簡(jiǎn)明Levy過(guò)程課件

1、1Levy Processes-From Probability to FinanceAnatoliy Swishchuk,Mathematical and Computational Finance Laboratory, Department of Mathematics and Statistics, U of C“Lunch at the Lab” TalkFebruary 3, 20052Outline Introduction: Probability and Stochastic Processes The Structure of Levy Processes Applicat

2、ions to Finance The talk is based on the paper by David Applebaum (University of Sheffield, UK), Notices of the AMS, Vol. 51, No 11.3Introduction: Probability Theory of Probability: aims to model and to measure the Chance The tools: Kolmogorovs theory of probability axioms (1930s) Probability can be

3、 rigorously founded on measure theory4Introduction: Stochastic Processes Theory of Stochastic Processes: aims to model the interaction of Chance and Time Stochastic Processes: a family of random variables (X(t), t=0) defined on a probability space (Omega, F, P) and taking values in a measurable spac

4、e (E,G) X(t) is a (E,G) measurable mapping from Omega to E: a random observation made on E at time t5Importance of Stochastic Processes Not only mathematically rich objects Applications: physics, engineering, ecology, economics, finance, etc. Examples: random walks, Markov processes, semimartingales

5、, measure-valued diffusions, Levy Processes, etc.6Importance of Levy Processes There are many important examples: Brownian motion, Poisson Process, stable processes, subordinators, etc. Generalization of random walks to continuous time The simplest classes of jump-diffusion processes A natural model

6、s of noise to build stochastic integrals and to drive SDE Their structure is mathematically robust Their structure contains many features that generalize naturally to much wider classes of processes, such as semimartingales, Feller-Markov processes, etc.7Main Original Contributors to the Theory of L

7、evy Processes: 1930s-1940s Paul Levy (France) Alexander Khintchine (Russia) Kiyosi Ito (Japan)8Paul Levy (1886-1971)9Main Original Papers Levy P. Sur les integrales dont les elements sont des variables aleatoires independentes, Ann. R. Scuola Norm. Super. Pisa, Sei. Fis. e Mat., Ser. 2 (1934), v. II

8、I, 337-366; Ser. 4 (1935), 217-218 Khintchine A. A new derivation of one formula by Levy P., Bull. Moscow State Univ., 1937, v. I, No 1, 1-5 Ito K. On stochastic processes, Japan J. Math. 18 (1942), 261-30110Definition of Levy Processes X(t) X(t) has independent and stationary increments Each X(0)=0

9、 w.p.1 X(t) is stochastically continuous, i. e, for all a0 and for all s=0, P (|X(t)-X(s)|a)-0 when t-s11The Structure of Levy Processes: The Levy-Khintchine Formula If X(t) is a Levy process, then its characteristic function equals to where12Examples of Levy Processes Brownian motion: characteristi

10、c (0,a,0) Brownian motion with drift (Gaussian processes): characteristic (b,a,0) Poisson process: characteristic (0,0,lambdaxdelta1), lambda-intensity, delta1-Dirac mass concentrated at 1 The compound Poisson process Interlacing processes=Gaussian process +compound Poisson process Stable processes

11、Subordinators Relativistic processes13Simulation of Standard Brownian Motion14Simulation of the Poisson Process15Stable Levy Processes Stable probability distributions arise as the possible weak limit of normalized sums of i.i.d. r.v. in the central limit theorem Example: Cauchy Process with density

12、 (index of stability is 1)16Simulation of the Cauchy Process17Subordinators A subordinator T(t) is a one-dimensional Levy process that is non-decreasing Important application: time change of Levy process X(t) : Y(t):=X(T(t) is also a new Levy process18Simulation of the Gamma Subordinator19The Levy-I

13、to Decomposition: Structure of the Sample Paths of Levy Processes20Application to Finance. I. Replace Brownian motion in BSM model with a more general Levy process (P. Carr, H. Geman, D. Madan and M. Yor) Idea: 1) small jumps term describes the day-to-day jitter that causes minor fluctuations in sto

14、ck prices; 2) big jumps term describes large stock price movements caused by major market upsets arising from, e.g., earthquakes, etc.21Main Problems with Levy Processes in Finance. Market is incomplete, i.e., there may be more than one possible pricing formula One of the methods to overcome it: ent

15、ropy minimization Example: hyperbolic Levy process (E. Eberlain) (with no Brownian motion part); a pricing formula have been developed that has minimum entropy22Hyperbolic Levy Process: Characteristic Function23Bessel Function of the Third Kind(!) The Bessel function of the third kind or Hankel func

16、tion Hn(x) is a (complex) combination of the two solutions of Bessel DE: the real part is the Bessel function of the first kind, the complex part the Bessel function of the second kind (very complicated!) 24Bessel Differential Equation25Application of Levy Processes in Finance. II. BSM formula conta

17、ins the constant of volatility One of the methods to improve it: stochastic volatility models (SDE for volatility) Example: stochastic volatility is an Ornstein-Uhlenbeck process driven by a subordinator T(t) (O. Barndorff-Nielsen and N. Shephard)26Stochastic Volatility Model Using Levy Process27References on Levy Processes (Books) D. Applebaum, Levy Processes and Stochastic Calculus, Cambridge University Press, 2004 O.E. Barndorff-Nielsen, T. Mikosch and S. Resnick (Eds.), Levy Processes: Theory and Applications, Birk