Lecture #31, 32: The Ornstein-Uhlenbeck Process as a Model of Volatility
|
|
- Georgiana Madlyn Nelson
- 6 years ago
- Views:
Transcription
1 Saisics 441 (Fall 214) November 19, 21, 214 Prof Michael Kozdron Lecure #31, 32: The Ornsein-Uhlenbeck Process as a Model of Volailiy The Ornsein-Uhlenbeck process is a di usion process ha was inroduced as a model of he velociy of a paricle undergoing Brownian moion We know from Newonian physics ha he velociy of a (classical) paricle in moion is given by he ime derivaive of is posiion However, if he posiion of a paricle is described by Brownian moion, hen he ime derivaive does no exis The Ornsein-Uhlenbeck process is an aemp o overcome his di culy by modelling he velociy direcly Furhermore, jus as Brownian moion is he scaling limi of simple random walk, he Ornsein-Uhlenbeck process is he scaling limi of he Ehrenfes urn model which describes he di usion of paricles hrough a permeable membrane In recen years, however, he Ornsein-Uhlenbeck process has appeared in finance as a model of he volailiy of he underlying asse price process Suppose ha he price of a sock {S, } is modelled by geomeric Brownian moion wih volailiy and drif µ so ha S saisfies he SDE ds = S db + µs d However, marke daa indicaes ha implied volailiies for di eren srike prices and expiry daes of opions are no consan Insead, hey appear o be smile shaped (or frown shaped) Perhaps he mos naural approach is o allow for he volailiy funcion of ime so ha S saisfies he SDE ds = ()S db + µs d () o be a deerminisic This was already suggesed by Meron in 1973 Alhough i does explain he di eren implied volailiy levels for di eren expiry daes, i does no explain he smile shape for di eren srike prices Insead, Hull and Whie in 1987 proposed o use a sochasic volailiy model where he underlying sock price {S, } saisfies he SDE and he variance process {v, ds = p v S db + µs d } is given by geomeric Brownian moion dv = c 1 v db + c 2 v d wih c 1 and c 2 known consans The problem wih his model is ha geomeric Brownian moion ends o increase exponenially which is an undesirable propery for volailiy Marke daa also indicaes ha volailiy exhibis mean-revering behaviour This lead Sein and Sein in 1991 o inroduce he mean-revering Ornsein-Uhlenbeck process saisfying dv = db + a(b v )d where a, b, and are known consans This process, however, allows negaive values of v 2 1
2 In 1993 Heson overcame his di culy by considering a more complex sochasic volailiy model Before invesigaing he Heson model, however, we will consider he Ornsein- Uhlenbeck process separaely and prove ha negaive volailiies are allowed hereby verifying ha he Sein and Sein sock price model is flawed We say ha he process {X, } is an Ornsein-Uhlenbeck process if X saisfies he Ornsein-Uhlenbeck sochasic di erenial equaion given by dx = db + ax d (21) where and a are consans and {B, } is a sandard Brownian moion Remark Someimes (21) is called he Langevin equaion, especially in physics conexs Remark The Ornsein-Uhlenbeck SDE is very similar o he SDE for geomeric Brownian moion; he only di erence is he absence of X in he db erm of (21) However, his sligh change makes (21) more challenging o solve The rick for solving (21) is o muliply boh sides by he inegraing facor e a and o compare wih d(e a X ) The chain rule ells us ha d(e a X )=e a dx + X d(e a )=e a dx ae a X d (22) and muliplying (21) by e a gives so ha subsiuing (23) ino (22) gives e a dx = e a db + ae a X d (23) d(e a X )= e a db + ae a X d ae a X d = e a db Since d(e a X )= e a db,wecannowinegraeoconcludeha and so e a X X = X = e a X + e as db s e a( s) db s (24) Observe ha he inegral in (24) is a Wiener inegral Definiion 81 ells us ha e a( s) db s N, e ( s) ds = N, e 1 In paricular, choosing X = x o be consan implies ha X = e a x + e a( s) db s N xe a, 2 (e 1) 2 2
3 Acually, we can generalize his slighly If we choose X N(x, 2 )independenlyof {B, }, henexercise312ellsusha X = e a X + e a( s) db s N xe a, 2 e 2 (e 1) + = N xe a, e 2 Exercise 21 Suppose ha {X, } is an Ornsein-Uhlenbeck process given by (24) wih X = Ifs<,compueCov(X s,x ) We say ha he process {X, saisfies he SDE } is a mean-revering Ornsein-Uhlenbeck process if X dx = db +(b X )d (25) where and b are consans and {B, } is a sandard Brownian moion The rick for solving he mean-revering Ornsein-Uhlenbeck process is similar Tha is, we muliply by e and compare wih d(e (b X )) The chain rule ells us ha and muliplying (25) by e gives so ha subsiuing (27) ino (26) gives d(e (b X )) = e dx + e (b X )d (26) e dx = e db + e (b X )d (27) d(e (b X )) = e db e (b X )d + e (b X )d = e db Since d(e (b X )) = e db,wecannowinegraeoconcludeha and so e (b X ) (b X )= X =(1 e )b + e X + e s db s Exercise 22 Suppose ha X N(x, 2 )isindependenof{b, disribuion of X given by (28) e s db s (28) } Deerminehe Exercise 23 Use an appropriae inegraing facor o solve he mean-revering Ornsein- Uhlenbeck SDE considered by Sein and Sein, namely dx = db + a(b X )d Assuming ha X = x is consan, deermine he disribuion of X and conclude ha P{X < } > forevery> Hin: X has a normal disribuion This hen explains our earlier claim ha he Sein and Sein model is flawed 2 3
4 As previous noed, Heson inroduced a sochasic volailiy model in 1993 ha overcame his di culy Assume ha he asse price process {S, } saisfies he SDE ds = p v S db (1) + µs d where he variance process {v, } saisfies dv = p v db (2) + a(b v )d (29) and he wo driving Brownian moions {B (1), } and {B (2), } are correlaed wih rae, ie, dhb (1),B (2) i = d The p v erm in (29) is needed o guaranee posiive volailiy when he process ouches zero he sochasic par becomes zero and he non-sochasic par will push i up The parameer a measures he speed of he mean-reversion, b is he average level of volailiy, and is he volailiy of volailiy Marke daa suggess ha he correlaion rae is ypically negaive The negaive dependence beween reurns and volailiy is someimes called he leverage e ec Heson s model involves a sysem of sochasic di erenial equaions The key ool for analyzing such a sysem is he mulidimensional version of Iô s formula Theorem 24 (Version V) Suppose ha {X, } and {Y, } are di usions defined by he sochasic di erenial equaions and dx = a 1 (, X,Y )db (1) dy = a 2 (, X,Y )db (2) + b 1 (, X,Y )d + b 2 (, X,Y )d, respecively, where {B (1), } and {B (2), } are each sandard one-dimensional Brownian moions If f 2 C 1 ([, 1)) C 2 (R 2 ), hen df(, X,Y )= f(, X,Y )d + f 1 (, X,Y )dx f 11(, X,Y )dhxi where he parial derivaives are defined as + f 2 (, X,Y )dy f 22(, X,Y )dhy i + f 12 (, X,Y )dhx, Y i f(, x, y) f(, x, f 1(, x, f(, x, y), f 11(, x, y) f(, x, f 2 (, x, f(, x, y), f 22(, x, f(, x, y), f 12(, x, y) f(, x, y), and dhx, Y i is compued according o he rule dhx, Y i =(dx )(dy )=a 1 (, X,Y )a 2 (, X,Y )dhb (1),B (2) i 2 4
5 Remark In a ypical problem involving he mulidimensional version of Iô s formula, he quadraic covariaion process hb (1),B (2) i will be specified However, wo paricular examples are worh menioning If B (1) = B (2),hendhB (1),B (2) i =d, whereasifb (1) and B (2) are independen, hen dhb (1),B (2) i = Exercise 25 Suppose ha f(, x, y) = xy Using Version V of Iô s formula (Theorem 24), verify ha he produc rule for di usions is given by d(x Y )=X dy + Y dx +dhx, Y i Thus, our goal in he nex few lecures is o price a European call opion assuming ha he underlying sock price follows Heson s model of geomeric Brownian moion wih a sochasic volailiy, namely 8 >< ds = p v S db (1) + µs d, dv = p v db (2) + a(b v )d, >: dhb (1),B (2) i = d 2 5
Chapter 14 Wiener Processes and Itô s Lemma. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull
Chaper 14 Wiener Processes and Iô s Lemma Copyrigh John C. Hull 014 1 Sochasic Processes! Describes he way in which a variable such as a sock price, exchange rae or ineres rae changes hrough ime! Incorporaes
More informationStochastic Modelling in Finance - Solutions to sheet 8
Sochasic Modelling in Finance - Soluions o shee 8 8.1 The price of a defaulable asse can be modeled as ds S = µ d + σ dw dn where µ, σ are consans, (W ) is a sandard Brownian moion and (N ) is a one jump
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationUnit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationHomework 10 (Stats 620, Winter 2017) Due Tuesday April 18, in class Questions are derived from problems in Stochastic Processes by S. Ross.
Homework (Sas 6, Winer 7 Due Tuesday April 8, in class Quesions are derived from problems in Sochasic Processes by S. Ross.. A sochasic process {X(, } is said o be saionary if X(,..., X( n has he same
More informationElements of Stochastic Processes Lecture II Hamid R. Rabiee
Sochasic Processes Elemens of Sochasic Processes Lecure II Hamid R. Rabiee Overview Reading Assignmen Chaper 9 of exbook Furher Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A Firs Course
More informationf(s)dw Solution 1. Approximate f by piece-wise constant left-continuous non-random functions f n such that (f(s) f n (s)) 2 ds 0.
Advanced Financial Models Example shee 3 - Michaelmas 217 Michael Tehranchi Problem 1. Le f : [, R be a coninuous (non-random funcion and W a Brownian moion, and le σ 2 = f(s 2 ds and assume σ 2
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationI. Return Calculations (20 pts, 4 points each)
Universiy of Washingon Spring 015 Deparmen of Economics Eric Zivo Econ 44 Miderm Exam Soluions This is a closed book and closed noe exam. However, you are allowed one page of noes (8.5 by 11 or A4 double-sided)
More informationLoss of martingality in asset price models with lognormal stochastic volatility
Loss of maringaliy in asse price models wih lognormal sochasic volailiy BJourdain July 7, 4 Absrac In his noe, we prove ha in asse price models wih lognormal sochasic volailiy, when he correlaion coefficien
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationDiebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationNature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.
Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationGEM4 Summer School OpenCourseWare
GEM4 Summer School OpenCourseWare hp://gem4.educommons.ne/ hp://www.gem4.org/ Lecure: Thermal Forces and Brownian Moion by Ju Li. Given Augus 11, 2006 during he GEM4 session a MIT in Cambridge, MA. Please
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More information2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/2007 2-17 2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationψ(t) = V x (0)V x (t)
.93 Home Work Se No. (Professor Sow-Hsin Chen Spring Term 5. Due March 7, 5. This problem concerns calculaions of analyical expressions for he self-inermediae scaering funcion (ISF of he es paricle in
More informationt + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that
ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so
More information7 The Itô/Stratonovich dilemma
7 The Iô/Sraonovich dilemma The dilemma: wha does he idealizaion of dela-funcion-correlaed noise mean? ẋ = f(x) + g(x)η() η()η( ) = κδ( ). (1) Previously, we argued by a limiing procedure: aking noise
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationCHAPTER 2: Mathematics for Microeconomics
CHAPTER : Mahemaics for Microeconomics The problems in his chaper are primarily mahemaical. They are inended o give sudens some pracice wih he conceps inroduced in Chaper, bu he problems in hemselves offer
More informationwhere the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
More informationOnline Appendix to Solution Methods for Models with Rare Disasters
Online Appendix o Soluion Mehods for Models wih Rare Disasers Jesús Fernández-Villaverde and Oren Levinal In his Online Appendix, we presen he Euler condiions of he model, we develop he pricing Calvo block,
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3
More informationACE 562 Fall Lecture 8: The Simple Linear Regression Model: R 2, Reporting the Results and Prediction. by Professor Scott H.
ACE 56 Fall 5 Lecure 8: The Simple Linear Regression Model: R, Reporing he Resuls and Predicion by Professor Sco H. Irwin Required Readings: Griffihs, Hill and Judge. "Explaining Variaion in he Dependen
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationOption pricing and implied volatilities in a 2-hypergeometric stochastic volatility model
Opion pricing and implied volailiies in a 2-hypergeomeric sochasic volailiy model Nicolas Privaul Qihao She Division of Mahemaical Sciences School of Physical and Mahemaical Sciences Nanyang Technological
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More information13.3 Term structure models
13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationModule 3: The Damped Oscillator-II Lecture 3: The Damped Oscillator-II
Module 3: The Damped Oscillaor-II Lecure 3: The Damped Oscillaor-II 3. Over-damped Oscillaions. This refers o he siuaion where β > ω (3.) The wo roos are and α = β + α 2 = β β 2 ω 2 = (3.2) β 2 ω 2 = 2
More informationTHE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant).
THE WAVE EQUATION 43. (S) Le u(x, ) be a soluion of he wave equaion u u xx = 0. Show ha Q43(a) (c) is a. Any ranslaion v(x, ) = u(x + x 0, + 0 ) of u(x, ) is also a soluion (where x 0, 0 are consans).
More informationChapter 5. Heterocedastic Models. Introduction to time series (2008) 1
Chaper 5 Heerocedasic Models Inroducion o ime series (2008) 1 Chaper 5. Conens. 5.1. The ARCH model. 5.2. The GARCH model. 5.3. The exponenial GARCH model. 5.4. The CHARMA model. 5.5. Random coefficien
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More informationHOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.
HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =
More informationDEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND
DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Asymmery and Leverage in Condiional Volailiy Models Michael McAleer WORKING PAPER
More informationExcel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand
Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338
More informationSolutionbank Edexcel AS and A Level Modular Mathematics
Page of 4 Soluionbank Edexcel AS and A Level Modular Mahemaics Exercise A, Quesion Quesion: Skech he graphs of (a) y = e x + (b) y = 4e x (c) y = e x 3 (d) y = 4 e x (e) y = 6 + 0e x (f) y = 00e x + 0
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationIMA Preprint Series # 2056
MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING By Gerard Awanou IMA Preprin Series # 056 ( July 005 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 400 Lind Hall 07 Church Sree S.E.
More informationThe Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite
American Journal of Operaions Research, 08, 8, 8-9 hp://wwwscirporg/journal/ajor ISSN Online: 60-8849 ISSN Prin: 60-8830 The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More informationAnnouncements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple
More informationADVANCED MATHEMATICS FOR ECONOMICS /2013 Sheet 3: Di erential equations
ADVANCED MATHEMATICS FOR ECONOMICS - /3 Shee 3: Di erenial equaions Check ha x() =± p ln(c( + )), where C is a posiive consan, is soluion of he ODE x () = Solve he following di erenial equaions: (a) x
More informationAn Introduction to Backward Stochastic Differential Equations (BSDEs) PIMS Summer School 2016 in Mathematical Finance.
1 An Inroducion o Backward Sochasic Differenial Equaions (BSDEs) PIMS Summer School 2016 in Mahemaical Finance June 25, 2016 Chrisoph Frei cfrei@ualbera.ca This inroducion is based on Touzi [14], Bouchard
More informationFinancial Econometrics Introduction to Realized Variance
Financial Economerics Inroducion o Realized Variance Eric Zivo May 16, 2011 Ouline Inroducion Realized Variance Defined Quadraic Variaion and Realized Variance Asympoic Disribuion Theory for Realized Variance
More informationA general continuous auction system in presence of insiders
A general coninuous aucion sysem in presence of insiders José M. Corcuera (based on join work wih G. DiNunno, G. Farkas and B. Oksendal) Faculy of Mahemaics Universiy of Barcelona BCAM, Basque Cener for
More informationLecture 4 Kinetics of a particle Part 3: Impulse and Momentum
MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an
More informationBasic Circuit Elements Professor J R Lucas November 2001
Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive and passive elemens. Some Definiions/explanaions
More informationLaplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More informationFrom Particles to Rigid Bodies
Rigid Body Dynamics From Paricles o Rigid Bodies Paricles No roaions Linear velociy v only Rigid bodies Body roaions Linear velociy v Angular velociy ω Rigid Bodies Rigid bodies have boh a posiion and
More information4.1 - Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationMEI STRUCTURED MATHEMATICS 4758
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Cerificae of Educaion Advanced General Cerificae of Educaion MEI STRUCTURED MATHEMATICS 4758 Differenial Equaions Thursday 5 JUNE 006 Afernoon
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationSolutions Problem Set 3 Macro II (14.452)
Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.
More informationTHE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI
THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More informationMath 2214 Solution Test 1B Fall 2017
Mah 14 Soluion Tes 1B Fall 017 Problem 1: A ank has a capaci for 500 gallons and conains 0 gallons of waer wih lbs of sal iniiall. A soluion conaining of 8 lbsgal of sal is pumped ino he ank a 10 galsmin.
More informationACE 562 Fall Lecture 4: Simple Linear Regression Model: Specification and Estimation. by Professor Scott H. Irwin
ACE 56 Fall 005 Lecure 4: Simple Linear Regression Model: Specificaion and Esimaion by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Simple Regression: Economic and Saisical Model
More informationSection 7.4 Modeling Changing Amplitude and Midline
488 Chaper 7 Secion 7.4 Modeling Changing Ampliude and Midline While sinusoidal funcions can model a variey of behaviors, i is ofen necessary o combine sinusoidal funcions wih linear and exponenial curves
More informationA proof of Ito's formula using a di Title formula. Author(s) Fujita, Takahiko; Kawanishi, Yasuhi. Studia scientiarum mathematicarum H Citation
A proof of Io's formula using a di Tile formula Auhor(s) Fujia, Takahiko; Kawanishi, Yasuhi Sudia scieniarum mahemaicarum H Ciaion 15-134 Issue 8-3 Dae Type Journal Aricle Tex Version auhor URL hp://hdl.handle.ne/186/15878
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More information1 st order ODE Initial Condition
Mah-33 Chapers 1-1 s Order ODE Sepember 1, 17 1 1 s order ODE Iniial Condiion f, sandard form LINEAR NON-LINEAR,, p g differenial form M x dx N x d differenial form is equivalen o a pair of differenial
More informationCHEMICAL KINETICS: 1. Rate Order Rate law Rate constant Half-life Temperature Dependence
CHEMICL KINETICS: Rae Order Rae law Rae consan Half-life Temperaure Dependence Chemical Reacions Kineics Chemical ineics is he sudy of ime dependence of he change in he concenraion of reacans and producs.
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationEmpirical Process Theory
Empirical Process heory 4.384 ime Series Analysis, Fall 27 Reciaion by Paul Schrimpf Supplemenary o lecures given by Anna Mikusheva Ocober 7, 28 Reciaion 7 Empirical Process heory Le x be a real-valued
More informationk B 2 Radiofrequency pulses and hardware
1 Exra MR Problems DC Medical Imaging course April, 214 he problems below are harder, more ime-consuming, and inended for hose wih a more mahemaical background. hey are enirely opional, bu hopefully will
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationNon-uniform circular motion *
OpenSax-CNX module: m14020 1 Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform
More informationTHE MYSTERY OF STOCHASTIC MECHANICS. Edward Nelson Department of Mathematics Princeton University
THE MYSTERY OF STOCHASTIC MECHANICS Edward Nelson Deparmen of Mahemaics Princeon Universiy 1 Classical Hamilon-Jacobi heory N paricles of various masses on a Euclidean space. Incorporae he masses in he
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationLecture 4: Processes with independent increments
Lecure 4: Processes wih independen incremens 1. A Wienner process 1.1 Definiion of a Wienner process 1.2 Reflecion principle 1.3 Exponenial Brownian moion 1.4 Exchange of measure (Girsanov heorem) 1.5
More informationReferences are appeared in the last slide. Last update: (1393/08/19)
SYSEM IDEIFICAIO Ali Karimpour Associae Professor Ferdowsi Universi of Mashhad References are appeared in he las slide. Las updae: 0..204 393/08/9 Lecure 5 lecure 5 Parameer Esimaion Mehods opics o be
More informationDesigning Information Devices and Systems I Spring 2019 Lecture Notes Note 17
EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive
More informationLicenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A
Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In
More information