GRAEME WEST AND LYDIA WEST, FINANCIAL MODELLING AGENCY© Contents 1. Introduction 2 2. European Bond Options2 2.1. Different volatility measures3 3. Caplets and Floorlets3 4. Caps and Floors4 4.1. A call/put on rates is a put/call on a bond4 4.2. Greeks 5 5. Stripping Black caps into caplets7 6. Swaptions 10 6.1. Valuation 11 6.2. Greeks 12 7. Why Black is useless for exotics 13 8. Exercises 13 Date: July 11, 2011. 1 2 GRAEME WEST AND LYDIA WEST, FINANCIAL MODELLING AGENCY© Bibliography 15 1. Introduction We consider the Black Model for futures/forwards which is the market standard for quoting prices (via implied volatilities). Black[1976] considered the problem of writing options on commodity futures and this was the first \natural" extension of the Black-Scholes model. This model also is used to price options on interest rates and interest rate sensitive instruments such as bonds. Since the Black-Scholes analysis assumes constant (or deterministic) interest rates, and so forward interest rates are realised, it is difficult initially to see how this model applies to interest rate dependent derivatives. However, if f is a forward interest rate, it can be shown that it is consistent to assume that • The discounting process can be taken to be the existing yield curve. • The forward rates are stochastic and log-normally distributed. The forward rates will be log-normally distributed in what is called the T -forward measure, where T is the pay date of the option. -
Upper and Lower Snell Envelopes and Robust Partial Hedging
American Options in incomplete Markets: Upper and lower Snell Envelopes and robust partial Hedging DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät II Humboldt-Universität zu Berlin von Herr Dipl. Math. Erick Treviño Aguilar geboren am 24.03.1975 in México Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Christoph Markschies Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II: Prof. Dr. Wolfgang Coy Gutachter: 1. Prof. Dr. Hans Föllmer 2. Prof. Dr. Peter Imkeller 3. Prof. Dr. Frank Riedel eingereicht am: 14. Dezember 2007 Tag der mündlichen Prüfung: 9. Juni 2008 Abstract This thesis studies American options in an incomplete financial market and in continuous time. It is composed of two parts. In the first part we study a stochastic optimization problem in which a robust convex loss functional is minimized in a space of stochastic integrals. This problem arises when the seller of an American option aims to control the shortfall risk by using a partial hedge. We quantify the shortfall risk through a robust loss functional motivated by an extension of classical expected util- ity theory due to Gilboa and Schmeidler. In a general semimartingale model we prove the existence of an optimal strategy. Under additional compactness assumptions we show how the robust problem can be reduced to a non-robust optimization problem with respect to a worst-case probability measure. In the second part, we study the notions of the upper and the lower Snell envelope associated to an American option. We construct the envelopes for stable families of equivalent probability measures, the family of local martin- gale measures being an important special case. -
Pricing Bermudan Swaptions on the LIBOR Market Model Using the Stochastic Grid Bundling Method
Pricing Bermudan Swaptions on the LIBOR Market Model using the Stochastic Grid Bundling Method Stef Maree,∗ Jacques du Toity Abstract We examine using the Stochastic Grid Bundling Method (SGBM) to price a Bermu- dan swaption driven by a one-factor LIBOR Market Model (LMM). Using a well- known approximation formula from the finance literature, we implement SGBM with one basis function and show that it is around six times faster than the equivalent Longstaff–Schwartz method. The two methods agree in price to one basis point, and the SGBM path estimator gives better (higher) prices than the Longstaff–Schwartz prices. A closer examination shows that inaccuracies in the approximation formula introduce a small bias into the SGBM direct estimator. 1 Introduction The LIBOR Market Model (LMM) is an interest rate market model. It owes much of its popularity to the fact that it is consistent with Black’s pricing formulas for simple interest rate derivatives such as caps and swaptions, see, for example, [3]. Under LMM all forward rates are modelled as individual processes, which typically results in a high dimensional model. Consequently when pricing more complex instruments under LMM, Monte Carlo simulation is usually used. When pricing products with early-exercise features, such as Bermudan swaptions, a method is required to determine the early-exercise policy. The most popular approach to this is the Longstaff–Schwartz Method (LSM) [8]. When time is discrete (as is usually the case in numerical schemes) Bellman’s backward induction principle says that the price of the option at any exercise time is the maximum of the spot payoff and the so-called contin- uation value. -
Approximations to the Lévy LIBOR Model By
Approximations to the Lévy LIBOR Model by Hassana Al-Hassan ([email protected]) Supervised by: Dr. Sure Mataramvura Co-Supervised by: Emeritus Professor Ronnie Becker Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town, South Africa A thesis submitted for the degree of Master of Science University of Cape Town May 27, 2014 The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or non- commercial research purposes only. Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University of Cape Town Plagiarism Declaration “I, Hassana AL-Hassan, know the meaning of plagiarism and declare that all of the work in the thesis, save for that which is properly acknowledged, is my own”. SIGNED Hassana Al-Hassan, May 27, 2014 i Declaration of Authorship I, Hassana AL-Hassan, declare that this thesis titled, “The LIBOR Market Model Versus the Levy LIBOR Market Model” and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other quali- fication at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. -
Interest Rate Models: Paradigm Shifts in Recent Years
Interest Rate Models: Paradigm shifts in recent years Damiano Brigo Q-SCI, Managing Director and Global Head DerivativeFitch, 101 Finsbury Pavement, London Columbia University Seminar, New York, November 5, 2007 This presentation is based on the book "Interest Rate Models: Theory and Practice - with Smile, Inflation and Credit" by D. Brigo and F. Mercurio, Springer-Verlag, 2001 (2nd ed. 2006) http://www.damianobrigo.it/book.html Damiano Brigo, Q-SCI, DerivativeFitch, London Columbia University Seminar, November 5, 2007 Overview ² No arbitrage and derivatives pricing. ² Modeling suggested by no-arbitrage discounting. 1977: Endogenous short-rate term structure models ² Reproducing the initial market interest-rate curve exactly. 1990: Exogenous short rate models ² A general framework for no-arbitrage rates dynamics. 1990: HJM - modeling instantaneous forward rates ² Moving closer to the market and consistency with market formulas 1997: Fwd market-rates models calibration and diagnostics power ² 2002: Volatility smile extensions of Forward market-rates models Interest rate models: Paradigms shifts in recent years 1 Damiano Brigo, Q-SCI, DerivativeFitch, London Columbia University Seminar, November 5, 2007 No arbitrage and Risk neutral valuation Recall shortly the risk-neutral valuation paradigm of Harrison and Pliska's (1983), characterizing no-arbitrage theory: A future stochastic payo®, built on an underlying fundamental ¯nancial asset, paid at a future time T and satisfying some technical conditions, has as unique price at current time t -
Essays in Quantitative Finance Karlsson, Patrik
Essays in Quantitative Finance Karlsson, Patrik 2016 Document Version: Publisher's PDF, also known as Version of record Link to publication Citation for published version (APA): Karlsson, P. (2016). Essays in Quantitative Finance. Total number of authors: 1 General rights Unless other specific re-use rights are stated the following general rights apply: 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 any publication from the public portal for the purpose of private study or research. • 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 Read more about Creative commons licenses: https://creativecommons.org/licenses/ 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. LUND UNIVERSITY PO Box 117 221 00 Lund +46 46-222 00 00 Patrik Karlsson Essays in Quantitative Finance Karlsson Essays in Quantitative Patrik Essays in Quantitative Finance Patrik Karlsson 9 789177 Lund Lund University Economic 530602 Department of Economics ISBN 978-91-7753-060-2 Studies ISSN 0460-0029 199 Number 199 Essays in Quantitative Finance Patrik Karlsson DOCTORAL DISSERTATION by due permission of the School of Economics and Management, Lund University, Sweden. -
Activity Report 2014
Activity Report 2014 s a world-leading financial center building on a A rich history, Switzerland’s financial sector has the natural ambition of housing a world-leading research and training center in banking and finance. The Swiss Finance Institute is the product of this ambition. Established at the initiative of the Swiss Bankers Association, it is a private foundation created in 2006 with the support of the Swiss banking and finance community, the Swiss stock exchange, the Swiss Confederation, the Swiss National Science Foundation, and several Swiss universities with the aim of advancing research activities in finance and executive education in the banking and finance sector. The Swiss Finance Institute encompasses three pre-existing foundations – the International Center for Financial Asset Management and Engineering (FAME), the Swiss Banking School, and the Stiftung Banking und Finance an der Universität Zürich. This merger has led to the creation of one of the major European providers of research, doctoral training, and advanced executive education in banking and finance. This report gives an overview of the Swiss Finance Institute’s activities from January to December 2014. 2 Table of Contents A Word from the Board 4 Swiss Finance Institute Faculty 5 Research Highlights 6 Research Projects 8 PhD Program in Finance 12 PhD Graduate Placements 13 Education 14 Knowledge Center 19 Knowledge Transfer 20 Governing and Advisory Bodies 23 2014 Facts & Figures 25 Research Paper Series 2014 28 SFI Professors 34 SFI Adjunct Professors 66 Overview of Courses Offered in 2014 at the Swiss Finance Institute 68 Knowledge Transfer Events Provided by the Swiss Finance Institute during 2014 69 3 A Word from the Board The Year of Cohesion 2014 saw SFI strengthen collaboration amongst the SFI activity areas, SFI members, and external partners. -
A Random Field LIBOR Market Model
L’Institut bénéficie du soutien financier de l’Autorité des marchés financiers ainsi que du ministère des Finances du Québec Document de recherche DR 14-02 A Random Field LIBOR Market Model Novembre 2014 Ce document de recherche a été rédigée par : Tao L. Wu et Shengqiang Xu L'Institut canadien des dérivés n'assume aucune responsabilité liée aux propos tenus et aux opinions exprimées dans ses publications, qui n'engagent que leurs auteurs. De plus, l'Institut ne peut, en aucun cas être tenu responsable des conséquences dommageables ou financières de toute exploitation de l'information diffusée dans ses publications. A Random Field LIBOR Market Model Tao L. Wu∗and Shengqiang Xu January 1, 2014 A random field LIBOR market model (RFLMM) is proposed by extend- ing the LIBOR market model, with interest rate uncertainties modeled via a random field. First, closed-form formulas for pricing caplet and swaption are derived. Then the random field LIBOR market model is integrated with the lognormal-mixture model to capture the implied volatility skew/smile. Finally, the model is calibrated to cap volatility surface and swaption volatil- ities. Numerical results show that the random field LIBOR market model can potentially outperform the LIBOR market model in capturing caplet volatility smile and the pricing of swaptions, in addition to possessing other advantages documented in the previous literature (no need of frequent re- calibration or to specify the number of factors in advance). 1 Introduction In this paper we extend the LIBOR market model (LMM) by describing forward rate uncertainties as a random field. -
Interest Rate Volatility IV
The LMM methodology Dynamics of the SABR-LMM model Covariance structure of SABR-LMM Interest Rate Volatility IV. The SABR-LMM model Andrew Lesniewski Baruch College and Posnania Inc First Baruch Volatility Workshop New York June 16 - 18, 2015 A. Lesniewski Interest Rate Volatility The LMM methodology Dynamics of the SABR-LMM model Covariance structure of SABR-LMM Outline 1 The LMM methodology 2 Dynamics of the SABR-LMM model 3 Covariance structure of SABR-LMM A. Lesniewski Interest Rate Volatility The LMM methodology Dynamics of the SABR-LMM model Covariance structure of SABR-LMM The LMM methodology The main shortcoming of short rate models is that they do not allow for close calibration to the entire volatility cube. This is not a huge concern on a trading desk, where locally calibrated term structure models allow for accurate pricing and executing trades. It is, however, a concern for managers of large portfolios of fixed income securities (such as mortgage backed securities) which have exposures to various segments of the curve and various areas of the vol cube. It is also relevant in enterprise level risk management in large financial institutions where consistent risk aggregation across businesses and asset classes is important. A methodology that satisfies these requirements is the LIBOR market model (LMM) methodology, and in particular its stochastic volatility extensions. That comes at a price: LMM is less tractable than some of the popular short rate models. It also tends to require more resources than those models. We will discuss a natural extension of stochastic volatility LMM, namely the SABR-LMM model. -
Optimal Stopping, Smooth Pasting and the Dual Problem
Introduction Preliminaries Main Theorems Questions Applications Optimal Stopping, Smooth Pasting and the Dual Problem Saul Jacka and Dominic Norgilas, University of Warwick Imperial 21 February 2018 Saul Jacka and Dominic Norgilas, University of Warwick The Compensator of the Snell Envelope Introduction Preliminaries Main Theorems Questions Applications The general optimal stopping problem: Given a ltered probability space (Ω; (Ft ); F; P) and an adapted gains process G, nd def St = ess sup E[Gτ jFt ] optional τ≥t Recall, under very general conditions I S is the minimal supermartingale dominating G def I τt = inffs ≥ t : Ss = Gs g is optimal I for any t, S is a martingale on [t; τt ] I when Gt = g(Xt ) (1) for some (continuous-time) Markov Process X , St can be written as a function, v(Xt ). Saul Jacka and Dominic Norgilas, University of Warwick The Compensator of the Snell Envelope Introduction Preliminaries Main Theorems Questions Applications Remark 1.1 Condition Gt = g(Xt ) is less restrictive than might appear. With θ being the usual shift operator, can expand statespace of X by appending adapted functionals F with the property that Ft+s = f (Fs ; (θs ◦ Xu; 0 ≤ u ≤ t)): (2) The resulting process Y def= (X ; F ) is still Markovian. If X is strong Markov and F is right-cts then Y is strong Markov. e.g if X is a BM, Z t Z s 0 Yt = Xt ; Lt ; sup Xs ; exp − α(Xu)du g(Xs ) ds 0≤s≤t 0 0 is a Feller process on the ltration of X .