Models of Default Risk

Size: px
Start display at page:

Download "Models of Default Risk"

Transcription

1 Models of Defaul Risk Models of Defaul Risk 1/29

2 Inroducion We consider wo general approaches o modelling defaul risk, a risk characerizing almos all xed-income securiies. The srucural approach was developed by Black and Scholes (1973) and Meron (1974) and values a rm s defaul-risky deb as an explici funcion of he rm s capial srucure and he value and risk of a rm s asses. The reduced form approach simply assumes ha defaul is a Poisson process wih a ime-varying defaul inensiy and defaul recovery rae wihou explicily modeling a rm s asses and capial srucure. Examples of he reduced form approach include Jarrow, Lando, and Turnbull (1997), Madan and Unal (1998), and Du e and Singleon (1999). Models of Defaul Risk 2/29

3 Srucural Approach Assumpions Consider a model similar o Meron (1974) where a rm owns risky asses wih dae marke value A () and dynamics da=a = ( ) d + dz (1) where and are he expecaion and he sandard deviaion of he rae of reurn on asses and is he rae a which asses are paid ou as dividends o he rm s shareholders. Along wih shareholders equiy, he rm has issued a zero-coupon deb ha promises o pay he amoun B a dae T >, where T. The dae marke values of shareholders equiy and he deb are E () and D (; T ), respecively, so ha A () = E () + D (; T ). Models of Defaul Risk 3/29

4 Assumpions (coninued) A dae T, he rm pays B o he debholders if here is su cien rm asse value. Else, bankrupcy occurs and he debholders ake ownership of he rm s asses. Thus, he payo o debholders is D (T ; T ) = min [B; A (T )] (2) = B max [0; B A (T )] Le P (; T ) be he curren dae price of a defaul-free, zero-coupon bond ha pays $1 a dae T and assume ha he Vasicek (1977) model holds for he defaul-free erm srucure speci ed earlier in (9.41) o (9.43). Models of Defaul Risk 4/29

5 Marke Value of Deb Recognizing ha deb s payo in (2) equals he defaul-free value B less he value of a pu opion wrien on he rm s asses wih srike B, i is valued using opion pricing resuls in Chapers 9 and 10: D (; T ) = P (; T ) B P (; T ) BN ( h 2 ) + e AN ( h 1 ) = P (; T ) BN (h 2 ) + e AN ( h 1 ) (3) where h 1 = ln e A= (P (; T ) B) v 2 =v, h 2 = h 1 v, and v () is given in (9.61). Noe ha he deb s promised yield-o-mauriy is R (; T ) 1 ln [B=D (; T )] and is credi spread is 1 R (; T ) ln [1=P (; T )]. Models of Defaul Risk 5/29

6 Marke Value of Shareholders Equiy Given (3), shareholders equiy equals E () = A () D (; T ) (4) = A P (; T ) BN (h 2 ) e AN ( h 1 ) h i = A 1 e N ( h 1 ) P (; T ) BN (h 2 ) Equiy is similar o a call opion on he rm s asses since is payo is max [A (T ) B; 0]. However, i di ers if he rm pays dividends o equiyholders prior o he deb s mauriy, as re eced in he rs erm in he las line of (4). Models of Defaul Risk 6/29

7 Discussion of Srucural Models Meron (1974) analyzes he properies of deb and equiy formulas similar o equaions (3) and (4). Noe ha an equiy formula such as (4) is useful when rms have publicly raded equiy, since observaion of he marke value of equiy and is volailiy can be used o esimae A () and, which can hen be used o value D (; T ). There is now a vas lieraure on srucural models ha modify and exend he original Meron (1974) framework. Examples include Black and Cox (1976), Leland (1994), and Collin-Dufresne and Goldsein (2001). Models of Defaul Risk 7/29

8 The Reduced-Form Approach As before, le D (; T ) be he dae value of a defaul-risky, zero-coupon bond ha promises o pay B a is mauriy dae of T. Le () d be he insananeous probabiliy of defaul occurring during he inerval (; + d), so ha () is he physical defaul inensiy, or hazard rae. Then he bond s (physical) survival probabiliy from daes o is R E e (u)du (5) Models of Defaul Risk 8/29

9 A Zero-Recovery Bond Firs, consider a bond ha, if i defauls, has zero recovery value, so ha D (T ; T ) = B if here is no defaul or D (T ; T ) = 0 if defaul occurs over he inerval from daes o T. Applying risk-neural pricing, his bond s value, D Z (; T ), is D Z (; T ) = E b R T i he r (u)du D (T ; T ) (6) where r () is he dae insananeous defaul-free ineres rae, and b E [] is he dae risk-neural expecaions operaor. Models of Defaul Risk 9/29

10 Sae Variables and Pricing Kernel Suppose he defaul-free erm srucure and () depend on an n 1 vecor of sae variables, x, ha follows he process dx = a (; x) d + b (; x) dz (7) where x = (x 1 :::x n ) 0, a (; x) is an n 1 vecor, b (; x) is an n n marix, and dz = (dz 1 :::dz n ) 0 is an n 1 vecor of independen Brownian moions so ha dz i dz j = 0 for i 6= j. Assuming complee markes, he sochasic discoun facor for pricing he rm s defaul-risky bond is dm=m = r (; x) d (; x) 0 dz (; x) [dq (; x) d] (8) where (; x) is an n 1 vecor of he marke prices of risk associaed wih dz and (; x) is he marke price of risk associaed wih he acual defaul even. Models of Defaul Risk 10/29

11 Defaul as a Poisson Process The defaul even is recorded by dq, which if defaul occurs q () jumps from 0 (he no-defaul sae) o 1 (he absorbing defaul sae) a which ime dq = 1. The risk-neural defaul inensiy, b (; x), is hen given by b (; x) = [1 (; x)] (; x). Defaul is a doubly sochasic process, also referred o as a Cox process, because i depends on he Brownian moion vecor dz ha drives x and deermines how he likelihood of defaul, b (; x), changes over ime, bu i also depends on he Poisson process dq ha deermines he arrival of defaul. Hence, defaul risk re ecs wo ypes of risk premia, (; x) and (; x). Models of Defaul Risk 11/29

12 Value of he Zero-Recovery Bond Based on (5), we can solve for D Z (; T ): D Z (; T ) = b E e R T r (u)du e R T b(u)du B (9) = b E he R i T [r (u)+ (u)]du b B Noe ha (9) is similar o valuing a defaul-free bond excep ha he discoun rae r (u) + b (u), raher han jus r (u), is used. Given speci c funcional forms for r (; x), b (; x), x and (; x), (9) can be compued. Models of Defaul Risk 12/29

13 Specifying Recovery Values Suppose ha if he bond defauls a dae, where < T, bondholders recover an amoun w (; x) a dae. Then he risk-neural probabiliy densiy of defauling a is R e (u)dub b () (10) In (10), b h R () is discouned by exp b i (u) du because defaul a dae is condiioned on no having defauled previously. Models of Defaul Risk 13/29

14 Valuing a Bond wih Recovery Value Thus, he presen value of recovery, D R (; T ), is: Z T D R (; T ) = E b e Z T = E b e R r (u)du w () e R b (u)dub () d R [r (u)+ b (u)]dub () w () d Puing his ogeher wih (9) gives he bond s oal value: (11) D (; T ) = D Z (; T ) + D R (; T ) (12) = b E he + Z T R T e [r (s)+ b (s)]ds B R [r (s)+ b (s)]ds b () w () d Models of Defaul Risk 14/29

15 Recovery Proporional o Par Value Le us consider paricular speci caions for w (; x). Le he defaul dae be and assume w (; x) = (; x) B, where (; x) can be a consan, say,. Then (11) is D R (; T ) = B Z T k (; ) d (13) where k (; ) b E he R [r (u)+ b (u)]dub () i (14) has a closed-form soluion when r (u; x) and b (u; x) are a ne funcions of x and he vecor x in (7) has a risk-neural process ha is also a ne. (13) can be compued by numerical inegraion of k (; ). Models of Defaul Risk 15/29

16 Recovery Proporional o Par, Payable a Mauriy Assume ha if defaul occurs a dae, bondholders recover (; x) B a dae T, which is equivalen o w (; x) = (; x) P (; T ) B. Then (11) is D R (; T ) Z T = E b e Z T = E b e = b E e R T R [r (u)+ (u)]dub b R T () (; x) e r (u)du Bd R b(u)dub () (; x) e Z T r (u)du e R R T r (u)du Bd b(u)dub () (; x) d B (15) Models of Defaul Risk 16/29

17 Recovery Proporional o Par, Payable a Mauriy If (; x) =, noe ha R h T R exp b (u) dui b () d is he oal risk-neural h probabiliy of defaul from o T and R i T equal 1 exp b (u) du. Thus, D R (; T ) = E b R T R T e r (u)du 1 e b(u)du B = b E he R T r (u)du e R i T [r (u)+ (u)]du b B = BP (; T ) D Z (; T ) (16) Therefore, he oal value of he bond is D (; T ) = D Z (; T )+D R (; T ) = 1 D Z (; T )+BP (; T ) (17) so only a value for he zero-recovery bond is required. Models of Defaul Risk 17/29

18 Recovery Proporional o Marke Value Assume ha a defaul, bondholders lose a proporion L (; x) of he bond s value jus prior o defaul: D + ; T = w (; x) = D ; T [1 L (; x)] (18) Treaing he defaulable bond as a coningen claim and applying Iô s lemma: dd (; T ) =D (; T ) = ( D k D ) d + 0 D dz L (; x) dq (19) where D and he n 1 vecor D are given by he usual Iô s lemma expressions, he expeced jump size k D ( ) E [D ( + ; T ) D ( ; T )] =D ( ; T ) = L (; x), so ha he drif erm in (19) is D + (; x) L (; x). Models of Defaul Risk 18/29

19 Recovery Proporional o Marke Value (coninued) The risk-neural process for D (; T ) replaces D wih r (): dd (; T ) =D (; T ) = r (; x) + b (; x) b L (; x) d(20) + 0 D dbz b L (; x) dq where b L (; x) is he risk-neural loss given defaul. Similar o (11.17), D (; T ) sais es he PDE: 1 2 Trace b (; x) b (; x) 0 D xx +ba (; x) 0 D x R (; x) D +D = 0 (21) where D x is he n 1 vecor of rs derivaives, D xx is he n n marix of second derivaives, ba (; x) = a (; x) b (; x), and R (; x) r (; x) + b (; x) b L (; x). Models of Defaul Risk 19/29

20 Soluion for he Defaulable Bond Value The PDE (21) is sandard excep ha R (; x) replaces r (; x) in he sandard PDE. Thus, he Feynman-Kac soluion is D (; T ) = b E he R T R(u;x)du i B (22) where R (; x) r (; x) + b (; x) b L (; x) is he defaul-adjused discoun rae. The produc s (; x) b (; x) b L (; x) is he credi spread on an insananeous-mauriy, defaulable bond, and since b (; x) and b L (; x) are no individually ideni ed, a single funcional form can be speci ed for s (; x). Models of Defaul Risk 20/29

21 Examples Le x = (x 1 x 2 ) 0 be a wo-dimensional vecor, ba (; x) = ( 1 (x 1 x 1 ) 2 (x 2 x 2 )) 0, and b (; x) is a p p diagonal marix wih elemens of 1 x1 and 2 x2. Also assume r (; x) = x 1 () and b (; x) = x 2 (), so ha he defaul-free erm srucure q and b (; x) are independen. De ning r x 1 and , he CIR bond price is P (; T ) = A 1 () e B 1()r (), where (23) " # 2 1 e ( 1+ 1 ) 21 r = A 1 () ( ) (e 1 1) (24) 2 e 1 1 B 1 () ( ) (e 1 1) (25) Models of Defaul Risk 21/29

22 Examples (coninued) Also de ne x 2, hen based on (9) we have D Z (; T ) = b E he R i T [r (s)+ (s)]ds b B where = b E he R T r (s)ds i b E he R T b(s)ds i B = P (; T ) V (; T ) B (26) V (; T ) = A 2 () e B 2() b () (27) and where A 2 () is he same as A 1 () in (24), and B 2 () is he same as B 1 () in (25) excep ha q 2 replaces 1, 2 replaces 1, replaces r, and replaces 1. Models of Defaul Risk 22/29

23 Example: Recovery Proporional o Par, Payable a Mauriy If recovery is a xed proporion,, of par, payable a mauriy, hen from (17): D (; T ) = 1 D Z (; T ) + BP (; T ) = + 1 V (; T ) P (; T ) B (28) In (27), V (; T ) is analogous o a bond price in he sandard Cox, Ingersoll, and Ross erm srucure model and is inversely relaed o b () and sricly less han 1 whenever b () is sricly posiive, which can be ensured when Thus, (28) con rms ha he defaulable bond s value declines as is risk-neural defaul inensiy rises. Models of Defaul Risk 23/29

24 Example: Recovery Proporional o Marke Value Assume recovery is proporional o marke value and s (; x) b (; x) b L (; x) = x 2 and de ne s x 2. Then from (22): D (; T ) = b E he R i T [r (u)+s(u)]du B where = b E he R T r (u)du i b E he R T s(u)du i B = P (; T ) S (; T ) B (29) S (; T ) = A 2 () e B 2()s() and where A 2 () is he same as A 1 () in (24) and B 2 () is he same as B 1 () in (25) excep ha q 2 replaces 1, 2 replaces 1, s replaces r, and replaces 1. (30) D (; T ) is like P (; T ) bu R () = r () + s () replaces r (). Models of Defaul Risk 24/29

25 Coupon Bonds Suppose a defaulable coupon bond promises n cash ows, wih he i h promised cash ow being equal o c i and being paid a dae T i >. Then he value of his coupon bond in erms of our zero-coupon bond formulas is np i=1 D (; T i ) c i B (31) Models of Defaul Risk 25/29

26 Credi Defaul Swaps A credi defaul swap is a conrac in which one pary, he proecion buyer, makes periodic paymens unil he conrac s mauriy as long as a paricular issuer does no defaul. The oher pary, he proecion seller, receives hese paymens in reurn for paying he di erence beween he issuer s deb s par value and is recovery value if defaul occurs prior o he swap s mauriy. Le he conrac specify equal periodic paymens of c a fuure daes +, + 2,..., + n. Since hese paymens are coningen on defaul no occurring, heir value is c np D Z (; + i) (32) B i=1 where D Z (; T ) is given in (9). Models of Defaul Risk 26/29

27 Credi Defaul Swaps (coninued) Le w (; x) be he recovery value of he defaulable deb underlying he swap conrac. Then similar o (11), he value of he swap proecion is Z +n R be e [r (u)+ (u)]dub b () [B w ()] d (33) Suppose he proecion seller s paymen in he even of defaul is B w () = B B = B 1. Then (33) is B 1 Z +n k (; ) d (34) where k (; ) is de ned in (14). Given funcional forms for r (; x), b (; x), w (; x), and x, he value of he swap paymens, c, ha equaes (32) o (33) can be deermined. Models of Defaul Risk 27/29

28 Implemening a Reduced-Form Approach A general issue when implemening he reduced-form approach is deermining he proper curren values b (), s (), or w () ha may no be direcly observable. One or more of hese defaul variables migh be inferred by seing he acual marke prices of one or more of an issuer s bonds o heir heoreical formulas. Then, based on he implied values of b (), s (), or w (), one can deermine wheher a given bond of he same issuer is over- or underpriced relaive o oher bonds. Alernaively, hese implied defaul variables can be used o se he price of a new bond of he same issuer or a credi derivaive (such as a defaul swap) wrien on he issuer s deb. Models of Defaul Risk 28/29

29 Summary There are wo main branches of modeling defaulable xed-income securiies. The srucural approach models defaul based on he ineracion beween a rm s asses and is liabiliies. Poenially, i can improve our undersanding beween capial srucure and corporae bond and loan prices. In conras, he reduced-form approach absracs from speci c characerisics of a rm s nancial srucure, bu i permis a more exible modeling of defaul probabiliies and may beer describe acual he prices of an issuer s deb. Models of Defaul Risk 29/29

Jarrow-Lando-Turnbull model

Jarrow-Lando-Turnbull model Jarrow-Lando-urnbull model Characerisics Credi raing dynamics is represened by a Markov chain. Defaul is modelled as he firs ime a coninuous ime Markov chain wih K saes hiing he absorbing sae K defaul

More information

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE 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 information

Introduction to Black-Scholes Model

Introduction to Black-Scholes Model 4 azuhisa Masuda All righs reserved. Inroducion o Black-choles Model Absrac azuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York, NY 6-439 Email:

More information

The Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations

The Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations The Mahemaics Of Sock Opion Valuaion - Par Four Deriving The Black-Scholes Model Via Parial Differenial Equaions Gary Schurman, MBE, CFA Ocober 1 In Par One we explained why valuing a call opion as a sand-alone

More information

Principles of Finance CONTENTS

Principles of Finance CONTENTS Principles of Finance CONENS Value of Bonds and Equiy... 3 Feaures of bonds... 3 Characerisics... 3 Socks and he sock marke... 4 Definiions:... 4 Valuing equiies... 4 Ne reurn... 4 idend discoun model...

More information

Alexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009

Alexander 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 information

MAFS Quantitative Modeling of Derivative Securities

MAFS 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 information

Matematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.

Matematisk 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 information

Synthetic CDO s and Basket Default Swaps in a Fixed Income Credit Portfolio

Synthetic CDO s and Basket Default Swaps in a Fixed Income Credit Portfolio Synheic CDO s and Baske Defaul Swaps in a Fixed Income Credi Porfolio Louis Sco June 2005 Credi Derivaive Producs CDO Noes Cash & Synheic CDO s, various ranches Invesmen Grade Corporae names, High Yield

More information

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE 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 information

FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004

FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004 FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004 This exam has 50 quesions on 14 pages. Before you begin, please check o make sure ha your copy has all 50 quesions and all 14 pages.

More information

Black-Scholes Model and Risk Neutral Pricing

Black-Scholes Model and Risk Neutral Pricing Inroducion echniques Exercises in Financial Mahemaics Lis 3 UiO-SK45 Soluions Hins Auumn 5 eacher: S Oriz-Laorre Black-Scholes Model Risk Neural Pricing See Benh s book: Exercise 44, page 37 See Benh s

More information

Pricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6.

Pricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6. Pricing ulnerable American Opions April 16, 2007 Peer Klein and Jun (James) Yang imon Fraser Universiy Burnaby, B.C. 5A 16 pklein@sfu.ca (604) 268-7922 Pricing ulnerable American Opions Absrac We exend

More information

IJRSS Volume 2, Issue 2 ISSN:

IJRSS Volume 2, Issue 2 ISSN: A LOGITIC BROWNIAN MOTION WITH A PRICE OF DIVIDEND YIELDING AET D. B. ODUOR ilas N. Onyango _ Absrac: In his paper, we have used he idea of Onyango (2003) he used o develop a logisic equaion used in naural

More information

Change of measure and Girsanov theorem

Change 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 information

Tentamen i 5B1575 Finansiella Derivat. Måndag 27 augusti 2007 kl Answers and suggestions for solutions.

Tentamen 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 information

Equivalent Martingale Measure in Asian Geometric Average Option Pricing

Equivalent Martingale Measure in Asian Geometric Average Option Pricing Journal of Mahemaical Finance, 4, 4, 34-38 ublished Online Augus 4 in SciRes hp://wwwscirporg/journal/jmf hp://dxdoiorg/436/jmf4447 Equivalen Maringale Measure in Asian Geomeric Average Opion ricing Yonggang

More information

MA Advanced Macro, 2016 (Karl Whelan) 1

MA Advanced Macro, 2016 (Karl Whelan) 1 MA Advanced Macro, 2016 (Karl Whelan) 1 The Calvo Model of Price Rigidiy The form of price rigidiy faced by he Calvo firm is as follows. Each period, only a random fracion (1 ) of firms are able o rese

More information

Optimal Early Exercise of Vulnerable American Options

Optimal Early Exercise of Vulnerable American Options Opimal Early Exercise of Vulnerable American Opions March 15, 2008 This paper is preliminary and incomplee. Opimal Early Exercise of Vulnerable American Opions Absrac We analyze he effec of credi risk

More information

An Analytical Implementation of the Hull and White Model

An 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 information

May 2007 Exam MFE Solutions 1. Answer = (B)

May 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 information

UCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory

UCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory UCLA Deparmen of Economics Fall 2016 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and you are o complee each par. Answer each par in a separae bluebook. All

More information

Pricing FX Target Redemption Forward under. Regime Switching Model

Pricing FX Target Redemption Forward under. Regime Switching Model In. J. Conemp. Mah. Sciences, Vol. 8, 2013, no. 20, 987-991 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.12988/ijcms.2013.311123 Pricing FX Targe Redempion Forward under Regime Swiching Model Ho-Seok

More information

Pricing formula for power quanto options with each type of payoffs at maturity

Pricing formula for power quanto options with each type of payoffs at maturity Global Journal of Pure and Applied Mahemaics. ISSN 0973-1768 Volume 13, Number 9 (017, pp. 6695 670 Research India Publicaions hp://www.ripublicaion.com/gjpam.hm Pricing formula for power uano opions wih

More information

Hull-White one factor model Version

Hull-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 information

Macroeconomics II A dynamic approach to short run economic fluctuations. The DAD/DAS model.

Macroeconomics II A dynamic approach to short run economic fluctuations. The DAD/DAS model. Macroeconomics II A dynamic approach o shor run economic flucuaions. The DAD/DAS model. Par 2. The demand side of he model he dynamic aggregae demand (DAD) Inflaion and dynamics in he shor run So far,

More information

Computations in the Hull-White Model

Computations 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 information

Advanced Tools for Risk Management and Asset Pricing

Advanced Tools for Risk Management and Asset Pricing MSc. Finance/CLEFIN 214/215 Ediion Advanced Tools for Risk Managemen and Asse Pricing May 215 Exam for Non-Aending Sudens Soluions Time Allowed: 13 minues Family Name (Surname) Firs Name Suden Number (Mar.)

More information

(1 + Nominal Yield) = (1 + Real Yield) (1 + Expected Inflation Rate) (1 + Inflation Risk Premium)

(1 + Nominal Yield) = (1 + Real Yield) (1 + Expected Inflation Rate) (1 + Inflation Risk Premium) 5. Inflaion-linked bonds Inflaion is an economic erm ha describes he general rise in prices of goods and services. As prices rise, a uni of money can buy less goods and services. Hence, inflaion is an

More information

Option Valuation of Oil & Gas E&P Projects by Futures Term Structure Approach. Hidetaka (Hugh) Nakaoka

Option Valuation of Oil & Gas E&P Projects by Futures Term Structure Approach. Hidetaka (Hugh) Nakaoka Opion Valuaion of Oil & Gas E&P Projecs by Fuures Term Srucure Approach March 9, 2007 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion

More information

A pricing model for the Guaranteed Lifelong Withdrawal Benefit Option

A pricing model for the Guaranteed Lifelong Withdrawal Benefit Option A pricing model for he Guaraneed Lifelong Wihdrawal Benefi Opion Gabriella Piscopo Universià degli sudi di Napoli Federico II Diparimeno di Maemaica e Saisica Index Main References Survey of he Variable

More information

A Note on Forward Price and Forward Measure

A Note on Forward Price and Forward Measure C Review of Quaniaive Finance and Accouning, 9: 26 272, 2002 2002 Kluwer Academic Publishers. Manufacured in The Neherlands. A Noe on Forward Price and Forward Measure REN-RAW CHEN FOM/SOB-NB, Rugers Universiy,

More information

7 pages 1. Hull and White Generalized model. Ismail Laachir. March 1, Model Presentation 1

7 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 information

Available online at ScienceDirect

Available online at  ScienceDirect Available online a www.sciencedirec.com ScienceDirec Procedia Economics and Finance 8 ( 04 658 663 s Inernaional Conference 'Economic Scienific Research - Theoreical, Empirical and Pracical Approaches',

More information

Bond Prices and Interest Rates

Bond Prices and Interest Rates Winer erm 1999 Bond rice Handou age 1 of 4 Bond rices and Ineres Raes A bond is an IOU. ha is, a bond is a promise o pay, in he fuure, fixed amouns ha are saed on he bond. he ineres rae ha a bond acually

More information

STOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING

STOCHASTIC 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 information

Credit risk modelling

Credit risk modelling Risk Inernaional 0 An inroducion o credi risk modelling Abukar Ali from YieldCurve.com provides essenial background on he wo main models of credi defaul risk Models of credi risks have long exised in he

More information

Investment Reversibility and Agency Cost of Debt

Investment Reversibility and Agency Cost of Debt Invesmen Reversibiliy and Agency Cos of Deb Gusavo Manso Ocober 24, 2007 Absrac Previous research has argued ha deb financing affecs equiyholders invesmen decisions, producing subsanial inefficiency. This

More information

Online Appendix. Using the reduced-form model notation proposed by Doshi, el al. (2013), 1. and Et

Online Appendix. Using the reduced-form model notation proposed by Doshi, el al. (2013), 1. and Et Online Appendix Appendix A: The concep in a muliperiod framework Using he reduced-form model noaion proposed by Doshi, el al. (2013), 1 he yearly CDS spread S c,h for a h-year sovereign c CDS conrac can

More information

t=1 C t e δt, and the tc t v t i t=1 C t (1 + i) t = n tc t (1 + i) t C t (1 + i) t = C t vi

t=1 C t e δt, and the tc t v t i t=1 C t (1 + i) t = n tc t (1 + i) t C t (1 + i) t = C t vi Exam 4 is Th. April 24. You are allowed 13 shees of noes and a calculaor. ch. 7: 137) Unless old oherwise, duraion refers o Macaulay duraion. The duraion of a single cashflow is he ime remaining unil mauriy,

More information

Market Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009

Market 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 information

Financial Econometrics Jeffrey R. Russell Midterm Winter 2011

Financial Econometrics Jeffrey R. Russell Midterm Winter 2011 Name Financial Economerics Jeffrey R. Russell Miderm Winer 2011 You have 2 hours o complee he exam. Use can use a calculaor. Try o fi all your work in he space provided. If you find you need more space

More information

Option pricing and hedging in jump diffusion models

Option pricing and hedging in jump diffusion models U.U.D.M. Projec Repor 21:7 Opion pricing and hedging in jump diffusion models Yu Zhou Examensarbee i maemaik, 3 hp Handledare och examinaor: Johan ysk Maj 21 Deparmen of Mahemaics Uppsala Universiy Maser

More information

The Binomial Model and Risk Neutrality: Some Important Details

The Binomial Model and Risk Neutrality: Some Important Details The Binomial Model and Risk Neuraliy: Some Imporan Deails Sanjay K. Nawalkha* Donald R. Chambers** Absrac This paper reexamines he relaionship beween invesors preferences and he binomial opion pricing

More information

VaR and Low Interest Rates

VaR and Low Interest Rates VaR and Low Ineres Raes Presened a he Sevenh Monreal Indusrial Problem Solving Workshop By Louis Doray (U de M) Frédéric Edoukou (U de M) Rim Labdi (HEC Monréal) Zichun Ye (UBC) 20 May 2016 P r e s e n

More information

DEBT INSTRUMENTS AND MARKETS

DEBT 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 information

EVA NOPAT Capital charges ( = WACC * Invested Capital) = EVA [1 P] each

EVA NOPAT Capital charges ( = WACC * Invested Capital) = EVA [1 P] each VBM Soluion skech SS 2012: Noe: This is a soluion skech, no a complee soluion. Disribuion of poins is no binding for he correcor. 1 EVA, free cash flow, and financial raios (45) 1.1 EVA wihou adjusmens

More information

MORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

MORNING 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 information

Economic Growth Continued: From Solow to Ramsey

Economic Growth Continued: From Solow to Ramsey Economic Growh Coninued: From Solow o Ramsey J. Bradford DeLong May 2008 Choosing a Naional Savings Rae Wha can we say abou economic policy and long-run growh? To keep maers simple, le us assume ha he

More information

Single Premium of Equity-Linked with CRR and CIR Binomial Tree

Single Premium of Equity-Linked with CRR and CIR Binomial Tree The 7h SEAMS-UGM Conference 2015 Single Premium of Equiy-Linked wih CRR and CIR Binomial Tree Yunia Wulan Sari 1,a) and Gunardi 2,b) 1,2 Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences,

More information

Inventory Investment. Investment Decision and Expected Profit. Lecture 5

Inventory Investment. Investment Decision and Expected Profit. Lecture 5 Invenory Invesmen. Invesmen Decision and Expeced Profi Lecure 5 Invenory Accumulaion 1. Invenory socks 1) Changes in invenory holdings represen an imporan and highly volaile ype of invesmen spending. 2)

More information

On the Edge of Completeness

On the Edge of Completeness On he Edge of Compleeness May 2000 Jean-Paul LAURENT Professor, ISFA Acuarial School, Universiy of Lyon, Scienific Advisor, BNP Paribas Correspondence lauren.jeanpaul@online.fr On he Edge of Compleeness:

More information

Lecture Notes to Finansiella Derivat (5B1575) VT Note 1: No Arbitrage Pricing

Lecture Notes to Finansiella Derivat (5B1575) VT Note 1: No Arbitrage Pricing Lecure Noes o Finansiella Deriva (5B1575) VT 22 Harald Lang, KTH Maemaik Noe 1: No Arbirage Pricing Le us consider a wo period marke model. A conrac is defined by a sochasic payoff X a bounded sochasic

More information

CHAPTER 3 How to Calculate Present Values. Answers to Practice Questions

CHAPTER 3 How to Calculate Present Values. Answers to Practice Questions CHAPTER 3 How o Calculae Presen Values Answers o Pracice Quesions. a. PV $00/.0 0 $90.53 b. PV $00/.3 0 $9.46 c. PV $00/.5 5 $ 3.5 d. PV $00/. + $00/. + $00/. 3 $40.8. a. DF + r 0.905 r 0.050 0.50% b.

More information

Supplement to Chapter 3

Supplement 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 information

FIXED INCOME MICHAEL MONOYIOS

FIXED INCOME MICHAEL MONOYIOS FIXED INCOME MICHAEL MONOYIOS Absrac. The course examines ineres rae or fixed income markes and producs. These markes are much larger, in erms of raded volume and value, han equiy markes. We firs inroduce

More information

Brownian motion. Since σ is not random, we can conclude from Example sheet 3, Problem 1, that

Brownian motion. Since σ is not random, we can conclude from Example sheet 3, Problem 1, that Advanced Financial Models Example shee 4 - Michaelmas 8 Michael Tehranchi Problem. (Hull Whie exension of Black Scholes) Consider a marke wih consan ineres rae r and wih a sock price modelled as d = (µ

More information

Erratic Price, Smooth Dividend. Variance Bounds. Present Value. Ex Post Rational Price. Standard and Poor s Composite Stock-Price Index

Erratic 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 information

1. Interest Rate Gap. Duration

1. 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

FAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004

FAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004 FAIR VALUATION OF INSURANCE LIABILITIES Pierre DEVOLDER Universié Caholique de Louvain 03/ 09/004 Fair value of insurance liabiliies. INTRODUCTION TO FAIR VALUE. RISK NEUTRAL PRICING AND DEFLATORS 3. EXAMPLES

More information

Basic Economic Scenario Generator: Technical Specications. Jean-Charles CROIX ISFA - Université Lyon 1

Basic 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 information

INFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES.

INFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES. INFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES. Join work wih Ying JIAO, LPMA, Universié Paris VII 6h World Congress of he Bachelier Finance Sociey, June 24, 2010. This research is par of he Chair

More information

Lecture: Autonomous Financing and Financing Based on Market Values I

Lecture: Autonomous Financing and Financing Based on Market Values I Lecure: Auonomous Financing and Financing Based on Marke Values I Luz Kruschwiz & Andreas Löffler Discouned Cash Flow, Secion 2.3, 2.4.1 2.4.3, Ouline 2.3 Auonomous financing 2.4 Financing based on marke

More information

CURRENCY TRANSLATED OPTIONS

CURRENCY TRANSLATED OPTIONS CURRENCY RANSLAED OPIONS Dr. Rober ompkins, Ph.D. Universiy Dozen, Vienna Universiy of echnology * Deparmen of Finance, Insiue for Advanced Sudies Mag. José Carlos Wong Deparmen of Finance, Insiue for

More information

Tentamen i 5B1575 Finansiella Derivat. Torsdag 25 augusti 2005 kl

Tentamen 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 information

Macroeconomics. Part 3 Macroeconomics of Financial Markets. Lecture 8 Investment: basic concepts

Macroeconomics. Part 3 Macroeconomics of Financial Markets. Lecture 8 Investment: basic concepts Macroeconomics Par 3 Macroeconomics of Financial Markes Lecure 8 Invesmen: basic conceps Moivaion General equilibrium Ramsey and OLG models have very simple assumpions ha invesmen ino producion capial

More information

CHAPTER CHAPTER18. Openness in Goods. and Financial Markets. Openness in Goods, and Financial Markets. Openness in Goods,

CHAPTER CHAPTER18. Openness in Goods. and Financial Markets. Openness in Goods, and Financial Markets. Openness in Goods, Openness in Goods and Financial Markes CHAPTER CHAPTER18 Openness in Goods, and Openness has hree disinc dimensions: 1. Openness in goods markes. Free rade resricions include ariffs and quoas. 2. Openness

More information

Elton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 21

Elton, 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 information

LIDSTONE IN THE CONTINUOUS CASE by. Ragnar Norberg

LIDSTONE IN THE CONTINUOUS CASE by. Ragnar Norberg LIDSTONE IN THE CONTINUOUS CASE by Ragnar Norberg Absrac A generalized version of he classical Lidsone heorem, which deals wih he dependency of reserves on echnical basis and conrac erms, is proved in

More information

AMS Q03 Financial Derivatives I

AMS Q03 Financial Derivatives I AMS Q03 Financial Derivaives I Class 08 Chaper 3 Rober J. Frey Research Professor Sony Brook Universiy, Applied Mahemaics and Saisics frey@ams.sunysb.edu Lecure noes for Class 8 wih maerial drawn mainly

More information

The macroeconomic effects of fiscal policy in Greece

The macroeconomic effects of fiscal policy in Greece The macroeconomic effecs of fiscal policy in Greece Dimiris Papageorgiou Economic Research Deparmen, Bank of Greece Naional and Kapodisrian Universiy of Ahens May 22, 23 Email: dpapag@aueb.gr, and DPapageorgiou@bankofgreece.gr.

More information

Econ 546 Lecture 4. The Basic New Keynesian Model Michael Devereux January 2011

Econ 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 information

Analyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective

Analyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive AFIR Colloquium Madrid, Spain June 22, 2 Alexander Bohner and Nadine Gazer Universiy

More information

Fundamental Basic. Fundamentals. Fundamental PV Principle. Time Value of Money. Fundamental. Chapter 2. How to Calculate Present Values

Fundamental Basic. Fundamentals. Fundamental PV Principle. Time Value of Money. Fundamental. Chapter 2. How to Calculate Present Values McGraw-Hill/Irwin Chaper 2 How o Calculae Presen Values Principles of Corporae Finance Tenh Ediion Slides by Mahew Will And Bo Sjö 22 Copyrigh 2 by he McGraw-Hill Companies, Inc. All righs reserved. Fundamenal

More information

Interest Rate Products

Interest Rate Products Chaper 9 Ineres Rae Producs Copyrigh c 2008 20 Hyeong In Choi, All righs reserved. 9. Change of Numeraire and he Invariance of Risk Neural Valuaion The financial heory we have developed so far depends

More information

Valuing Real Options on Oil & Gas Exploration & Production Projects

Valuing Real Options on Oil & Gas Exploration & Production Projects Valuing Real Opions on Oil & Gas Exploraion & Producion Projecs March 2, 2006 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion 2. Wha

More information

Proceedings of the 48th European Study Group Mathematics with Industry 1

Proceedings of the 48th European Study Group Mathematics with Industry 1 Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 1 ADR Opion Trading Jasper Anderluh and Hans van der Weide TU Delf, EWI (DIAM), Mekelweg 4, 2628 CD Delf jhmanderluh@ewiudelfnl, JAMvanderWeide@ewiudelfnl

More information

Pricing and hedging contingent claims using variance and higher-order moment futures

Pricing and hedging contingent claims using variance and higher-order moment futures Pricing and hedging coningen claims using variance and higher-order momen fuures by Leonidas S. Rompolis * and Elias Tzavalis * Absrac This paper suggess perfec hedging sraegies of coningen claims under

More information

HEDGING VOLATILITY RISK

HEDGING VOLATILITY RISK HEDGING VOLAILIY RISK Menachem Brenner Sern School of Business New York Universiy New York, NY 00, U.S.A. Email: mbrenner@sern.nyu.edu Ernes Y. Ou ABN AMRO, Inc. Chicago, IL 60604, U.S.A. Email: Yi.Ou@abnamro.com

More information

HEDGING SYSTEMATIC MORTALITY RISK WITH MORTALITY DERIVATIVES

HEDGING SYSTEMATIC MORTALITY RISK WITH MORTALITY DERIVATIVES HEDGING SYSTEMATIC MORTALITY RISK WITH MORTALITY DERIVATIVES Workshop on moraliy and longeviy, Hannover, April 20, 2012 Thomas Møller, Chief Analys, Acuarial Innovaion OUTLINE Inroducion Moraliy risk managemen

More information

Mathematical methods for finance (preparatory course) Simple numerical examples on bond basics

Mathematical methods for finance (preparatory course) Simple numerical examples on bond basics Mahemaical mehods for finance (preparaory course) Simple numerical examples on bond basics . Yield o mauriy for a zero coupon bond = 99.45 = 92 days (=0.252 yrs) Face value = 00 r 365 00 00 92 99.45 2.22%

More information

Technological progress breakthrough inventions. Dr hab. Joanna Siwińska-Gorzelak

Technological progress breakthrough inventions. Dr hab. Joanna Siwińska-Gorzelak Technological progress breakhrough invenions Dr hab. Joanna Siwińska-Gorzelak Inroducion Afer The Economis : Solow has shown, ha accumulaion of capial alone canno yield lasing progress. Wha can? Anyhing

More information

Adding and Subtracting Black-Scholes: A New Approach to Approximating Derivative Prices in Continuous-Time Models

Adding and Subtracting Black-Scholes: A New Approach to Approximating Derivative Prices in Continuous-Time Models Adding and Subracing Black-Scholes: A New Approach o Approximaing Derivaive Prices in Coninuous-Time Models Dennis Krisensen Columbia Universiy and CREATES Anonio Mele London School of Economics Firs draf:

More information

Final Exam Answers Exchange Rate Economics

Final Exam Answers Exchange Rate Economics Kiel Insiu für Welwirhschaf Advanced Sudies in Inernaional Economic Policy Research Spring 2005 Menzie D. Chinn Final Exam Answers Exchange Rae Economics This exam is 1 ½ hours long. Answer all quesions.

More information

Supplement to Models for Quantifying Risk, 5 th Edition Cunningham, Herzog, and London

Supplement to Models for Quantifying Risk, 5 th Edition Cunningham, Herzog, and London Supplemen o Models for Quanifying Risk, 5 h Ediion Cunningham, Herzog, and London We have received inpu ha our ex is no always clear abou he disincion beween a full gross premium and an expense augmened

More information

DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Krzysztof Jajuga Wrocław University of Economics

DYNAMIC 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 information

The Market for Volatility Trading; VIX Futures

The Market for Volatility Trading; VIX Futures he Marke for olailiy rading; IX uures Menachem Brenner ern chool of Business New York Universiy New York, NY, U..A. Email: mbrenner@sern.nyu.edu el: 998 33, ax: 995 473 Jinghong hu chool of Inernaional

More information

AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES

AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES Inernaional Journal of Pure and Applied Mahemaics Volume 76 No. 4 212, 549-557 ISSN: 1311-88 (prined version url: hp://www.ijpam.eu PA ijpam.eu AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM

More information

A Method for Estimating the Change in Terminal Value Required to Increase IRR

A Method for Estimating the Change in Terminal Value Required to Increase IRR A Mehod for Esimaing he Change in Terminal Value Required o Increase IRR Ausin M. Long, III, MPA, CPA, JD * Alignmen Capial Group 11940 Jollyville Road Suie 330-N Ausin, TX 78759 512-506-8299 (Phone) 512-996-0970

More information

The intersection of market and credit risk q

The intersection of market and credit risk q Journal of Banking & Finance 24 (2000) 271±299 www.elsevier.com/locae/econbase The inersecion of marke and credi risk q Rober A. Jarrow a,1, Suar M. Turnbull b, * a Johnson Graduae School of Managemen,

More information

Financial Econometrics (FinMetrics02) Returns, Yields, Compounding, and Horizon

Financial Econometrics (FinMetrics02) Returns, Yields, Compounding, and Horizon Financial Economerics FinMerics02) Reurns, Yields, Compounding, and Horizon Nelson Mark Universiy of Nore Dame Fall 2017 Augus 30, 2017 1 Conceps o cover Yields o mauriy) Holding period) reurns Compounding

More information

CHAPTER CHAPTER26. Fiscal Policy: A Summing Up. Prepared by: Fernando Quijano and Yvonn Quijano

CHAPTER CHAPTER26. Fiscal Policy: A Summing Up. Prepared by: Fernando Quijano and Yvonn Quijano Fiscal Policy: A Summing Up Prepared by: Fernando Quijano and vonn Quijano CHAPTER CHAPTER26 2006 Prenice Hall usiness Publishing Macroeconomics, 4/e Olivier lanchard Chaper 26: Fiscal Policy: A Summing

More information

Parameters of the IRB Approach. 1. Class of exposures to central governments and central banks, exposures to institutions or corporate exposures

Parameters of the IRB Approach. 1. Class of exposures to central governments and central banks, exposures to institutions or corporate exposures Annex 13 Parameers of he IRB Approach I. The PD value 1. Class of exposures o cenral governmens and cenral bans, exposures o insiuions or corporae exposures a) The PD value for an exposure o an insiuion

More information

1 Purpose of the paper

1 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 information

Provide a brief review of futures markets. Carefully review alternative market conditions and which marketing

Provide a brief review of futures markets. Carefully review alternative market conditions and which marketing Provide a brief review of fuures markes. Carefully review alernaive marke condiions and which markeing sraegies work bes under alernaive condiions. Have an open and ineracive discussion!! 1. Sore or Wai

More information

EQUILIBRIUM ASSET PRICING MODELS

EQUILIBRIUM ASSET PRICING MODELS EQUILIBRIUM ASSET PRICING MODELS 2 Asse pricing derived rom heory o consumpion and invesmen behavior 2 Pricing equaions oen ake he orm o PV models: 4 Asse value equals expeced sum o discouned uure CFs

More information

HEDGING VOLATILITY RISK

HEDGING VOLATILITY RISK HEDGING VOLAILIY RISK Menachem Brenner Sern School of Business New York Universiy New York, NY 00, U.S.A. Email: mbrenner@sern.nyu.edu el: 998 033 Fax: 995 473 Ernes Y. Ou Archeus Capial Managemen New

More information

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE OF ACTUARIES OF INDIA INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 9 h November 2010 Subjec CT6 Saisical Mehods Time allowed: Three Hours (10.00 13.00 Hrs.) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1. Please read he insrucions

More information

Systemic Risk Illustrated

Systemic Risk Illustrated Sysemic Risk Illusraed Jean-Pierre Fouque Li-Hsien Sun March 2, 22 Absrac We sudy he behavior of diffusions coupled hrough heir drifs in a way ha each componen mean-revers o he mean of he ensemble. In

More information

Market and Information Economics

Market and Information Economics Marke and Informaion Economics Preliminary Examinaion Deparmen of Agriculural Economics Texas A&M Universiy May 2015 Insrucions: This examinaion consiss of six quesions. You mus answer he firs quesion

More information