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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. -
Introduction to Lévy Processes
Introduction to Lévy Processes Huang Lorick [email protected] Document type These are lecture notes. Typos, errors, and imprecisions are expected. Comments are welcome! This version is available at http://perso.math.univ-toulouse.fr/lhuang/enseignements/ Year of publication 2021 Terms of use This work is licensed under a Creative Commons Attribution 4.0 International license: https://creativecommons.org/licenses/by/4.0/ Contents Contents 1 1 Introduction and Examples 2 1.1 Infinitely divisible distributions . 2 1.2 Examples of infinitely divisible distributions . 2 1.3 The Lévy Khintchine formula . 4 1.4 Digression on Relativity . 6 2 Lévy processes 8 2.1 Definition of a Lévy process . 8 2.2 Examples of Lévy processes . 9 2.3 Exploring the Jumps of a Lévy Process . 11 3 Proof of the Levy Khintchine formula 19 3.1 The Lévy-Itô Decomposition . 19 3.2 Consequences of the Lévy-Itô Decomposition . 21 3.3 Exercises . 23 3.4 Discussion . 23 4 Lévy processes as Markov Processes 24 4.1 Properties of the Semi-group . 24 4.2 The Generator . 26 4.3 Recurrence and Transience . 28 4.4 Fractional Derivatives . 29 5 Elements of Stochastic Calculus with Jumps 31 5.1 Example of Use in Applications . 31 5.2 Stochastic Integration . 32 5.3 Construction of the Stochastic Integral . 33 5.4 Quadratic Variation and Itô Formula with jumps . 34 5.5 Stochastic Differential Equation . 35 Bibliography 38 1 Chapter 1 Introduction and Examples In this introductive chapter, we start by defining the notion of infinitely divisible distributions. We then give examples of such distributions and end this chapter by stating the celebrated Lévy-Khintchine formula. -
Superprocesses and Mckean-Vlasov Equations with Creation of Mass
Sup erpro cesses and McKean-Vlasov equations with creation of mass L. Overb eck Department of Statistics, University of California, Berkeley, 367, Evans Hall Berkeley, CA 94720, y U.S.A. Abstract Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of sup erpro cesses. The solutions can b e approxi- mated by a sequence of non-interacting sup erpro cesses or by the mean- eld of multityp e sup erpro cesses with mean- eld interaction. The lat- ter approximation is asso ciated with a propagation of chaos statement for weakly interacting multityp e sup erpro cesses. Running title: Sup erpro cesses and McKean-Vlasov equations . 1 Intro duction Sup erpro cesses are useful in solving nonlinear partial di erential equation of 1+ the typ e f = f , 2 0; 1], cf. [Dy]. Wenowchange the p oint of view and showhowtheyprovide sto chastic solutions of nonlinear partial di erential Supp orted byanFellowship of the Deutsche Forschungsgemeinschaft. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 equation of McKean-Vlasovtyp e, i.e. wewant to nd weak solutions of d d 2 X X @ @ @ + d x; + bx; : 1.1 = a x; t i t t t t t ij t @t @x @x @x i j i i=1 i;j =1 d Aweak solution = 2 C [0;T];MIR satis es s Z 2 t X X @ @ a f = f + f + d f + b f ds: s ij s t 0 i s s @x @x @x 0 i j i Equation 1.1 generalizes McKean-Vlasov equations of twodi erenttyp es. -
A Switching Self-Exciting Jump Diffusion Process for Stock Prices Donatien Hainaut, Franck Moraux
A switching self-exciting jump diffusion process for stock prices Donatien Hainaut, Franck Moraux To cite this version: Donatien Hainaut, Franck Moraux. A switching self-exciting jump diffusion process for stock prices. Annals of Finance, Springer Verlag, 2019, 15 (2), pp.267-306. 10.1007/s10436-018-0340-5. halshs- 01909772 HAL Id: halshs-01909772 https://halshs.archives-ouvertes.fr/halshs-01909772 Submitted on 31 Oct 2018 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. A switching self-exciting jump diusion process for stock prices Donatien Hainaut∗ ISBA, Université Catholique de Louvain, Belgium Franck Morauxy Univ. Rennes 1 and CREM May 25, 2018 Abstract This study proposes a new Markov switching process with clustering eects. In this approach, a hidden Markov chain with a nite number of states modulates the parameters of a self-excited jump process combined to a geometric Brownian motion. Each regime corresponds to a particular economic cycle determining the expected return, the diusion coecient and the long-run frequency of clustered jumps. We study rst the theoretical properties of this process and we propose a sequential Monte-Carlo method to lter the hidden state variables. -
On Two Dimensional Markov Processes with Branching Property^)
ON TWO DIMENSIONAL MARKOV PROCESSES WITH BRANCHING PROPERTY^) BY SHINZO WATANABE Introduction. A continuous state branching Markov process (C.B.P.) was introduced by Jirina [8] and recently Lamperti [10] determined all such processes on the half line. (A quite similar result was obtained independently by the author.) This class of Markov processes contains as a special case the diffusion processes (which we shall call Feller's diffusions) studied by Feller [2]. The main objective of the present paper is to extend Lamperti's result to multi-dimensional case. For simplicity we shall consider the case of 2-dimensions though many arguments can be carried over to the case of higher dimensions(2). In Theorem 2 below we shall characterize all C.B.P.'s in the first quadrant of a plane and construct them. Our construction is in an analytic way, by a similar construction given in Ikeda, Nagasawa and Watanabe [5], through backward equations (or in the terminology of [5] through S-equations) for a simpler case and then in the general case by a limiting procedure. A special attention will be paid to the case of diffusions. We shall show that these diffusions can be obtained as a unique solution of a stochastic equation of Ito (Theorem 3). This fact may be of some interest since the solutions of a stochastic equation with coefficients Holder continuous of exponent 1/2 (which is our case) are not known to be unique in general. Next we shall examine the behavior of sample functions near the boundaries (xj-axis or x2-axis). -
Particle Filtering-Based Maximum Likelihood Estimation for Financial Parameter Estimation
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Spiral - Imperial College Digital Repository Particle Filtering-based Maximum Likelihood Estimation for Financial Parameter Estimation Jinzhe Yang Binghuan Lin Wayne Luk Terence Nahar Imperial College & SWIP Techila Technologies Ltd Imperial College SWIP London & Edinburgh, UK Tampere University of Technology London, UK London, UK [email protected] Tampere, Finland [email protected] [email protected] [email protected] Abstract—This paper presents a novel method for estimating result, which cannot meet runtime requirement in practice. In parameters of financial models with jump diffusions. It is a addition, high performance computing (HPC) technologies are Particle Filter based Maximum Likelihood Estimation process, still new to the finance industry. We provide a PF based MLE which uses particle streams to enable efficient evaluation of con- straints and weights. We also provide a CPU-FPGA collaborative solution to estimate parameters of jump diffusion models, and design for parameter estimation of Stochastic Volatility with we utilise HPC technology to speedup the estimation scheme Correlated and Contemporaneous Jumps model as a case study. so that it would be acceptable for practical use. The result is evaluated by comparing with a CPU and a cloud This paper presents the algorithm development, as well computing platform. We show 14 times speed up for the FPGA as the pipeline design solution in FPGA technology, of PF design compared with the CPU, and similar speedup but better convergence compared with an alternative parallelisation scheme based MLE for parameter estimation of Stochastic Volatility using Techila Middleware on a multi-CPU environment. -
A Mathematical Theory of Network Interference and Its Applications
INVITED PAPER A Mathematical Theory of Network Interference and Its Applications A unifying framework is developed to characterize the aggregate interference in wireless networks, and several applications are presented. By Moe Z. Win, Fellow IEEE, Pedro C. Pinto, Student Member IEEE, and Lawrence A. Shepp ABSTRACT | In this paper, we introduce a mathematical I. INTRODUCTION framework for the characterization of network interference in In a wireless network composed of many spatially wireless systems. We consider a network in which the scattered nodes, communication is constrained by various interferers are scattered according to a spatial Poisson process impairments such as the wireless propagation effects, and are operating asynchronously in a wireless environment network interference, and thermal noise. The effects subject to path loss, shadowing, and multipath fading. We start introduced by propagation in the wireless channel include by determining the statistical distribution of the aggregate the attenuation of radiated signals with distance (path network interference. We then investigate four applications of loss), the blocking of signals caused by large obstacles the proposed model: 1) interference in cognitive radio net- (shadowing), and the reception of multiple copies of the works; 2) interference in wireless packet networks; 3) spectrum same transmitted signal (multipath fading). The network of the aggregate radio-frequency emission of wireless net- interference is due to accumulation of signals radiated by works; and 4) coexistence between ultrawideband and nar- other transmitters, which undesirably affect receiver nodes rowband systems. Our framework accounts for all the essential in the network. The thermal noise is introduced by the physical parameters that affect network interference, such as receiver electronics and is usually modeled as additive the wireless propagation effects, the transmission technology, white Gaussian noise (AWGN). -
Hierarchical Dirichlet Process Hidden Markov Model for Unsupervised Bioacoustic Analysis
Hierarchical Dirichlet Process Hidden Markov Model for Unsupervised Bioacoustic Analysis Marius Bartcus, Faicel Chamroukhi, Herve Glotin Abstract-Hidden Markov Models (HMMs) are one of the the standard HDP-HMM Gibbs sampling has the limitation most popular and successful models in statistics and machine of an inadequate modeling of the temporal persistence of learning for modeling sequential data. However, one main issue states [9]. This problem has been addressed in [9] by relying in HMMs is the one of choosing the number of hidden states. on a sticky extension which allows a more robust learning. The Hierarchical Dirichlet Process (HDP)-HMM is a Bayesian Other solutions for the inference of the hidden Markov non-parametric alternative for standard HMMs that offers a model in this infinite state space models are using the Beam principled way to tackle this challenging problem by relying sampling [lO] rather than Gibbs sampling. on a Hierarchical Dirichlet Process (HDP) prior. We investigate the HDP-HMM in a challenging problem of unsupervised We investigate the BNP formulation for the HMM, that is learning from bioacoustic data by using Markov-Chain Monte the HDP-HMM into a challenging problem of unsupervised Carlo (MCMC) sampling techniques, namely the Gibbs sampler. learning from bioacoustic data. The problem consists of We consider a real problem of fully unsupervised humpback extracting and classifying, in a fully unsupervised way, whale song decomposition. It consists in simultaneously finding the structure of hidden whale song units, and automatically unknown number of whale song units. We use the Gibbs inferring the unknown number of the hidden units from the Mel sampler to infer the HDP-HMM from the bioacoustic data. -
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 → ∞. -
Convergence Rates for the Full Gaussian Rough Paths
Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2014, Vol. 50, No. 1, 154–194 DOI: 10.1214/12-AIHP507 © Association des Publications de l’Institut Henri Poincaré, 2014 www.imstat.org/aihp Convergence rates for the full Gaussian rough paths Peter Friza,b,1 and Sebastian Riedelc,2 aFakultät II, Institut für Mathematik, TU Berlin, MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany bWeierstrass Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany. E-mail: [email protected] cFakultät II, Institut für Mathematik, TU Berlin, MA 7-4, Straße des 17. Juni 136, 10623 Berlin, Germany. E-mail: [email protected] Received 5 August 2011; revised 23 April 2012; accepted 22 June 2012 Abstract. Under the key assumption of finite ρ-variation, ρ ∈[1, 2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), ρ = 1resp.ρ = 1/(2H), we recover and extend the respective results of (Trans. Amer. Math. Soc. 361 (2009) 2689–2718) and (Ann. Inst. Henri Poincasé Probab. Stat. 48 (2012) 518–550). In particular, we establish an a.s. rate − − − k (1/ρ 1/2 ε),anyε>0, for Wong–Zakai and Milstein-type approximations with mesh-size 1/k. When applied to fBM this answers a conjecture in the afore-mentioned references. Résumé. Nous établissons des vitesses fines de convergence presque sûre pour les approximations des chemins rugueux Gaussiens, sous l’hypothèse que la fonction de covariance du processus Gaussien sous-jacent ait une ρ-variation finie, ρ ∈[1, 2). -
Long Term Risk: a Martingale Approach
Long Term Risk: A Martingale Approach Likuan Qin∗ and Vadim Linetskyy Department of Industrial Engineering and Management Sciences McCormick School of Engineering and Applied Sciences Northwestern University Abstract This paper extends the long-term factorization of the pricing kernel due to Alvarez and Jermann (2005) in discrete time ergodic environments and Hansen and Scheinkman (2009) in continuous ergodic Markovian environments to general semimartingale environments, with- out assuming the Markov property. An explicit and easy to verify sufficient condition is given that guarantees convergence in Emery's semimartingale topology of the trading strate- gies that invest in T -maturity zero-coupon bonds to the long bond and convergence in total variation of T -maturity forward measures to the long forward measure. As applications, we explicitly construct long-term factorizations in generally non-Markovian Heath-Jarrow- Morton (1992) models evolving the forward curve in a suitable Hilbert space and in the non-Markovian model of social discount of rates of Brody and Hughston (2013). As a fur- ther application, we extend Hansen and Jagannathan (1991), Alvarez and Jermann (2005) and Bakshi and Chabi-Yo (2012) bounds to general semimartingale environments. When Markovian and ergodicity assumptions are added to our framework, we recover the long- term factorization of Hansen and Scheinkman (2009) and explicitly identify their distorted probability measure with the long forward measure. Finally, we give an economic interpre- tation of the recovery theorem of Ross (2013) in non-Markovian economies as a structural restriction on the pricing kernel leading to the growth optimality of the long bond and iden- tification of the physical measure with the long forward measure. -
A Short History of Stochastic Integration and Mathematical Finance
A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes – Monograph Series Vol. 45 (2004) 75–91 c Institute of Mathematical Statistics, 2004 A short history of stochastic integration and mathematical finance: The early years, 1880–1970 Robert Jarrow1 and Philip Protter∗1 Cornell University Abstract: We present a history of the development of the theory of Stochastic Integration, starting from its roots with Brownian motion, up to the introduc- tion of semimartingales and the independence of the theory from an underlying Markov process framework. We show how the development has influenced and in turn been influenced by the development of Mathematical Finance Theory. The calendar period is from 1880 to 1970. The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copen- hagen, who effectively created a model of Brownian motion while studying time series in 1880 [81].2; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1, 2, 11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 [21](See [64] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately influential.