Variance Swap. by Fabrice Douglas Rouah

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Variance Swap. by Fabrice Douglas Rouah"

Transcription

1 Variance wap by Fabrice Douglas Rouah In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach described by Demeer e al. [] and ohers. We also show how a simpler version can be derived, using he forward price as he hreshold in he payo decomposiion ha is used in he derivaion. he variance swap has a payo equal o N var R K var () where N var is he noional, R is he realized annual variance of he sock over he life of he swap, and K var E[ R ] is he delivery (srike) variance. he objecive is o nd he value of K var : ock Price DE he variance swap sars by assuming a sock price evoluion similar o Black- choles, bu wih ime-varying volailiy parameer d d + dw : Consider f() ln and apply Iō s Lemma d ln d + dw so ha d d ln : () he Variance In equaion () ake he average variance from o V Z " Z d Z # d d ln " Z # d ln : (3)

2 he variance swap rae K var is he fair value of he variance; ha is, i is he expeced value of he average variance under he risk neural measure. Hence " Z # K var E[V ] E d (4) # E " Z d ln r E ln : he erm d represens he rae of reurn of he underlying, so under he risk neural measure he average expeced reurn over [; ] is he annual risk free rae r imes he ime period, namely r h. Mos i of he res of his noe will be devoed o nding an expression for E ln : 3 Log Conrac he log conrac has he payo funcion f( ) ln + : (5) Noe ha f ( ) ( ) and f ( ). 4 Payo Funcion Decomposiion Any payo funcion f( ) as a funcion of he underlying erminal price > can be decomposed as follows f( ) f( ) + f ( )( ) + (6) Z f (K)(K ) + dk + Z f (K)( K) + dk where > is an arbirary hreshold. ee he Noe on for a derivaion of equaion (6) using hree di eren approaches. Apply equaion (6) o he log conrac (5) o ge ln + ln + + ( ) + Z K (K ) + dk + Z K ( K) + dk:

3 Cancel from boh sides and re-arrange he erms o obain ln Z + K ( Z K (K ) + dk + (7) K) + dk: his is equaion (8) of Demeer e al. []. ake expecaions on boh sides of equaion (7), bringing he expecaions inside he inegrals where needed E ln E[ ] + (8) Z K E (K ) + dk + er + e r Z P (K)dK + er K Z Z K E ( K C(K)dK K) + dk where P (K) e r E [(K ) + ] is he pu price, C(K) e r E [( K) + ] is he call price, and where E[ ] e r F is he ime- forward price of he underlying a ime zero when he underlying price is. Now wrie ln ln + ln which implies ha E ln ln E ln : (9) ubsiue equaion (8) ino (9) o obain E ln ln e r + () e r Z 5 Fair Value of Variance P (K)dK + er K Z K C(K)dK: Recall equaion (4) for he fair value K var r E ln : () 3

4 h i ubsiue equaion () for E ln o obain he fair value of variance a incepion, namely, a ime. K var r e r ln + () e r Z P (K)dK + er K his is equaion (9) of Demeer e al. []. Z 6 Fair Value Using Forward Price K C(K)dK omeimes K var is wrien in a simpli ed form. o see his, le he hreshold in equaion () be de ned as he forward price, e r F. Afer some minor algebra, we arrive a ( Z K var F Z ) er K P (K)dK + F K C(K)dK : (3) 7 Mark-o-Marke Value of a Variance wap In his secion we use he noaion of Jacquier and laoui [6]. A incepion of he variance swap, he swap srike is se o he expeced value of he fuure variance, so he swap has value zero. Going forward, however, he value of he swap can become non-zero. o see his, rs denoe he denoe he average expeced variance over he ime inerval (; ) as " Z # Kvar ; E udu where E [] denoes he expecaion a ime. A incepion ( ) we wrie " Z # E udu K ; var which is K var ha appears in equaion (4), (), (), and (3), he value a incepion of he variance swap srike. he value a ime of he variance swap srike, denoed, is he ime- expeced value " Z # e r( ) E udu Kvar ; : (4) ) 4

5 A incepion his expeced value is zero, bu a ime i is no necessarily so. We wrie equaion (4) by breaking up he inegral, which produces " e r( ) E Z udu + Z # udu Kvar ; (5) ( " e r( ) ; + Z # ) E udu Kvar ; e r( ) ; + K; var Kvar ; e r( ) ; Kvar ; + Kvar ; K ; var R where ; udu is he realized variance a ime, which is known. he las equaion in (5) for is one ha is ofen encounered, such as ha which appears in ecion 8.6 of Flavell [3], for example. I indicaes ha he ime mark-o-marke value of he variance swap is a weighed average of wo componens. he erm ; Kvar ;, which represens he "accrued value" of he variance swap. Indeed, his is he realized variance up o ime minus he conraced srike.. he erm K ; var K ; var, which represens he di erence in fair srikes calculaed a ime, and calculaed a ime. Hence, a ime, o obain, we need o calculae R udu, which involves only variance ha has already been realized. We also need o calculae Kvar ;. In he same way ha equaions () or (3) are used wih opions of mauriy o obain Kvar ;, hose same equaions can be used wih opions of mauriy ( ) o obain K ; var : 8 Consan Vega of a Variance wap Exhibi of Demeer e al. [] shows ha a porfolio of opions weighed inversely by he square of heir srikes has a vega which becomes independen of he spo price as he number of opions increases. his can be demonsraed by seing up a porfolio of weighed opions C de ned as Z w(k)c (; K; ) dk (6) where w(k) is he weigh associaed wih he opions and C(; K; ) are heir Black-choles prices. his porfolio is similar o ha appearing in equaion (3). However, since we are concerned wih vanna, which is idenical for calls and pus, we do no have o spli up he porfolio ino calls and pus eiher will 5

6 do. his implies ha we can de ne C(; K; ) o be eiher calls or pus, and we don need o spli up he inegral in equaion (6) ino wo inegrals. o explain heir Exhibi, Demeer e al. [] demonsrae ha he vanna of he porfolio, is zero only when he weighs are inversely proporional o K. 8. Vega of he Porfolio he he porfolio is idenical for vanilla calls and pus so he vega of Z K; ) dk: he Black-choles vega of an individual opion p exp d where d p ln(x) + and x K. Hence we can wrie equaion (7) as p V p Z Change he variable of inegraion o x. and we can wrie p Z V p w(k) exp d dk: Hence dx dk so ha dk dx w(x) exp d dx: (8) In his las equaion, d depends on x only. Hence when we di ereniae V wih respec o we only need o di ereniae he erm w(x). his derivaive is, by he 8. Vanna of he Porfolio w(x) w(x) x: ubsiuing equaion (9) ino (8) and di ereniaing wih respec o, he sensiiviy of he porfolio vega p p w(x) + d dx: he erm inside he square brackes can be wrien w(x) w(k) : 6

7 @ implies ha w + or ha w w K. he soluion o his di erenial equaion is w(k) / K : Hence, when he weighs are chosen o be inversely proporional o K, he porfolio vega is insensiive o he spo price so ha is vanna is zero. his is illusraed in he following gure, which reproduces par of Exhibi of Demeer e al. []. A porfolio of calls wih srikes ranging from $6 o $4 in incremens of $ is formed, and he Black-choles vega of he porfolio is calculaed by weighing each call equally (doed line) and by K (solid line). he ineres rae is se o zero, he spo price is se o $, he mauriy is 6 monhs and he annual volailiy is %. 4 Weighed by inverse srike Equally weighed Porfolio Vega ock Price he vega of he equally-weighed porfolio (doed line) is clearly no consan bu increases wih he sock price. he vega of he srike-weighed porfolio, on he oher hand, is a in he $6 o $4 region, which indicaes ha is vanna is zero here. 7

8 9 Volailiy wap his is a swap on volailiy insead of on variance, so he payo is N vol ( R K vol ) () where N vol is he noional, R is he realized annual volailiy, and K vol is he srike volailiy. he values of N vol and K vol can be obained by wriing equaion () as N vol ( R K vol ) N vol R Kvol N vol R + K vol K vol R Kvol : () 9. Naive Esimae of rike Volailiy Comparing he las erm in equaion () wih equaion (), we see ha N vol K vol N var and K var K vol from which we obain he naive esimaes K vol p K var and N vol N var K vol. 9. Vega Noional and Convexiy he payo of a volailiy swap is linear in realized volailiy, bu he payo of a variance swap is convex in realized volailiy. he noional on volailiy, N vol is usually called vega noional. his is because N vol represens he change in he payo of he swap wih a poin change in volailiy. his is bes illusraed wih an example. uppose ha he variance noional is N var $;, and ha he fair esimae of volaily is 5, so ha he variance srike is K var 5. he srike on he volailiy is K vol 5, and he vega noional is N vol $; 5 $5;. In he following gure we plo he payo from he volailiy swap and from he variance swap, as R varies from o 5: 8

9 5 Variance Payoff NoionalVariance x (Realized Vol^ K^) Volailiy Payoff NoionalVega x (Realized Vol K) 5 Payoff in $M Realized Volailiy in Percen he payo from he volailiy is Volailiy Payo N vol ( R K vol ) $5; ( R 5) ; which is linear in R. When he realized volailiy increases by a single poin, he payo increases by exacly N vol $5;. his is represened by he dashed line. he payo from he variance is Variance Payo N var R K var $; R 5 ; which is convex in R and is represened by he solid line. uppose ha he realized variance experiences a wo-poin increase from R 38 o R 4. hen he volailiy payo increases from $65; o $75; an increase of $; which is exacly N vol. he variance payo increases from $89; o $975; an increase of $56;. 9

10 9.3 Convexiy Adjusmen he convexiy bias described in [] is he di erence beween he las and rs erm in equaion (). Wihou loss of generaliy we can se N vol. Hence Convexiy Bias R K vol ( R K vol ) K vol ( R K vol ) : K vol We approximae he expeced value of volailiy by he square roo of he expeced value of variance. Hence we use he approximaion E [ R ] p E [ R ]. We can nd a beer approximaion o he volailiy swap by considering a second order aylor series expansion of p x p p x x + p (x x ) + x 8 p (x x x 3 ) : () e x R v and x E R v in equaion () o obain p v p v + p v (v v) + 8 p v 3 (v v) : (3) ake expecaions and he middle erm on he righ hand side of equaion (3) drops ou o become E p v p v + 8 p v 3 E (v v) : In he original noaion q E [ R ] E [ R ] + V ar R 8E [ : R ]3 Hence, he loss of accuracy brough on by approximaing E [ R ] wih p E [ R ] can be miigaed by adding he adjusmen erm V ar[ R] 8E[ R] 3 : Oher Issues. Implemening he Variance wap Formula o come. Requires an approximae due o he fac ha marke prices of pus and calls are no available on a coninuum of srikes, bu are insead available a discree srikes, ofen in incremens of $5 or $.5.. Variance wap Greeks o come. form. Many of he Greeks for variance swaps are available in analyical

11 .3 Variance wap as Model-Free Implied Volailiy o come..4 he VIX o come. he VIX index is a 3-day variance swap on he &P 5 Index, wih convexiy adjusmen. Variance wap in Heson s Model In he Heson (993) model he sock price and sock price volailiy v each follow heir own di usion, and hese di usions are driven by correlaed Brownian moion. Hence d d + p v dz () dv ( v )d + p v dz () h i wih E dz () dz() d. he volailiy v follows a CIR process, and i is sraighforward o show ha he expeced value of v, given v s (s < ) is E [v j v s ] v s e ( s) ( + e s) + (v s ) e ( s) : ee, for example, Brigo and Mercurio []. As explained by Gaheral [4], a variance swap requires an esimae of he fuure variance over he (; ) ime period, namely of he oal (inegraed) variance w R v d. A fair esimae of w is is condiional expecaion E [w j v ]. his is given by " Z # E [w j v ] E v d v Z Z E [v j v ] d + (v ) e d + e (v ) : ince v represens he oal variance over (; ), i mus be scaled by in order o represen a fair esimae of annual variance (assuming ha is expressed in years.) Hence he srike variance for a variance swap is given by E [w j v ] e his is he expression on page 38 of Gaheral [4]. (v ) + :

12 References [] Brigo, D., and F. Mercurio (6). Ineres Rae Models - heory and Pracice: Wih mile, In aion, and Credi. econd Ediion. New York, NY: pringer. [] Demeer, K., Derman, E., Kamal, M., and J. Zou (999). "A Guide o Volailiy and Variance waps," Journal of Derivaives, ummer 999; 6, 4, pp 9-3. [3] Flavell, R. (). waps and Oher Derivaives. econd Ediion. New York, NY: John Wiley & ons. [4] Gaheral, J. (6) he Volailiy urface: A Praciioner s Guide. New York, NY: John Wiley & ons. [5] Heson,.L. (993). "A Closed-Form oluion for Opions wih ochasic Volailiy wih Applicaions o Bond and Currency Opions." Review of Financial udies, Vol. 6, pp [6] Jacquier, A., and. laoui (7). "Variance Dispersion and Correlaion waps," Working Paper, Universiy of London and AXA Invesmen Managemen.

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Math 201 Lecture 12: Cauchy-Euler Equations

Math 201 Lecture 12: Cauchy-Euler Equations Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process, Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ

More information

PRICING and STATIC REPLICATION of FX QUANTO OPTIONS

PRICING and STATIC REPLICATION of FX QUANTO OPTIONS PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of

More information

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

WHAT ARE OPTION CONTRACTS?

WHAT ARE OPTION CONTRACTS? WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be

More information

A general decomposition formula for derivative prices in stochastic volatility models

A general decomposition formula for derivative prices in stochastic volatility models A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 2013 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

Part 1: White Noise and Moving Average Models

Part 1: White Noise and Moving Average Models Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

Pricing exotic options. with an implied integrated variance

Pricing exotic options. with an implied integrated variance Pricing exoic opions wih an implied inegraed variance Ruh Kaila Ocober 13, 212 Absrac In his paper, we consider he pricing of exoic opions using wo opion-implied densiies: he densiy of he implied inegraed

More information

CBOE VIX PREMIUM STRATEGY INDEX (VPD SM ) CAPPED VIX PREMIUM STRATEGY INDEX (VPN SM )

CBOE VIX PREMIUM STRATEGY INDEX (VPD SM ) CAPPED VIX PREMIUM STRATEGY INDEX (VPN SM ) CBOE VIX PREIU STRATEGY INDEX (VPD S ) CAPPED VIX PREIU STRATEGY INDEX (VPN S ) The seady growh of CBOE s volailiy complex provides a unique opporuniy for invesors inen on capuring he volailiy premium.

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return Calculation of U.S. Treasury Constant Maturity Indices Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Section 7.1 Angles and Their Measure

Section 7.1 Angles and Their Measure Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

More information

Research Article Optimal Geometric Mean Returns of Stocks and Their Options

Research Article Optimal Geometric Mean Returns of Stocks and Their Options Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

THE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS

THE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Foreign Exchange and Quantos

Foreign Exchange and Quantos IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in

More information

Description of the CBOE S&P 500 BuyWrite Index (BXM SM )

Description of the CBOE S&P 500 BuyWrite Index (BXM SM ) Descripion of he CBOE S&P 500 BuyWrie Index (BXM SM ) Inroducion. The CBOE S&P 500 BuyWrie Index (BXM) is a benchmark index designed o rack he performance of a hypoheical buy-wrie sraegy on he S&P 500

More information

Technical Appendix to Risk, Return, and Dividends

Technical Appendix to Risk, Return, and Dividends Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,

More information

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009 ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

More information

Differential Equations and Linear Superposition

Differential Equations and Linear Superposition Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

More information

Stochastic Calculus and Option Pricing

Stochastic Calculus and Option Pricing Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes

More information

An Interest Rate Swap Volatility Index and Contract

An Interest Rate Swap Volatility Index and Contract Anonio Mele QUASaR Yoshiki Obayashi Applied Academics LLC Firs draf: November 10, 2009. This version: June 26, 2012. ABSTRACT Ineres rae volailiy and equiy volailiy evolve heerogeneously over ime, comoving

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

and Decay Functions f (t) = C(1± r) t / K, for t 0, where

and Decay Functions f (t) = C(1± r) t / K, for t 0, where MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

1. The graph shows the variation with time t of the velocity v of an object.

1. The graph shows the variation with time t of the velocity v of an object. 1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

More information

Chapter 2 Kinematics in One Dimension

Chapter 2 Kinematics in One Dimension Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

More information

9. Capacitor and Resistor Circuits

9. Capacitor and Resistor Circuits ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Week #9 - The Integral Section 5.1

Week #9 - The Integral Section 5.1 Week #9 - The Inegral Secion 5.1 From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,

More information

Machine Learning in Pairs Trading Strategies

Machine Learning in Pairs Trading Strategies Machine Learning in Pairs Trading Sraegies Yuxing Chen (Joseph) Deparmen of Saisics Sanford Universiy Email: osephc5@sanford.edu Weiluo Ren (David) Deparmen of Mahemaics Sanford Universiy Email: weiluo@sanford.edu

More information

Debt Portfolio Optimization Eric Ralaimiadana

Debt Portfolio Optimization Eric Ralaimiadana Deb Porfolio Opimizaion Eric Ralaimiadana 27 May 2016 Inroducion CADES remi consiss in he defeasance of he Social Securiy and Healh public deb To achieve is ask, he insiuion has been assigned a number

More information

Technical Description of S&P 500 Buy-Write Monthly Index Composition

Technical Description of S&P 500 Buy-Write Monthly Index Composition Technical Descripion of S&P 500 Buy-Wrie Monhly Index Composiion The S&P 500 Buy-Wrie Monhly (BWM) index is a oal reurn index based on wriing he nearby a-he-money S&P 500 call opion agains he S&P 500 index

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 Life tables, force of mortality and standard life insurance products Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

More information

THE PRESSURE DERIVATIVE

THE PRESSURE DERIVATIVE Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations. Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

More information

( ) in the following way. ( ) < 2

( ) in the following way. ( ) < 2 Sraigh Line Moion - Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and man-made. Wrie down several of hem. Horizonal cars waer

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

Carol Alexander ICMA Centre, University of Reading. Aanand Venkatramanan ICMA Centre, University of Reading

Carol Alexander ICMA Centre, University of Reading. Aanand Venkatramanan ICMA Centre, University of Reading Analyic Approximaions for Spread Opions Carol Alexander ICMA Cenre, Universiy of Reading Aanand Venkaramanan ICMA Cenre, Universiy of Reading 15h Augus 2007 ICMA Cenre Discussion Papers in Finance DP2007-11

More information

Chapter 2: Principles of steady-state converter analysis

Chapter 2: Principles of steady-state converter analysis Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya. Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

More information

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his

More information

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Two generic frameworks for credit index volatility products and their application to credit index options

Two generic frameworks for credit index volatility products and their application to credit index options Two generic frameworks for credi index volailiy producs and heir applicaion o credi index opions Taou k Bounhar Lauren Luciani Firs version : March 2008 This version : 14 Ocober 2008 Absrac The exension

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 2010 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

More information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convolution. Recommended Problems. x2[n] 1 2[n] 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

More information

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

More information

Fourier Series Approximation of a Square Wave

Fourier Series Approximation of a Square Wave OpenSax-CNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier

More information

FX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS

FX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS FX OPTION PRICING: REULT FROM BLACK CHOLE, LOCAL VOL, QUAI Q-PHI AND TOCHATIC Q-PHI MODEL Absrac Krishnamurhy Vaidyanahan 1 The paper suggess a new class of models (Q-Phi) o capure he informaion ha he

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed. Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

More information

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance Finance Leers, 003, (5), 6- Skewness and Kurosis Adjused Black-Scholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance

More information

Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions to Nonfarm Payroll Employment: 1964 to 2011 Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

More information