

 Ethan Rich
 2 years ago
 Views:
Transcription
1 Pricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Homepage: Firs draf: April 998 I am graeful o Peer Honore for helpful commens.
2 he problem Consider a xedincome derivaive wih a single payo a ime which depends on he erm srucure. In paricular, we will look a opions on zerocoupon bonds and ineresrae caps. For a call opion on a zerocoupon bond mauring a ime, he ime payo and hence value of he derivaive isgiven by V max P (; ), K; : () By he noarbirage heorem, he price oday ( ) is V E e, R Q r sds V ; () where he expecaion is aken under he riskneural disribuion (also called he Qmeasure). hus, he price depends on he sochasic process for he shor rae and he conracual specicaion of he securiy (i.e., how he payo is linked o he erm srucure). he price V in equaion () is given by he expecaion of he produc of wo dependen random variables, and calculaing his expecaion is ofen quie dicul. he purpose of his noe is presening a changeofmeasure echnique which considerably simplies he evaluaion of V. Specically, weare going o calculae V as V P (;)E Q (V ); (3) where Q is a new probabiliy measure (disribuion), he socalled forwardrisk adjused measure. his echnique was inroduced in he xedincome lieraure by Jamshidian (99). Model seup and noaion Our ermsrucure model is a general onefacor HJM model, see Heah, Jarrow and Moron (99) or Lund (998) for an exposiion. Under he Qmeasure, forward raes are governed by where df(; ),(; ) P (; )d + (; )dw Q ; (4) Z P (; ), (; u)du: (5) Bond prices evolve according o he SDE dp (; )r P(; )d + P (; )P (; )dw Q ; (6) so P (; )isheime volailiy of he zero mauring a ime.
3 3 he forwardrisk adjused measure Under cerain regulariy condiions, he price of he derivaive securiy follows he SDE dv r V d + V ()V dw Q : (7) his means ha, under he riskneural disribuion, he expeced rae of reurn equals he shor rae (jus like any oher securiy), and he reurn volailiy is V (). So far, neiher V nor V () are known, bu his is no essenial for he following argumens. In fac, he only hing ha maers is ha he price process has he form (7) since his faciliaes pricing by he forwardrisk adjused measure. We begin by dening he deaed price process F V P(; ) (8) for [;]. We can inerpre F as he price of V in unis of he mauriy bond price (i.e., as a relaive price). Using Io's lemma, i can be shown ha df P ( P, V )F d +( V, P )F dw Q (9) where P and V are shorhand noaion for P (; ) and V (), respecively. he proof of (9) is given in appendix A. Furhermore, we dene a new probabiliy measure, Q, such ha W Q W Q, Z P (u; )du; [;]; () is a Brownian moion under Q. and W Q is In dierenial form he relaionship beween W Q dw Q dw Q, P (; )d dw Q, P d: () he new probabiliy measure is known as he forwardrisk adjused measure. I is very imporan o noe ha here is a dieren measure for each (payo dae). If we subsiue () ino (9), we obain he dynamics of F under he new probabiliy measure Q. Sraighforward calculaions give df, P ( V, P )F d +( V, P )F dw Q + P d ( V, P )F dw Q ; (3) We have, implicily, used a similar echnique when dening he riskneural measure earlier. Specically, ifw is a Brownian moion under he original (rue) probabiliy measure, we dene Q such ha W Q W Q +Z (u)du () is a Brownian moion under Q. Noe ha (u) is he marke price.
4 as he wo erms wih d cancel ou. hus under Q, he drif is zero and F is a maringale. he new probabiliy measure was dened in order o obain his resul since he maringale propery implies ha F E Q (F ): (4) Moreover, by deniion P (;), so a mauriy we have F V, and using (4), he curren ( ) price of he derivaive securiy can now be calculaed as V P (;)F P(;)E Q (F ) P(;)E Q (V ); (5) which isp(;) imes he expeced payo under Q. Generally, he laer calculaion is a lo simpler han direc evaluaion of he expecaion under Q, as in equaion () above. Wih (5) a hand, he only remaining ask is deermining he disribuion of he payo under he forwardrisk adjused measure. We conclude his secion by noing ha f(; ) is a maringale under Q. o see his, subsiue () ino he forwardrae SDE (4), df(; ),(; ) P (; )d + (; )dw Q + P (; )d (; )dw Q : (6) his propery urns ou be very useful when pricing ahemoney ineresrae caps, cf. he second example in he nex secion. 4 wo examples For concreeness, we use he exended Vasicek model which is a special case of he onefacor HJM model wih and (; )e,(, ) ; (7) P (; ),Z (; u)du e,(, ), : (8) he exended Vasicek model is a Markovian HJM model, cf. Lund (998), bu he following pricing formulas for bond opions [equaion (3)] and ineresrae caps [equaion (4)] do no depend on he Markov propery. Noe ha we are using he maringale propery in he \opposie" direcion (i.e., backwards) in equaions (4) and (5). Normally, we know F and use he maringale propery ocompue he expeced value a ime. his line of reasoning is implici in he weak form of marke eciency [Fama (97)] where we argue ha he bes forecas of he fuure sock price is he sock price oday. In he conex of pricing derivaives, we use our knowledge abou he Q disribuion of he payo V, combined wih he maringale propery off (he relaive price), o compue he curren (relaive) price of he derivaive securiy. 3
5 4. Call opion on a zerocoupon bond In he rs example, he xedincome derivaive is a call opion on a zerocoupon bond mauriy a ime. he opion expires (maures) a ime < wih he following payo: C max ; (9) P (; ), K; where K is he srike (exercise) price of he opion. In order o price his securiy, we need he disribuion of C under he forwardrisk adjused measure. Since C only depends P (; ) and since P (;), we can calculae he expecaion of C from he disribuion of he relaive price, F (; ; )P(; )P (; ); () which is also he forward price of he mauriy bond for delivery a ime. Using he resuls of secion 3, he SDE for F (; ; ) under Q is given by df (; ; ) f P (; ), P (; )g F (; ; )dw Q F (; ; )F (; ; )dw Q : () Since bond prices are always sricly posiive, he logarihm of F (; ; )iswelldened, and a simple applicaion of Io's lemma gives d log F (; ; ), F (; ; )d + F (; ; )dw Q : () Afer inegraing from o we have Z log F (;; ) log P (; ) log F (;; ), (; ; F )d +Z F (; ; )dw Q : (3) he rs equaliy in (3) follows because P (;). Moreover, if F (; ; ) is deerminisic, i.e. if he model is Gaussian, i follows from (3) ha log P (; ) is condiionally normally disribued wih variance! F (; ) Z and mean F (; ) log F (;; ), F (; ; )d; (4) Z F (; ; )d log F (;; ), (;! F ): (5) For he exended Vasicek model, F (; ; )isgiven by F (; ; ) e,(, ), ; (6) ), e,(, ) e,(, ) e,(, 4
6 cf. equaion (8), and he variance! F (; ) can be calculaed as! F (; ) e,(, ), e,(, ), A e,(, ) e, B (, )Var Q (r ); (7) since he las parenhesis in he second line can be recognized as he condiional variance of r, see equaion (4) in Jamshidian (989). he funcion B() is he \facor loading" (sochasic duraion) for he Vasicek model. 3 Finally, he price of he call opion, denoed C(;K), is given by: C(;K) P(;)E (C ) P(;)Z log K P (;)Z log K,P (;)KZ e x, K A p!f e,(x, F )! F dx e x p e,(x, F )! F dx!f log K p!f e,(x, F )! F dx P (; )N(d ),P(;)KN(d ); (3) where N() is he cumulaive normal disribuion funcion,! F is shorhand noaion for!(; ), and! d log P (; ) P(;),log K +! F! F (3) d d,! F : (3) he calculaion is compleely analogous o he BlackScholes model for call opions on sock prices, so we skip he inermediae seps leading o he nal expression for C(;K) in equaion (3). 4 3 An alernaive derivaion for he exended Vasicek model can be based on he formula log P (; ) A(; )+B(,)r ; (8) cf. Lund (998), and since A(; ) is deerminisic, he variance of log P (; ) under Q is Var Q (log P (; )) B (, ) Var Q (r ): (9) Of course, wih his approach wewould sill need o deermine he variance of r. 4 Hin for your own derivaion: if x is N (; ), he runcaed mean of exp(x) is Z L e x p e,(x, ) dx 5
7 4. Ineresrae caps Consider a derivaive wih he following payo a ime : C max(r, K; ): (34) his corresponds o a simple ineresrae cap. 5 by: he price (oday) of he cap is given C(;K) P(;)E Q (C ) P(;)E Q [max(r, K; )] : (35) In order o calculae (35) we mus deermine he disribuion of r under Q (he forwardrisk adjused measure). Firs, noe ha r f(;); (36) so we can obain he disribuion of r from f(;). Second, under Q he mauriy forward rae is a maringale, as shown in equaion (6) in secion 3. his means ha r f(;) f(;)+z (; )dw Q : (37) If (; ) is deerminisic, r is condiionally normally disribued (a ime ) wih mean f(;) and variance Var Q (r ) Z (; )d v (;): (38) For he exended Vasicek model, his becomes Z v (;) e,(,), e, d : (39) Z L exp + p e +, (x,, ) dx,l +(+ ) N since he inegrand in he second line equals exp, + ; (33) imes he densiy funcion of a normal disribuion wih mean + and variance. 5 In he real world, caps are more complicaed. he underlying ineres rae is no he shor rae, bu (say) he hreemonh (LIBOR) ineres rae. Moreover, a cap conrac for years on he hreemonh rae is a porfolio of 4 socalled caples (singlepaymen caps), and he paymen of he i'h caple is :5 max[r 3M ((i, )4), K; )], where R 3M () is he hreemonh ineres rae a ime. he paymens are made in arrear, which means ha he i'h paymen is made a ime i4 (hree monhs afer he xing dae). Of course, he \real world" cap can be priced by he same principles as he simple cap described in his secion, ha is by a suiable applicaion of he he forwardrisk adjused measure for each caple. Needless o say, he algebra become more involved, bu ha is he only real dierence. 6
8 Finally, we compue he expeced payo under Q and hence he price of he cap. For reason of space, we concenrae on he ahemoney cap 6 where K f(;), and C(;f(;)) P (;)E Q [max(r, f(;);)] P (;) v(;) p ; (4) where v(;) is dened in (38) for any Gaussian onefacor HJM model, and in (39) for he exended Vasicek model. he second line in (4) follows by noing ha he payo can be wrien as max(r, f(;);) max Z (; )dw Q ;! (4) sz which has he same disribuion as (; )d max(x; ) v(;) max(x; ); (4) where x is N (; ), ha is normally disribued wih zero mean and uni variance. he expeced value of max(x; ) is given by E[max(x; )] Z Z x p, e x dx p e,u du p : (43) he second line follows by a change of variables from x o u x. his complees he proof of (4). 6 he price formula for a cap wih arbirary exercise price, K, (in he exended Vasicek model) can be found in Longsa (995). 7
9 Appendix A: proof of equaion (9) o simplify he noaion, we wrie he SDE for V subscrips, and P (; ) wihou he ime dv rv d + V VdW Q (44) dp rpd + P PdW Q : (45) Noe ha since V and P are driven by he same Brownian moion, changes in V and P are perfecly correlaed. he objecive is o deermine he SDE for he funcion F (V; P) VP. Firs, we compue he requisie parial derivaives of F wih respec o V and P P F (V; F (V; V P 3 F (V; P) P : (5) Second, an applicaion of Io's lemma gives us df P rv, V P rp + V P 3 P P, P F, V P F P V V, V P P P P V V P P d + dw Q (5) d +( V F, P F)dW Q (5) P ( P, V )Fd+( V, P )FdW Q (53) which is equaion (9) in secion 3. his complees he proof. 8
10 References Fama, E.F. (97), \Ecien Capial Markes: A Review of heory and Empirical ess," Journal of Finance, 5, 383{47. Heah, D., R. Jarrow and A. Moron (99), \Bond Pricing and he erm Srucure of Ineres Raes," Economerica, 6, 77{5. Jamshidian, F. (989), \An Exac Bond Opion Formula," Journal of Finance, 44, 5{9. Jamshidian, F. (99), \Bond and Opion Evaluaion in he Gaussian Ineres Rae Model," Research in Finance, 9, 3{7. Longsa, F.A. (995), \Hedging Ineres Rae Risk wih Opions on Average Ineres Raes," Journal of Fixed Income, March 995, 37{45. Lund, J. (998), \Review of Coninuousime ermsrucure Models Par II: ArbirageFree Models," Lecure Noes, Deparmen of Finance, Aarhus School of Business, April
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 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationThe 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 informationStochastic Calculus, Week 10. Definitions and Notation. TermStructure Models & Interest Rate Derivatives
Sochasic Calculus, Week 10 TermSrucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Termsrucure models 3. Ineres rae derivaives Definiions and Noaion Zerocoupon
More informationDuration 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 UVAF38 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 informationMTH6121 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= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationStochastic 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 information12. Market LIBOR Models
12. Marke LIBOR Models As was menioned already, he acronym LIBOR sands for he London Inerbank Offered Rae. I is he rae of ineres offered by banks on deposis from oher banks in eurocurrency markes. Also,
More informationModeling 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 informationForeign Exchange and Quantos
IEOR E4707: Financial Engineering: ConinuousTime 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 informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationValuation 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 informationT ϕ t ds t + ψ t db t,
16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationModelling of Forward Libor and Swap Rates
Modelling of Forward Libor and Swap Raes Marek Rukowski Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology, 661 Warszawa, Poland Conens 1 Inroducion 2 2 Modelling of Forward Libor
More informationA 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, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationPRICING 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 informationChapter 6 Interest Rates and Bond Valuation
Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Longerm debloosely, bonds wih a mauriy of one year or more Shorerm debless han a year o mauriy, also called unfunded deb Bondsricly
More informationHedging 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 buyside of a forward/fuures
More informationVariance Swap. by Fabrice Douglas Rouah
Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com 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
More informationEuropean option prices are a good sanity check when analysing bonds with exotic embedded options.
European opion prices are a good saniy check when analysing bonds wih exoic embedded opions. I s an old exam quesion. Arbiragefree economy where ZCB prices are driven 1D BM, i.e. dp (, T ) = r()p (,
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationPROFIT 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 informationStochastic 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 BlackScholes
More informationRepresenting 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 informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationJournal 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 YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationConceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...
Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The BlackScholes Shl Ml Moel... pricing opions an calculaing
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationThe yield curve, and spot and forward interest rates Moorad Choudhry
he yield curve, and spo and forward ineres raes Moorad Choudhry In his primer we consider he zerocoupon or spo ineres rae and he forward rae. We also look a he yield curve. Invesors consider a bond yield
More information2.5 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 informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationDYNAMIC 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 informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationEURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONEFACTOR MODEL
EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONEFACTOR MODEL MARC HENRARD Absrac. In his noe we give pricing formlas for differen insrmens linked o rae fres erodollar fres. We provide
More informationInductance 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 informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\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 informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationA 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 informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationarxiv:submit/1578408 [qfin.pr] 3 Jun 2016
Derivaive pricing for a mulicurve exension of he Gaussian, exponenially quadraic shor rae model Zorana Grbac and Laura Meneghello and Wolfgang J. Runggaldier arxiv:submi/578408 [qfin.pr] 3 Jun 206 Absrac
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationOptimal 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 informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationDependent 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 information4. 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 informationSkewness and Kurtosis Adjusted BlackScholes Model: A Note on Hedging Performance
Finance Leers, 003, (5), 6 Skewness and Kurosis Adjused BlackScholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and BlackMeronScholes
More informationStochastic Volatility Models: Considerations for the Lay Actuary 1. Abstract
Sochasic Volailiy Models: Consideraions for he Lay Acuary 1 Phil Jouber Coomaren Vencaasawmy (Presened o he Finance & Invesmen Conference, 191 June 005) Absrac Sochasic models for asse prices processes
More informationNikkei Stock Average Volatility Index Realtime Version Index Guidebook
Nikkei Sock Average Volailiy Index Realime Version Index Guidebook Nikkei Inc. Wih he modificaion of he mehodology of he Nikkei Sock Average Volailiy Index as Nikkei Inc. (Nikkei) sars calculaing and
More informationA 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 informationINVESTMENT GUARANTEES IN UNITLINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE
INVESMEN UARANEES IN UNILINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN
More informationYTM 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 informationValuation of Credit Default Swaptions and Credit Default Index Swaptions
Credi Defaul Swapions Valuaion of Credi Defaul Swapions and Marek Rukowski School of Mahemaics and Saisics Universiy of New Souh Wales Sydney, Ausralia Recen Advances in he Theory and Pracice of Credi
More informationPrincipal 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 informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationWHAT 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 informationRelative 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 informationChapter 9 Bond Prices and Yield
Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value
More informationPRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES
PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN
More informationDynamic Option Adjusted Spread and the Value of Mortgage Backed Securities
Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder 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 informationCredit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006
i Credi risk T. Bielecki, M. Jeanblanc and M. Rukowski Lecure of M. Jeanblanc Preliminary Version LISBONN JUNE 26 ii Conens Noaion vii 1 Srucural Approach 3 1.1 Basic Assumpions.....................................
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler 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 informationPricing BlackScholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension
Pricing Blackcholes Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zulang Liao a, and HsingHua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationThe 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 informationOn the paper Is Itô calculus oversold? by A. Izmailov and B. Shay
On he paper Is Iô calculus oversold? by A. Izmailov and B. Shay M. Rukowski and W. Szazschneider March, 1999 The main message of he paper Is Iô calculus oversold? by A. Izmailov and B. Shay is, we quoe:
More informationAn Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price
An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationFORWARD AND FUTURES CONTRACTS
Page1 C H A P T E R 2 FORWARD AND FUTURES CONTRACTS 2.1 INTRODUCTION The main purpose of forward and fuures conracs is he managemen of risk. The exposure o risk as a resul of ransacing in he spo marke
More informationTable 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 informationTimeinhomogeneous Lévy Processes in CrossCurrency Market Models
Timeinhomogeneous Lévy Processes in CrossCurrency Marke Models Disseraion zur Erlangung des Dokorgrades der Mahemaischen Fakulä der AlberLudwigsUniversiä Freiburg i. Brsg. vorgeleg von Naaliya Koval
More informationRandom Walk in 1D. 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 informationFX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI QPHI AND STOCHASTIC QPHI MODELS
FX OPTION PRICING: REULT FROM BLACK CHOLE, LOCAL VOL, QUAI QPHI AND TOCHATIC QPHI MODEL Absrac Krishnamurhy Vaidyanahan 1 The paper suggess a new class of models (QPhi) o capure he informaion ha he
More informationOpenGamma Quantitative Research Multicurves: Variations on a Theme
OpenGamma Quaniaive Research Mulicurves: Variaions on a Theme Marc Henrard marc@opengamma.com OpenGamma Quaniaive Research n. 6 Ocober 2012 Absrac The mulicurves framework is ofen implemened in a way
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include socalled implici or embedded opions.
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationAn accurate analytical approximation for the price of a Europeanstyle arithmetic Asian option
An accurae analyical approximaion for he price of a Europeansyle arihmeic Asian opion David Vyncke 1, Marc Goovaers 2, Jan Dhaene 2 Absrac For discree arihmeic Asian opions he payoff depends on he price
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationTechnical 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 informationOptimalCompensationwithHiddenAction and LumpSum Payment in a ContinuousTime Model
Appl Mah Opim (9) 59: 99 46 DOI.7/s45895 OpimalCompensaionwihHiddenAcion and LumpSum Paymen in a ConinuousTime Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationIntroduction to Arbitrage Pricing
Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, 661 Warszawa, Poland
More informationPricing Futures and Futures Options with Basis Risk
Pricing uures and uures Opions wih Basis Risk ChouWen ang Assisan professor in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan TingYi Wu PhD candidae in
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationEfficient Pricing of Energy Derivatives
Efficien Pricing of Energy Derivaives Anders B. Trolle EPFL and Swiss Finance Insiue March 1, 2014 Absrac I presen a racable framework, firs developed in Trolle and Schwarz (2009), for pricing energy derivaives
More informationNew Pricing Framework: Options and Bonds
arxiv:1407.445v [qfin.pr] 14 Oc 014 New Pricing Framework: Opions and Bonds Nick Laskin TopQuark Inc. Torono, ON, M6P P Absrac A unified analyical pricing framework wih involvemen of he sho noise random
More informationResearch 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 informationA Simple Approach to CAPM, Option Pricing and Asset Valuation
A imple Approach o CAPM, Opion Pricing and Asse Valuaion Riccardo Cesari (*) Universià di Bologna, Dip. Maemaes, viale Filopani, 5 406 Bologna, Ialy Email: rcesari@economia.unibo.i Carlo D Adda Universià
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More information