Interest Rate Derivatives: More Advanced Models. Chapter 24. The Two-Factor Hull-White Model (Equation 24.1, page 571) Analytic Results
|
|
- Anis Osborne
- 5 years ago
- Views:
Transcription
1 Ineres Rae Dervaves: More Advanced s Chaper 4 4. The Two-Facor Hull-Whe (Equaon 4., page 57) [ θ() ] σ 4. dx = u ax d dz du = bud σdz where x = f () r and he correlaon beween dz and dz s ρ The shor rae revers o a level dependen on u, and u self s mean reverng Analyc Resuls Bond prces and European opons on zero-coupon bonds can be calculaed analycally when f(r) = r 4.3 Opons on Coupon-Bearng Bonds We canno use he same procedure for opons on coupon-bearng bonds as we do n he case of one-facor models If we mae he approxmae assumpon ha he coupon-bearng bond prce s lognormal, we can use Blac s model The approprae volaly s calculaed from he volales of and correlaons beween he underlyng zero-coupon bond prces 4.4 Volaly Srucures In he one-facor Ho-Lee or Hull-Whe model he forward rae S.D.s are eher consan or declne exponenally. All forward raes are nsananeously perfecly correlaed In he wo-facor model many dfferen forward rae S.D. paerns and correlaon srucures can be obaned Example Gvng Humped Volaly Srucure (Fgure 4., page 57) a=, b=0., σ =0.0, σ =0.065, ρ=
2 4.7 Transformaon of he General dx = [ θ() u ax] d σ dz du = bud σ dz where x = f ( r) and he correlaon beween dz and dz s ρ We defne y = x u ( b a) so ha dy = [ θ() ay] d σ3dz3 du = bud σ dz 4.8 Transformaon of he General connued σ ρσ σ σ3 = σ ( b a) b a The correlaon beween dz dz3 s ρσ σ ( b a) σ 3 and Aracve Feaures of he I s Marov so ha a recombnng 3- dmensonal ree can be consruced The volaly srucure s saonary Volaly and correlaon paerns smlar o hose n he real world can be ncorporaed no he model 4.9 HJM : Noaon P(,T ): prce a me of a dscoun bond wh prncpal of $ maurng a T Ω : vecor of pas and presen values of neres raes and bond prces a me ha are relevan for deermnng bond prce volales a ha me v(,t,ω ): volaly of P(,T) 4.0 Noaon connued 4. ng Bond Prces 4. ƒ(,t,t ): forward rae as seen a for he perod beween T and T F(,T): nsananeous forward rae as seen a for a conrac maurng a T r(): shor-erm rs-free neres rae a dz(): Wener process drvng erm srucure movemens dpt (, ) = rptd () (, ) vt (,, Ω ) PTdz (, ) () We can choose any v funcon provdng v (,, Ω ) = 0 for all
3 ng Forward Raes Equaon 4.7, page 575) 4.3 Tree For a General 4.4 df(, T) = m(, T, Ω) d s(, T, Ω) dz() We mus have mt (,, Ω) = st (,, Ω) s (, τ, Ω) dτ Smlar resuls hold when here s more han one facor T A non-recombnng ree means ha he process for r s non-marov The LIBOR Mare Noaon The LIBOR mare model s a model consruced n erms of he forward raes underlyng caple prces : h rese dae F (): forward rae beween mes and m (): ndex for nex rese dae a me ς (): volaly of F () a me v (): volaly of P(, ) a me δ : Volaly Srucure 4.7 In Theory he Λ s can be deermned from Cap Prces 4.8 We assume a saonary volaly srucure where he volaly of F () depends only on he number of accrual perods beween he nex rese dae and of m() ] [.e., s a funcon only Defne Λ as he volaly of F ( ) when m( ) = If σ s he volaly for he (, provdes a perfec f o cap prces we mus have σ = Ths allows he Λ' s o be deermned nducvely = Λ δ ) caple.if he model
4 Example 4. (Page 579) If Blac volales for he frs hree caples are 4%, %, and 0%, hen Λ 0 =4.00% Λ =9.80% Λ =5.3% Example 4. (Page 579) n σ n (%) Λ n- (%) n σ n (%) Λ n- (%) The Process for F n a One- Facor LIBOR Mare df = Κ Λ m ( ) F dz The drf depends on he world chosen In a world ha s forward rs - neural wh respec o P (, ), he drf s zero 4. Rollng Forward Rs- Neuraly (Equaon 4.6, page 579) I s ofen convenen o choose a world ha s always FRN wr a bond maurng a he nex rese dae. In hs case, we can dscoun from o a he δ rae observed a me. The process for F s 4. df F F = δ Λ Λ δ F j= m() m() m() d Λ m() dz The LIBOR Mare and HJM Mone Carlo Implemenaon of BGM Cap (Equaon 4.8, page 580) In he lm as he me beween reses ends o zero, he LIBOR mare model wh rollng forward rs neuraly becomes he HJM model n he radonal rs-neural world We assume no change o he drf beween rese daes so ha F( j ) = F( j)exp j= δf ( j) ΛjΛj Λ j jl δ Λ δ j ε δ j j
5 Mulfacor Versons of BGM BGM can be exended so ha here are several componens o he volaly A facor analyss can be used o deermne how he volaly of F s spl no componens Rache Caps, Scy Caps, and Flex Caps A plan vanlla cap depends only on one forward rae. Is prce s no dependen on he number of facors. Rache caps, scy caps, and flex caps depend on he jon dsrbuon of wo or more forward raes. Ther prces end o ncrease wh he number of facors 4.7 Valung European Opons n he LIBOR Mare There s a good analyc approxmaon ha can be used o value European swap opons n he LIBOR mare model. See pages 58 o Calbrang he LIBOR Mare In heory he LMM can be exacly calbraed o cap prces as descrbed earler In pracce we proceed as for he one-facor models n Chaper 3 and mnmze a funcon of he form n = ( U V ) P where U s he mare prce of he h calbrang nsrumen, V s he model prce of he h calbrang nsrumen and P s a funcon ha penalzes bg changes or curvaure n a and σ Types of Morgage-Baced Secures (MBSs) 4.9 Opon-Adjused Spread (OAS) 4.30 Pass-Through Collaeralzed Morgage Oblgaon (CMO) Ineres Only (IO) Prncpal Only (PO) To calculae he OAS for an neres rae dervave we value assumng ha he nal yeld curve s he Treasury curve a spread We use an erave procedure o calculae he spread ha maes he dervave s model prce = mare prce. Ths s he OAS.
Correlation of default
efaul Correlaon Correlaon of defaul If Oblgor A s cred qualy deeroraes, how well does he cred qualy of Oblgor B correlae o Oblgor A? Some emprcal observaons are efaul correlaons are general low hough hey
More informationFloating rate securities
Caps and Swaps Floang rae secures Coupon paymens are rese perodcally accordng o some reference rae. reference rae + ndex spread e.g. -monh LIBOR + 00 bass pons (posve ndex spread 5-year Treasury yeld 90
More informationSection 6 Short Sales, Yield Curves, Duration, Immunization, Etc.
More Tuoral a www.lledumbdocor.com age 1 of 9 Secon 6 Shor Sales, Yeld Curves, Duraon, Immunzaon, Ec. Shor Sales: Suppose you beleve ha Company X s sock s overprced. You would ceranly no buy any of Company
More informationPricing and Valuation of Forward and Futures
Prcng and Valuaon of orward and uures. Cash-and-carry arbrage he prce of he forward conrac s relaed o he spo prce of he underlyng asse, he rsk-free rae, he dae of expraon, and any expeced cash dsrbuons
More informationMORNING SESSION. Date: Wednesday, May 4, 2016 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Exam QFICORE MORNING SESSION Dae: Wednesday, May 4, 016 Tme: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucons 1. Ths examnaon has a oal of 100 pons. I consss of a
More informationThe Financial System. Instructor: Prof. Menzie Chinn UW Madison
Economcs 435 The Fnancal Sysem (2/13/13) Insrucor: Prof. Menze Chnn UW Madson Sprng 2013 Fuure Value and Presen Value If he presen value s $100 and he neres rae s 5%, hen he fuure value one year from now
More information7 pages 1. Hull and White Generalized model. Ismail Laachir. March 1, Model Presentation 1
7 pages 1 Hull and Whie Generalized model Ismail Laachir March 1, 212 Conens 1 Model Presenaion 1 2 Calibraion of he model 3 2.1 Fiing he iniial yield curve................... 3 2.2 Fiing he caple implied
More informationAmerican basket and spread options. with a simple binomial tree
Amercan baske and spread opons wh a smple bnomal ree Svelana orovkova Vre Unverse Amserdam Jon work wh Ferry Permana acheler congress, Torono, June 22-26, 2010 1 Movaon Commody, currency baskes conss of
More informationOnline appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory
Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory APPNDX A: Dervng he sandard CA forula We wsh o fnd an expresson for he rsky value of a
More informationConvexity Adjustments in Inflation linked Derivatives using a multi-factor version of the Jarrow and Yildirim (2003) Model
Imperal College of Scence, echnology and edcne Deparmen of ahemacs Convexy Adjusmens n Inflaon lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Hongyun L Sepember 007 Submed o Imperal
More informationPricing Model of Credit Default Swap Based on Jump-Diffusion Process and Volatility with Markov Regime Shift
Assocaon for Informaon Sysems AIS Elecronc brary (AISe) WICEB 13 Proceedngs Wuhan Inernaonal Conference on e-busness Summer 5-5-13 Prcng Model of Cred Defaul Swap Based on Jump-Dffuson Process and Volaly
More informationA valuation model of credit-rating linked coupon bond based on a structural model
Compuaonal Fnance and s Applcaons II 247 A valuaon model of cred-rang lnked coupon bond based on a srucural model K. Yahag & K. Myazak The Unversy of Elecro-Communcaons, Japan Absrac A cred-lnked coupon
More informationNormal Random Variable and its discriminant functions
Normal Random Varable and s dscrmnan funcons Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped sngle prooype
More informationAn Implementation of the Displaced Diffusion, Stochastic Volatility Extension of the LIBOR Market Model
Maser Thess Deparmen o Busness Sudes Auhor: Chrsan Sørensen Advsor: Elsa Ncolao An Implemenaon o he Dsplaced Duson, Sochasc Volaly Exenson o he LIBOR Mare Model A Comparson o he Sandard Model Handelshøjsolen,
More information1 Purpose of the paper
Moneary Economics 2 F.C. Bagliano - Sepember 2017 Noes on: F.X. Diebold and C. Li, Forecasing he erm srucure of governmen bond yields, Journal of Economerics, 2006 1 Purpose of he paper The paper presens
More informationRecall from last time. The Plan for Today. INTEREST RATES JUNE 22 nd, J u n e 2 2, Different Types of Credit Instruments
Reall from las me INTEREST RATES JUNE 22 nd, 2009 Lauren Heller Eon 423, Fnanal Markes Smple Loan rnpal and an neres paymen s pad a maury Fxed-aymen Loan Equal monhly paymens for a fxed number of years
More informationStochastic Local Volatility
Sochasc Local Volaly CAROL ALEXANDER ICMA Cenre, Unversy of Readng LEONARDO M. NOGUEIRA ICMA Cenre, Unversy of Readng and Banco Cenral do Brasl Frs verson: Sepember 004 hs verson: March 008 ICMA Cenre
More informationA Solution to the Time-Scale Fractional Puzzle in the Implied Volatility
Arcle A Soluon o he Tme-Scale Fraconal Puzzle n he Impled Volaly Hdeharu Funahash 1, * and Masaak Kjma 1 Mzuho Secures Co. Ld., Tokyo 1-4, Japan Maser of Fnance Program, Tokyo Meropolan Unversy, Tokyo
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE SOLUTIONS Interest Theory
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Ineres Theory Ths page ndcaes changes made o Sudy Noe FM-09-05. January 4, 04: Quesons and soluons 58 60 were added. June, 04
More informationMay 2007 Exam MFE Solutions 1. Answer = (B)
May 007 Exam MFE Soluions. Answer = (B) Le D = he quarerly dividend. Using formula (9.), pu-call pariy adjused for deerminisic dividends, we have 0.0 0.05 0.03 4.50 =.45 + 5.00 D e D e 50 e = 54.45 D (
More informationConvexity adjustments in inflation-linked derivatives
Cung edge Inflaon Convexy adjusmens n nflaon-lnked dervaves Dorje Brody, John Crosby and Hongyun L value several ypes of nflaon-lnked dervaves usng a mul-facor verson of he Hughson (1998) and Jarrow &
More informationSTOCHASTIC LOCAL VOLATILITY
STOCHASTIC OCA VOATIITY Carol Alexander Char of Rsk Managemen and Drecor of Research ISMA Cenre, Busness School, The Unversy of Readng PO Box 4, Readng, RG6 6BA Uned Kngdom c.alexander@smacenre.rdg.ac.uk
More informationTentamen i 5B1575 Finansiella Derivat. Måndag 27 augusti 2007 kl Answers and suggestions for solutions.
Tenamen i 5B1575 Finansiella Deriva. Måndag 27 augusi 2007 kl. 14.00 19.00. Answers and suggesions for soluions. 1. (a) For he maringale probabiliies we have q 1 + r d u d 0.5 Using hem we obain he following
More informationEstimating intrinsic currency values
Esmang nrnsc currency values Forex marke praconers consanly alk abou he srenghenng or weakenng of ndvdual currences. In hs arcle, Jan Chen and Paul Dous presen a new mehodology o quanfy hese saemens n
More informationExtended One-Factor Short-Rate Models
CHAPTER 5 Exended One-Facor Shor-Rae Model 5.1. Ho Le Model Definiion 5.1 (Ho Le model). In he Ho Le model, he hor rae i aumed o aify he ochaic differenial equaion dr() =θ()d + σdw (), σ>0, θ i deerminiic,
More informationA New N-factor Affine Term Structure Model of Futures Price for CO 2 Emissions Allowances: Empirical Evidence from the EU ETS
WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang A ew -facor Affne erm Srucure Model of Fuures Prce for CO Emssons Allowances: Emprcal Evdence from he EU ES KAI CHAG, SU-SHEG
More informationLab 10 OLS Regressions II
Lab 10 OLS Regressons II Ths lab wll cover how o perform a smple OLS regresson usng dfferen funconal forms. LAB 10 QUICK VIEW Non-lnear relaonshps beween varables nclude: o Log-Ln: o Ln-Log: o Log-Log:
More informationIn-Arrears Interest Rate Derivatives under the 3/2 Model
Modern Economy, 5, 6, 77-76 Publshed Onlne June 5 n ScRes. hp://www.scrp.org/journal/me hp://d.do.org/.46/me.5.6667 In-Arrears Ineres Rae Dervaves under he / Model Joanna Goard School of Mahemacs and Appled
More informationwhere lnp(, ) f(, ) = P(, ) = exp { f(, u)du} = exp{q(, )} Q(, ) = f(, u)du Heah, Jarrow, and Moron (1992) claimed ha under risk-neural measure, he dr
HJM Model HJM model is no a ransiional model ha bridges popular LIBOR marke model wih once popular shor rae models, bu an imporan framework ha encompasses mos of he ineres rae models in he marke. As he
More informationINSTITUTE OF ACTUARIES OF INDIA
INSIUE OF ACUARIES OF INDIA EAMINAIONS 23 rd May 2011 Subjec S6 Finance and Invesmen B ime allowed: hree hours (9.45* 13.00 Hrs) oal Marks: 100 INSRUCIONS O HE CANDIDAES 1. Please read he insrucions on
More informationMarket Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009
s Praciioner Course: Ineres Rae Models March 29, 2009 In order o value European-syle opions, we need o evaluae risk-neural expecaions of he form V (, T ) = E [D(, T ) H(T )] where T is he exercise dae,
More informationEconomics of taxation
Economcs of axaon Lecure 3: Opmal axaon heores Salane (2003) Opmal axes The opmal ax sysem mnmzes he excess burden wh a gven amoun whch he governmen wans o rase hrough axaon. Opmal axes maxmze socal welfare,
More informationEmpirical analysis on China money multiplier
Aug. 2009, Volume 8, No.8 (Serial No.74) Chinese Business Review, ISSN 1537-1506, USA Empirical analysis on China money muliplier SHANG Hua-juan (Financial School, Shanghai Universiy of Finance and Economics,
More informationComputations in the Hull-White Model
Compuaions in he Hull-Whie Model Niels Rom-Poulsen Ocober 8, 5 Danske Bank Quaniaive Research and Copenhagen Business School, E-mail: nrp@danskebank.dk Specificaions In he Hull-Whie model, he Q dynamics
More informationSupplement to Chapter 3
Supplemen o Chaper 3 I. Measuring Real GD and Inflaion If here were only one good in he world, anchovies, hen daa and prices would deermine real oupu and inflaion perfecly: GD Q ; GD Q. + + + Then, he
More informationAlexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009
lexander L. Baranovski, Carsen von Lieres and ndré Wilch 8. May 2009/ Defaul inensiy model Pricing equaion for CDS conracs Defaul inensiy as soluion of a Volerra equaion of 2nd kind Comparison o common
More informationModels of Default Risk
Models of Defaul Risk Models of Defaul Risk 1/29 Inroducion We consider wo general approaches o modelling defaul risk, a risk characerizing almos all xed-income securiies. The srucural approach was developed
More informationPricing under the Real-World Probability Measure for Jump-Diffusion Term Structure Models
QUANTITATIV FINANC RSARCH CNTR QUANTITATIV FINANC RSARCH CNTR Research Paper 198 June 27 Prcng under he Real-World Probably Measure for Jump-Dffuson Term Srucure Models Ncola Bru-Lbera, Chrsna Nkopoulos-Sklbosos
More information(SPOT) LIBOR (RATE), denoted L ( t, strictly denote the continuouslycompounding. SHORT (INSTANTANEOUS SPOT) RATE, denoted r (t)
he Bak o halad Facal Isuos Polcy Group Quaave Models & Facal Egeerg eam Mahemacal Face Models Noe 4 ดร.พ มใจ นาคสก ล INERES RAE MODELS SPO VS. FORWARD INSANANEOUS RAES SPO INERES RAE deoed r u rae agreed
More informationDYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Krzysztof Jajuga Wrocław University of Economics
DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 2006 Krzyszof Jajuga Wrocław Universiy of Economics Ineres Rae Modeling and Tools of Financial Economerics 1. Financial Economerics
More informationNumerical Evaluation of European Option on a Non Dividend Paying Stock
Inernaonal Journal of Compuaonal cence and Mahemacs. IN 0974-389 olume Number 3 (00) pp. 6--66 Inernaonal Research Publcaon House hp://www.rphouse.com Numercal Evaluaon of European Opon on a Non Dvdend
More informationDEBT INSTRUMENTS AND MARKETS
DEBT INSTRUMENTS AND MARKETS Zeroes and Coupon Bonds Zeroes and Coupon Bonds Ouline and Suggesed Reading Ouline Zero-coupon bonds Coupon bonds Bond replicaion No-arbirage price relaionships Zero raes Buzzwords
More informationDynamic Relationship and Volatility Spillover Between the Stock Market and the Foreign Exchange market in Pakistan: Evidence from VAR-EGARCH Modelling
Dynamc Relaonshp and Volaly pllover Beween he ock Marke and he Foregn xchange marke n Paksan: vdence from VAR-GARCH Modellng Dr. Abdul Qayyum Dr. Muhammad Arshad Khan Inroducon A volale sock and exchange
More informationwhere r() = r(s)e a( s) + α() α(s)e a( s) + σ e a( u) dw(u) s α() = f M (0, ) + σ a (1 e a ) Therefore, r() condiional on F s is normally disribued wi
Hull-Whie Model Conens Hull-Whie Model Hull-Whie Tree Example: Hull-Whie Tree Calibraion Appendix: Ineres Rae Derivaive PDE Hull-Whie Model This secion is adaped from Brigo and Mercurio (006). As an exension
More informationChapter 9 - Level 2 - Course FM Solutions
Chaper 9 - Level 2 - Course FM Soluons ONLY CERTAIN PROBLEMS HAVE SOLUTIONS. THE REMAINING WILL BE ADDED OVER TIME.. (S08T3) Yvonne mus make a paymen of 80,000 a he end of one year. Addonally, she mus
More informationBasic Economic Scenario Generator: Technical Specications. Jean-Charles CROIX ISFA - Université Lyon 1
Basic Economic cenario Generaor: echnical pecicaions Jean-Charles CROIX IFA - Universié Lyon 1 January 1, 13 Conens Inroducion 1 1 Risk facors models 3 1.1 Convenions............................................
More informationHull-White one factor model Version
Hull-Whie one facor model Version 1.0.17 1 Inroducion This plug-in implemens Hull and Whie one facor models. reference on his model see [?]. For a general 2 How o use he plug-in In he Fairma user inerface
More informationMisspecification in term structure models of commodity prices: Implications for hedging price risk
19h Inernaional Congress on Modelling and Simulaion, Perh, Ausralia, 12 16 December 2011 hp://mssanz.org.au/modsim2011 Misspecificaion in erm srucure models of commodiy prices: Implicaions for hedging
More informationTime-Varying Correlations Between Credit Risks and Determinant Factors
me-varyng Correlaons Beween Cred Rsks and Deermnan Facors Frs & Correspondng Auhor: Ju-Jane Chang Asssan Professor n he Deparmen of Fnancal Engneerng and Acuaral Mahemacs, Soochow Unversy, awan 56, Sec.
More informationChain-linking and seasonal adjustment of the quarterly national accounts
Sascs Denmark Naonal Accouns 6 July 00 Chan-lnkng and seasonal adjusmen of he uarerly naonal accouns The mehod of chan-lnkng he uarerly naonal accouns was changed wh he revsed complaon of daa hrd uarer
More informationResearch Paper Series. No. 64. Yield Spread Options under the DLG Model. July, 2009
Research Paper Series No. 64 Yield Spread Opions under he LG Model Masaaki Kijima, Keiichi Tanaka and Tony Wong July, 2009 Graduae School of Social Sciences, Tokyo Meropolian Universiy Graduae School of
More informationAffine Term Structure Pricing with Bond Supply As Factors
by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 1 / 23 Affine Term Srucure Pricing wih Bond Supply As Facors by Fumio Hayashi Slides prepared for CIGS Conference 31
More informationWage Growth and the Measurement of Social Security s Financial Condition. Jagadeesh Gokhale Cato Institute
DRAFT Wage rowh and he Measuremen of Socal Secury s Fnancal Condon by Jagadeesh okhale Cao Insue Aprl 26 Jagadeesh okhale s Senor Fellow a he Cao Insue. The auhor hanks Alan Auerbach, Mchael Boskn, Jeffery
More informationGaining From Your Own Default
Ganng From Your Own Defaul Jon Gregory jon@ofranng.com Jon Gregory (jon@ofranng.com), Quan ongress US, 14 h July 2010 page 1 Regulaon s Easy () Wha don lke as a regulaor? Dfferen nsuons valung asses dfferenly
More informationNoise and Expected Return in Chinese A-share Stock Market. By Chong QIAN Chien-Ting LIN
Nose and Expeced Reurn n Chnese A-share Sock Marke By Chong QIAN Chen-Tng LIN 1 } Capal Asse Prcng Model (CAPM) by Sharpe (1964), Lnner (1965) and Mossn (1966) E ( R, ) R f, + [ E( Rm, ) R f, = β ] + ε
More informationarxiv: v4 [q-fin.mf] 6 Aug 2015
AFFINE LIBOR MODELS WITH MULTIPLE CURVES: THEORY, EXAMPLES AND CALIBRATION ZORANA GRBAC, ANTONIS PAPAPANTOLEON, JOHN SCHOENMAKERS, AND DAVID SKOVMAND arxv:45.245v4 q-fn.mf] 6 Aug 25 Absrac. We nroduce
More informationEcon 546 Lecture 4. The Basic New Keynesian Model Michael Devereux January 2011
Econ 546 Lecure 4 The Basic New Keynesian Model Michael Devereux January 20 Road map for his lecure We are evenually going o ge 3 equaions, fully describing he NK model The firs wo are jus he same as before:
More informationFugit (options) The terminology of fugit refers to the risk neutral expected time to exercise an
Fug (opons) INTRODUCTION The ermnology of fug refers o he rsk neural expeced me o exercse an Amercan opon. Invened by Mark Garman whle professor a Berkeley n he conex of a bnomal ree for Amercan opon hs
More informationCointegration between Fama-French Factors
1 Conegraon beween Fama-French Facors Absrac Conegraon has many applcaons n fnance and oher felds of scence researchng me seres and her nerdependences. The analyss s a useful mehod o analyse non-conegraon
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 05 h November 007 Subjec CT8 Financial Economics Time allowed: Three Hours (14.30 17.30 Hrs) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1) Do no wrie your
More informationTerms and conditions for the MXN Peso / US Dollar Futures Contract (Physically Delivered)
The Englsh verson of he Terms and Condons for Fuures Conracs s publshed for nformaon purposes only and does no consue legal advce. However, n case of any Inerpreaon conroversy, he Spansh verson shall preval.
More informationElton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 21
Elon, Gruber, Brown, and Goezmann oluions o Tex Problems: Chaper Chaper : Problem We can use he cash lows bonds A and B o replicae he cash lows o bond C. Le YA be he racion o bond A purchased and YB be
More informationAn Analytical Implementation of the Hull and White Model
Dwigh Gran * and Gauam Vora ** Revised: February 8, & November, Do no quoe. Commens welcome. * Douglas M. Brown Professor of Finance, Anderson School of Managemen, Universiy of New Mexico, Albuquerque,
More informationReal-World Pricing for a Modified Constant Elasticity of Variance Model
Real-World Prcng for a Modfed Consan Elascy of Varance Model Shane M. Mller 1 and Eckhard Plaen 2 June 1, 2009 Absrac hs paper consders a modfed consan elascy of varance (MCEV) model. hs model uses he
More informationdb t = r t B t dt (no Itô-correction term, as B has finite variation (FV), so ordinary Newton- Leibniz calculus applies). Now (again as B is FV)
ullin4b.ex pm Wed 21.2.2018 5. The change-of-numeraire formula Here we follow [BM, 2.2]. For more deail, see he paper Brigo & Mercurio (2001c) cied here, and H. GEMAN, N. El KAROUI and J. C. ROCHET, Changes
More informationMultifactor Term Structure Models
1 Multfactor Term Structure Models A. Lmtatons of One-Factor Models 1. Returns on bonds of all maturtes are perfectly correlated. 2. Term structure (and prces of every other dervatves) are unquely determned
More informationOnline appendices from The xva Challenge by Jon Gregory. APPENDIX 14A: Deriving the standard CVA formula.
Onlne appendces fro he xa Challenge by Jon Gregory APPNDX 4A: Dervng he sandard CA forla We wsh o fnd an expresson for he rsky vale of a need se of dervaves posons wh a ax ary dae Denoe he rsk-free vale
More informationMarket interest-rate models
Market interest-rate models Marco Marchioro www.marchioro.org November 24 th, 2012 Market interest-rate models 1 Lecture Summary No-arbitrage models Detailed example: Hull-White Monte Carlo simulations
More informationSTOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING
STOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING Tomasz R. Bielecki Deparmen of Mahemaics Norheasern Illinois Universiy, Chicago, USA T-Bielecki@neiu.edu (In collaboraion wih Marek Rukowski)
More informationMAFS Quantitative Modeling of Derivative Securities
MAFS 5030 - Quaniaive Modeling of Derivaive Securiies Soluion o Homework Three 1 a For > s, consider E[W W s F s = E [ W W s + W s W W s Fs We hen have = E [ W W s F s + Ws E [W W s F s = s, E[W F s =
More informationMATH 373 Test 4 Spring 2017 May 5, 2017
MATH 373 Tes 4 Spring 017 May 5, 017 1. The Bell Life Insurance Company has a wo year annuiy where i has promised o pay Elizabeh 5,000 a he end of each year for he nex wo years. Bell wans o absoluely mach
More informationPricing Inflation-Indexed Derivatives Using the Extended Vasicek Model of Hull and White
Pricing Inflaion-Indexed Derivaives Using he Exended Vasicek Model of Hull and Whie Alan Sewar Exeer College Universiy of Oxford A hesis submied in parial fulfillmen of he MSc in Mahemaical Finance April
More informationTHE TWO-PERIOD MODEL (CONTINUED)
GOVERNMENT AND FISCAL POLICY IN THE TWO-PERIOD MODEL (CONTINUED) MAY 25, 20 A Governmen in he Two-Period Model ADYNAMIC MODEL OF THE GOVERNMENT So far only consumers in our wo-period framework Inroduce
More informationA Markov Copulae Approach to Pricing and Hedging of Credit Index Derivatives and Ratings Triggered Step Up Bonds
A Markov Copulae Approach o Prcng and Hedgng of Cred Index Dervaves and Rangs Trggered Sep Up Bonds Tomasz R. Beleck, Andrea Vdozz, Luca Vdozz Absrac The paper presens seleced resuls from he heory of Markov
More informationBoğaziçi University Department of Economics Money, Banking and Financial Institutions L.Yıldıran
Chaper 3 INTEREST RATES Boğazç Unversy Deparmen of Economcs Money, Bankng and Fnancal Insuons L.Yıldıran Sylzed Fac abou Ineres Raes: Ineres raes Expanson Recesson Ineres raes affec economc acvy by changng
More informationTentamen i 5B1575 Finansiella Derivat. Torsdag 25 augusti 2005 kl
Tenamen i 5B1575 Finansiella Deriva. Torsdag 25 augusi 2005 kl. 14.00 19.00. Examinaor: Camilla Landén, el 790 8466. Tillåna hjälpmedel: Av insiuionen ulånad miniräknare. Allmänna anvisningar: Lösningarna
More informationTerm Structure Models: IEOR E4710 Spring 2005 c 2005 by Martin Haugh. Market Models. 1 LIBOR, Swap Rates and Black s Formulae for Caps and Swaptions
Term Srucure Models: IEOR E4710 Spring 2005 c 2005 by Marin Haugh Marke Models One of he principal disadvanages of shor rae models, and HJM models more generally, is ha hey focus on unobservable insananeous
More informationCorrelation Smile, Volatility Skew and Systematic Risk Sensitivity of Tranches
Correlaon Smle Volaly Skew and Sysemac Rsk Sensvy of ranches Alfred Hamerle Andreas Igl and lan Plank Unversy of Regensburg ay 0 Absac he classcal way of eang he correlaon smle phenomenon wh cred ndex
More informationResearch Article A General Gaussian Interest Rate Model Consistent with the Current Term Structure
Inernaional Scholarly Research Nework ISRN Probabiliy and Saisics Volume 212, Aricle ID 67367, 16 pages doi:1.542/212/67367 Research Aricle A General Gaussian Ineres Rae Model Consisen wih he Curren Term
More informationValuing Surrender Options in Korean Interest Indexed Annuities
Valung Surrender Opons n Korean Ineres Indexed Annues Chang Km * Absrac We presen surrender rae models wh explanaory varables such as he dfference beween reference raes and credng raes, polcy age snce
More informationMethodology of the CBOE S&P 500 PutWrite Index (PUT SM ) (with supplemental information regarding the CBOE S&P 500 PutWrite T-W Index (PWT SM ))
ehodology of he CBOE S&P 500 PuWre Index (PUT S ) (wh supplemenal nformaon regardng he CBOE S&P 500 PuWre T-W Index (PWT S )) The CBOE S&P 500 PuWre Index (cker symbol PUT ) racks he value of a passve
More informationMatematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.
Maemaisk saisik Tenamen: 8 5 8 kl 8 13 Maemaikcenrum FMS17/MASM19 Prissäning av Derivaillgångar, 9 hp Lunds ekniska högskola Soluion. 1. In he firs soluion we look a he dynamics of X using Iôs formula.
More informationDescription of the CBOE Russell 2000 BuyWrite Index (BXR SM )
Descripion of he CBOE Russell 2000 BuyWrie Index (BXR SM ) Inroducion. The CBOE Russell 2000 BuyWrie Index (BXR SM ) is a benchmark index designed o rack he performance of a hypoheical a-he-money buy-wrie
More informationDYNAMIC ECONOMETRIC MODELS Vol. 8 Nicolaus Copernicus University Toruń 2008
DYNAMIC ECONOMETRIC MODELS Vol. 8 Ncolaus Coperncus Unversy Toruń 2008 Por Fszeder Ncolaus Coperncus Unversy n Toruń Julusz Preś Szczecn Unversy of Technology Prcng of Weaher Opons for Berln Quoed on he
More informationAgricultural and Rural Finance Markets in Transition
Agrculural and Rural Fnance Markes n Transon Proceedngs of Regonal Research Commee NC-04 S. Lous, Mssour Ocober 4-5, 007 Dr. Mchael A. Gunderson, Edor January 008 Food and Resource Economcs Unversy of
More informationChange of measure and Girsanov theorem
and Girsanov heorem 80-646-08 Sochasic calculus I Geneviève Gauhier HEC Monréal Example 1 An example I Le (Ω, F, ff : 0 T g, P) be a lered probabiliy space on which a sandard Brownian moion W P = W P :
More informationOption-Implied Currency Risk Premia
Opon-Impled Currency Rsk Prema Jakub W. Jurek and Zhka Xu Absrac We use cross-seconal nformaon on he prces of G10 currency opons o calbrae a non-gaussan model of prcng kernel dynamcs and consruc esmaes
More informationOPTIMUM FISCAL AND MONETARY POLICY USING THE MONETARY OVERLAPPING GENERATION MODELS
Kuwai Chaper of Arabian Journal of Business and Managemen Review Vol. 3, No.6; Feb. 2014 OPTIMUM FISCAL AND MONETARY POLICY USING THE MONETARY OVERLAPPING GENERATION MODELS Ayoub Faramarzi 1, Dr.Rahim
More informationScholarship Project Paper 02/2012
Scholarshp Proec Paper 02/2012 HE DEERMINANS OF CREDI SPREAD CHANGES OF INVESMEN GRADE CORPORAE BONDS IN HAILAND BEWEEN JUNE 2006 AND FEBRUARY 2012: AN APPLICAION OF HE REGIME SWICHING MODEL reerapo Kongorann
More informationMORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quaniaive Finance and Invesmen Core Exam QFICORE MORNING SESSION Dae: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion
More informationOn Monte Carlo Simulation for the HJM Model Based on Jump
On Mone Carlo Simulaion for he HJM Model Based on Jump Kisoeb Park 1, Moonseong Kim 2, and Seki Kim 1, 1 Deparmen of Mahemaics, Sungkyunkwan Universiy 44-746, Suwon, Korea Tel.: +82-31-29-73, 734 {kisoeb,
More informationOnce we know he probabiliy densiy funcion (pdf) φ(s ) of S, a European call wih srike is priced a C() = E [e r d(s ) + ] = e r d { (S )φ(s ) ds } = e
Opion Basics Conens ime-dependen Black-Scholes Formula Black-76 Model Local Volailiy Model Sochasic Volailiy Model Heson Model Example ime-dependen Black-Scholes Formula Le s begin wih re-discovering he
More informationAdvances in Valuation Adjustments. Topquants Autumn 2015
Advances in Valuation Adjustments Topquants Autumn 2015 Quantitative Advisory Services EY QAS team Modelling methodology design and model build Methodology and model validation Methodology and model optimisation
More informationOnline Data, Fixed Effects and the Construction of High-Frequency Price Indexes
Onlne Daa, Fxed Effecs and he Consrucon of Hgh-Frequency Prce Indexes Jan de Haan* and Rens Hendrks** * ascs eherlands / Delf Unversy of Technology ** ascs eherlands EMG Worksho 23 Ams of he aer Exlan
More informationFITTING EXPONENTIAL MODELS TO DATA Supplement to Unit 9C MATH Q(t) = Q 0 (1 + r) t. Q(t) = Q 0 a t,
FITTING EXPONENTIAL MODELS TO DATA Supplemen o Un 9C MATH 01 In he handou we wll learn how o fnd an exponenal model for daa ha s gven and use o make predcons. We wll also revew how o calculae he SSE and
More informationFinancial Innovation and Asset Price Volatility. Online Technical Appendix
Fnancal Innovaon and Asse Prce Volaly Onlne Techncal Aendx Felx Kubler and Karl Schmedders In hs echncal aendx we derve all numbered equaons dslayed n he aer Equaons For he wo models n he aer, he frs se
More informationYou should turn in (at least) FOUR bluebooks, one (or more, if needed) bluebook(s) for each question.
UCLA Deparmen of Economics Spring 05 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and each par is worh 0 poins. Pars and have one quesion each, and Par 3 has
More informationErratic Price, Smooth Dividend. Variance Bounds. Present Value. Ex Post Rational Price. Standard and Poor s Composite Stock-Price Index
Erraic Price, Smooh Dividend Shiller [1] argues ha he sock marke is inefficien: sock prices flucuae oo much. According o economic heory, he sock price should equal he presen value of expeced dividends.
More informationONLINE APPENDIX The Effect of Uncertainty on Investment: Evidence from Texas Oil Drilling by Ryan Kellogg
ONLINE APPENDIX The Effec of Uncerainy on Invesmen: Evidence from Texas Oil Drilling by Ryan Kellogg Appendix 1: Consrucion of he ime series of implied fuures price volaily This appendix describes how
More information1. Interest Rate Gap. Duration
. Ineres Rae Gap. Duraion Mauriy Gap Problem. Mauriy Gap A bank invess $00 million in 3-year, 0% fixed rae bonds (assume hese are all asses) In he same ime, i issuses $90 million in -year, 0% percen fixed
More information