Pricing corporate bonds, CDS and options on CDS with the BMC model
|
|
- Martin Cain
- 5 years ago
- Views:
Transcription
1 Pricing corporae bonds, CDS and opions on CDS wih he BMC model D. Bloch Universié Paris VI, France Absrac Academics have always occuled he calibraion and hedging of exoic credi producs assuming ha credi models could be calibraed on vanilla producs. However, in mos markes one can only observe he five year CDS, forcing praciioners o make guesses and ignore he risk of defaul. We choose o address he calibraion and hedging of exoic credi producs by relaing he credi spread o he equiy volailiy surface in an affine model. We briefly describe a jump-diffusion model wih local inensiy funcion of ime and of he sock price. A change of measure using he cumulaive survival probabiliy is defined o simplify calculaion. We hen use i o price corporae bond and CDS prices and show ha we ge closed form soluions. We hen exend he approach o price opions on CDS. Calibraion of he model parameers o liquid credi and equiy informaion is discussed. 1 Inroducion We consider a financial marke made of asses whose prices are sochasic process (S(), 0) on a probabiliy space (Ω, F, P) where F is a righ coninuous filraion including all P negligible ses in F. F is he agens informaion se a ime corresponding o he knowledge of he marke prices up o ha dae, ha is F = σ(s(s),s ). We define he defaul asse such ha if defaul occur before mauriy hen paymen is no made. Le τ be a sopping ime and denoe he defaul probabiliy F () =P [τ <] wih mass funcion f() = F (). The survival probabiliy up o ime being η() =1 F () hen he mass funcion becomes f() = η().
2 86 Compuaional Finance and is Applicaions Definiion 1.1 We define he hazard rae or he local inensiy as 1 c() = lim P [ τ + d τ ] d 0 d We choose o model he defaul risk explicily by assuming ha he company may defaul a exponenially disribued ime τ wih inensiy parameer c such ha he survival probabiliy is η() =P [τ >]=e c. We define he increasing righ coninuous process of ime o defaul by N = I ( τ ). The process N has saionary and independen incremens and represens he number of jumps ha occurred before ime. We define he maringale process M associaed o he ime o defaul process under he P measure by dm = dn (1 N )c()d such ha he inensiy c() does no have o sop a defaul ime. We suppose ha a defaul ime he agens are informed so ha heir informaion se conains he σ-algebra M = σ(m s,s ). Assumpion 1 We assume ha he filraion G is he augmenaion of he σ- algebra σ(w s,m s ; s ). Therefore, G is he augmened filraion of (S s ; s ) generaed by he prices. Noe ha he filraion M is made of he ses τ s, s and τ>and ha G G, F F, G {τ>} = F {τ>}. More generally, for sochasic hazard rae c() =Λ(X ) where X solve an Io SDE (dx = µ(x )d + σ(x )dw ), if we assume he Brownian process W and he Jump process M o be independen hen he condiional survival probabiliy under he P 1 maringale measure is η(s, ) =P [τ > F s,τ >s]=e[e s Λ(Xu)du F s,τ >s] = E[e s Λ(Xu)du F s ]I {τ>s} Noe, his equaion is compuaionally equivalen o pricing a zero coupon bond, reaing he inensiy c() as he insananeous ineres rae. Assumpion 2 In he res of his aricle we will model he credi spread similarly o he way ineres rae was modelled in a Linear Gaussian model, ha is he sae process X is affine. Our approach is moivaed by he analyical racabiliy and richness of affine sae processes.
3 Compuaional Finance and is Applicaions 87 2 The jump-diffusion model We now incorporae he Jump process M o he underlying sock price S under he P 1 maringale measure. We assume ha he discouned sock price e r S is a maringale and ha i jumps o zero a defaul ime τ ds =(r + c())s d + σ S S dw P 1 S dn for <τ (2.1) Noe, in ha case he jump size is adaped and i is possible o hedge agains defaul wih one exra opion so ha he no-arbirage heory can be applied. Definiion 2.1 By analogy o he fixed income world, we define he price of a credi forward conrac as Ψ (X T )= Π (X T ) 1[e P T ()η T () = EP T r(s)ds e T E P 1 [e T c(s)ds X T F ] r(s)ds e T c(s)ds F ] where X T is a random variable paid a mauriy T. The credi forward conrac is expressed under he credi forward measure Q T,η as Ψ (X T )=E QT,η [X T F ] Due o he naure of he local inensiy and of he spo rae which could boh be sochasic, we made in [1] a change of variable and defined he condiional forward price Y T () in order o simplify numerical calculaion Y T () = S for <τ (2.2) P T ()η T () where he process Y T () seaime for he mauriy T is a maringale under he Q T,η probabiliy measure. This change of variable has for effec of removing he pah dependencies on he ineres rae r and he local inensiy c() from he underlying sock price. Definiion 2.2 We now incorporae he credi spread volailiy, making he local inensiy c() a funcion of ime and of he sock price S c(, S )=γ + φ(, S ) (2.3) Definiion 2.3 By analogy wih he fixed income world, we define he forward hazard rae f(, S,T) and he coninuous rae κ T (, S ) seaime for he mauriy T o saisfy η T () =E P 1 [e T c(u,su)du F ]=e T f(,s,u)du κt (,S)(T ) = e
4 88 Compuaional Finance and is Applicaions Hypoheses 1 We make he hypoheses ha we can choose a form for he local inensiy c(, S ) such ha, once inegraed, we ge back o he running inensiy κ T (, S ) on average. Wih his hypoheses simplificaions can be made and an analogy o he fixed income world defined. As ime o mauriy ends o zero, we recover he hazard rae c(, S ) = f(, S,) = lim T 0 κ T (, S ). Noe, a ime he survival probabiliy depends only on he sock price S. Also we require ha as S 0 hen η T () 0. As an example we defined in [1] a specific form for he Kappa funcion a fixed mauriy T which boh fi he daa and gives us an explici bijecion beween he underlying sock price S and he forward price Y T (). This form of Kappa funcion belongs o he class of affine models κ T (, S )= A(, T ) log(s )+B(, T ) (2.4) Therefore, he forward price Y T () can be re-wrien as Y T () = e P T () S1 A(,T ) (T ) B(,T ) (T ) Noe ha similarly o fixed income, affine models are subjec o negaive hazard rae. However, he probabiliy of geing negaive hazard rae is small. We define he variable X = log(s ) so ha is sochasic differenial equaion under he risk neural probabiliy measure P 1 is dx = ( r + c() 1 ) 2 σ2 S d + σs dw P 1 for <τ We define he log of he survival probabiliy f T (, X ) under he P 1 probabiliy measure as f T (, X ) = log(η T ()) = a(, T )X + b(, T ) for < τ where a(, T )=A(, T )(T ) and b(, T )=B(, T )(T ). 3 The no-arbirage consrain Duffie in [2] showed ha for any affine funcion Λ and any w R E [e s Λ(Xu)+wXsdu ]=e α(s )+β(s )X where coefficiens α(.) and β(.) mus saisfy a generalised Riccai ODE. Seing w =0we ge back o he survival probabiliy. I is equivalen o imposing consrains of No-Arbirage o he survival probabiliy. We consider he pricing equaion a ime 0 of a coningen claim C on a
5 Compuaional Finance and is Applicaions 89 process X under he credi forward measure Q T,η is C 0 = P T (0)η T (0)E QT,η [X T F 0 ]I {τ>0} For X T = 1, ha is paying one a mauriy, we ge C 0 = P T (0)η T (0) or equivalenly C 0 = E P 1[ e 0 r(s)ds e 0 c(s)ds E P 1 [e T r(s)ds e T c(s)ds ] F ] F 0 = P (0)η (0)E Q,η [P T ()η T ()] Equaing he wo prices, we ge he consrain of AOA for deerminisic ineres raes η T (0) = η (0)E Q,η [η T () F ] (3.5) which saes ha he survival probabiliy η T () under he probabiliy measure Q,η mus be a maringale. In order o deermine he consrains imposed by he No-Arbirage condiion, we give an approach which avoid having o calculae a sochasic drif. The variable which we choose o diffuse is he forward price of S P U ()η U () mauriy U, Y U () =. We ake he log of he forward price giving log(y U ()) = (1 a(, U))X + b(, U) log(p U ()) so ha he log of he sock price can be wrien as X = log(y U ()) b(, U) + log(p U ()) 1 a(, U) Now, under he Q U,η probabiliy measure, he diffusion of he forward price is dy U () Y U () = γ Y (, U)σ S dw QU,η where (1 a(, U)) = γ Y (, U). Using he relaion dyu () d log(y U ()) = Y U () 1 2 γ2 Y (, U)σ2 Sd we ge once inegrae beween [0,] he rajecory log(y U ()) = log(y U (0)) 1 γy 2 (s, U)σ 2 Sds 2 + γ Y (s, U)σ S dws QU,η 0 0 (3.6) Now, from he definiion of he log of he survival probabiliy and for U<T, we ge f T (, X )=a(, T )X b(, T ) = a(, T ) [ log(yu ()) b(, U) + log(p U ()) ] b(, T ) 1 a(, U) Seing = U, wegef T (U, X U ) = a(u, T )log(y U (U)) b(u, T ) where log(y U (U)) = log(s U ). We use he previous calculaion on he log of he forward price (3.6) o ge he rajecory of he log of sock process a ime U, haisx U =
6 90 Compuaional Finance and is Applicaions log(y U (U)). We hen plug i back ino he definiion of he survival probabiliy and ge f T (U, X U )=a(u, T ) [ log(y U (0)) U 0 U ] γ Y (s, U)σ S dws QU,η b(u, T ) 0 γ 2 Y (s, U)σ 2 Sds Taking he exponenial of he log of he survival probabiliy which is no a maringale and calculaing is expecaion, we ge E QU,η [η T (U)] = e a(u,t )log(yu (0)) 1 2 a(u,t )γy (U,T ) U 0 γ2 Y (s,u)σ2 S ds b(u,t ). To ge a maringale, we apply he consrain of AOA o he funcion b(u, T ) so ha he survival probabiliy becomes η T (U) = η T (0) 1 η U (0) e 2 a2 (U,T ) U 0 γ2 Y (s,u)σ2 S ds+a(u,t ) ( U 1 a(s,u) )σ 0 SdWs QU,η. We can now apply he AOA equaion (3.5) wih = U o deduce he relaion beween he funcions a(.,.) and b(.,.). Assuming ha we have calibraed he model on oday s CDS curve, hen he values a(0,.) and b(0,.) are known. We are lef wih wo degrees of freedom, ha is he funcions a(u, T ) and b(u, T ),o make he link beween he wo mauriies U and T. + a(u, T )log(y U (0)) 1 2 a(u, T )γ Y (U, T ) U 0 γ2 Y (s, U)σ2 S ds = ( a(0,t) a(0,u) ) log(s 0 ) b(0,t)+b(0,u)+b(u, T ) (3.7) Remark 3.1 Due o he assumpions of non-saionariy of he model, he survival probabiliy η T (U) is no a maringale under he Q U,η measure. Therefore, he funcions a(, T ) and b(, T ) are chosen in such a way as o recover he saionariy, ensuring he maringale propery of he survival probabiliy over ime. 4 The pricing equaion Similarly o he firm model, we assumed ha he price of he corporae bond is a coningen claim on he sock price S. This is because in realiy he probabiliy of defaul is funcion among ohers of he financial healh of he company measured via is raing raio and he underlying sock volailiy. Now he credi informaion embedded in he corporae bond is ransferred ino he underlying sock price. We now consider he probabiliy space (Ω, G, P) where G is he augmened filraion of (S s ; s ) generaed by he prices. The pricing equaion a ime of a
7 Compuaional Finance and is Applicaions 91 coningen claim C on he sock price S under he P 1 -measure is C = E P 1 [e T r(s)ds Φ(S T )I {τ>t} G ]I {τ>} Re-expressing he pricing equaion of a coningen claim C under he Brownian filraion F gives C = E P 1 [e T r(s)ds e T c(s)ds Φ(S T ) F ]I {τ>} Working under he credi forward measure Q T,η and denoing by B T () = P T ()η T () he corporae bond wihou recovery rae, he coningen claim C becomes C = B T ()E QT,η [Φ(S T ) F ]I {τ>} (4.8) We now incorporae he recovery rae in he pricing of he coningen claim. Depending on he erms of he conrac, when defaul occur a new payoff is se. We consider he no necessarily predicable funcion R () o represen he rebae and we define he payoff as φ T = R ()I {τ T } +Φ(S T )I {τ>t} I is equivalen o a payoff wih a barrier having a rebae. Examples of rebae are R =Φ(S 0 ) ˆR or R =Φ(S τ ) ˆR. We can now re-wrie he coningen claim under he P 1 measure and he augmened filraion G as C = E P 1 [e T r(s)ds φ T G ]I {τ>} Again, we work under he Brownian filraion F and expand he expecaion. The coningen claim under he forward probabiliy measure P T,1 is C = P T ()E P T,1 [e T + P T ()E P T,1 [ ( 1 e T c(s)ds Φ(S T ) F ]I {τ>} c(s)ds) R F ]I {τ>} Now, assuming he rebae funcion R () o be predicable, he coningen claim under he credi forward measure Q T,η becomes C = B T ()E QT,η [Φ(S T ) F ]I {τ>} + P T () ˆR ( 1 η T () ) We define he corporae bond B T () wih recovery funcion R () and nominal N, a coningen claim wih payoff saisfying he pricing equaion φ T = R I {τ T } + NI {τ>t} B T () =P T ()η T ()N + P T () ˆR ( 1 η T () ) In general, if assume ha he risk free bond P T ()N is a log-normally disribued radable asse hen he disribuion of he corporae bond BT () is no lognormal and i is no a radable asse. Therefore, we choose no o use he corporae bond BT () as numeraire bu is simplified version B T ().
8 92 Compuaional Finance and is Applicaions 5 Solving he equaion Our aim is now o define under he risk neural probabiliy measure P 1 he dynamic of he corporae bond B T (, S ) which we assumed log-normally disribued, concenraing on he relevan pars ha is is volailiy db T (, S ) B T (, S ) = σ B(, S,T)dW P 1 + ( κ T (, S )+κ T (, S )+r ) d +... The corporae bond B T (, S ) is a funcion of boh he ime and he underlying sock price S where κ T (, S ) is he derivaive of he Kappa funcion wih respec o ime. Applying Io s Lemma on B = B T (, S ),wege db B = ( κ T (, S )+κ T (, S )+r ) d (T ) κ T () S [Sσ sdw P ]+... (5.9) Therefore, he volailiy of he corporae bond reurn is σ B (, S,T)= (T κ )σ s S T () S. We choose o apply Io s lemma on he opion price V = V (, S, B) as a funcion of he sock price S and he corporae bond price B. Now,hewo dimensional PDE over S and B saisfies LV = V (σ s σ B (, S)) 2 S 2 2 V S 2 =0 Doing he change of variable V (, Y T ()) = 1 B V (, S, B) where Y T () = S B T (,S ) is a maringale under he QT,η probabiliy measure, he wo dimensional PDE simplifies o a one dimensional one LV = V σ2 Y Y T () 2 2 V Y 2 =0 (5.10) where σy 2 =(σ s σ B (, Y T ())) 2. For he class of affine models for he Kappa κt (,S) funcion, we ge he derivaive erm S = A(,T ) S so ha he corporae bond volailiy becomes σ B (, S )=σ s (T ) A(, T ) which is a deerminisic funcion. Obviously, he advanage of his form is ha i can be divided by any number and i will remain a deerminisic funcion. The volailiy of he forward price now becomes σ Y = σ s (1 (T ) A(, T )). We can inerpre he parameer A(, T ) as he liaison parameer beween he credi spread and he sock price. I acs as a credi-dela. Inuiively, he closer we ge o mauriy he smaller becomes he effec of he credi spread and in he limi we ge back o he sock price volailiy. Clearly, when we are long he Sock price, hen he credi risk has for effec o lower he volailiy. This is exacly wha we expec of a jump-diffusion process for he underlying variable.
9 Compuaional Finance and is Applicaions 93 6 Pricing CDS Company ABC is long he defaul proecion of mauriy T and pays, condiional on he survival of he reference name, company XYZ fixed paymen X a ime T i,i=1,..., N where N is he number of paymen up o ime T,hais XI {Ti<τ} a T i. We can wrie he fixed leg of he defaul swap as Fixed Leg = X T ()E [ T e u rds P [τ >u]du F ] On he oher hand, company XY Z will pay recovery value R condiional on defaul of he reference name, ha is (1 R)I {<τ <TN } a τ. We can wrie he floa leg of he defaul swap as Floaing Leg = E [ T (1 R)e u rds P [τ du]du F ] The marke quoes when hey exis, a erm srucure of par credi spreads X T j,par for mauriies T j, j =1,..., M. For each par credi spreads X T j,par, here is a lis of fixed paymen daes T i,i=1,..., N. For any choice of he survival funcion η T () = e κt (,S)(T ),hemass funcion becomes κ (, S,T)= κ(,s,t ) T dηt () dt = ( κ(, S,T)+κ (, S,T)(T ) ) η T () where. Therefore, he floaing leg becomes T Floaing Leg =(1 ˆR) P u () ( κ(, S,u)+κ (, S,u)(u ) ) η u ()du so ha he defaul swap rae is X T,par =(1 ˆR) T P u () ( κ(, S,u)+κ (, S,u)(u ) ) η u ()du N i=1 P T i ()η Ti ()(T i T i 1 ) (6.11) We use ha formula o calibrae he model parameers on all he marke quoes of he erm srucure of par credi spreads X T j,par for mauriies T j,j=1,.., M when hey exis. We choose o approximae he inegral in he Floaing Leg wih a sum of P inegrals over he inervals (T i 1,T i ] of lengh δ wih he consrain ha all fixed paymen T i are muliple ineger of ha lengh. Now for T 0 = and T P = T, he par defaul swap rae becomes P Ti X T,par =(1 ˆR) i=1 T i 1 P u () ( κ(, S,u)+κ (, S,u)(u )) ) η u ()du M Ti i=1 T i 1 P u ()η u ()du 7 Opions on CDS In he previous secion we defined in equaion (6.11) he par defaul swap rae X T,par () of mauriy T. We now consider ha Company ABC has he righ bu
10 94 Compuaional Finance and is Applicaions Figure 1: CDS Curve beween 23/03/2003 and 30/03/2008 and he resuling Impled Smile for various mauriies. no he obligaion o ener he forward CDS of mauriy T and par rae TX e a ime T. The payoff which is he value of ha righ a ime T is ( XT,par(T ) TX) N e + P Ti (T )η Ti (T )(T i T i 1 ) i=1 The opion C() on CDS can hen be wrien as C() =E QT,η [ ( X T,par(T ) TX e ) N + P Ti (T )η Ti (T )(T i T i 1 ) F ] Replace he par defaul swap rae X T,par(T ) in he payoff funcion, we ge ((1 ˆR) T T P u(t ) dηu(t ) du du T X e N +.Therefore, he opion on CDS becomes i=1 P T i (T )η Ti (T )(T i T i 1 )) C() =E QT,η[( T ( ˆR 1) P u (T ) dη u(t ) T du du N ) + F ] TX e P Ti (T )η Ti (T )(T i T i 1 ) i=1 i=1 8 Calibraion We illusrae our approach by fiing our model (c.f. Equaion (6.11)) o a downward sloping CDS yield curve calculaed on he 23/03/2003. The high yield curve (c.f. Figure 1) has been chosen o emphasise he impac of he surviving probabiliies and he resuling implied volailiy surface. Using he sock price equaion (2.1) we calibrae he remaining degree of libery σ S o he a-he-money erm srucure of volailiy. Then from he coningen claim (4.8), we ge he equivalen implied volailiy surface. As an example and
11 Compuaional Finance and is Applicaions 95 o simplify inerpreaion, we consider he a-he-money volailiy fla a 37%. We hen plo he skew a various mauriies in (c.f. Figure 1). Obviously he advanage of our model is ha we can generae a full volailiy surface from closed form soluion. 9 Conclusion We have presened an affine sae process for he evoluion of he credi spread and defined a change of measure which allowed us o simplify calculaion. We required for ha model o be calibraed o oday s marke daa, ha is, he clean risky discoun curve and he a-he-money erm srucure of volailiy. Because of he analyical racabiliy of he affine process, we can also calibrae he defaul parameers of he model o he liquid shor mauriies equiy opions, he medium mauriies o boh he corporae bond prices and he quoed par credi swap raes and he longer mauriies o converible bond prices. References [1] Bloch, D.&Miralles, P., Credi reamen in converible bond model. Technical Repor, DrKW, London, [2] D uffie,d.t.&schachermayer, W., Affine processes and applicaions o finance. Working Paper, Graduae School of Business, 2001, Sanford Univer- siy, 2001.
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 informationJarrow-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 informationThe 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 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 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 informationIntroduction 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 informationBlack-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 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 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 informationPricing 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 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 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 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 informationEquivalent 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 informationSynthetic 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 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 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 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 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 informationA UNIFIED PDE MODELLING FOR CVA AND FVA
AWALEE A UNIFIED PDE MODELLING FOR CVA AND FVA By Dongli W JUNE 2016 EDITION AWALEE PRESENTATION Chaper 0 INTRODUCTION The recen finance crisis has released he counerpary risk in he valorizaion of he derivaives
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 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 informationOption 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 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 informationHeath Jarrow Morton Framework
CHAPTER 7 Heah Jarrow Moron Framework 7.1. Heah Jarrow Moron Model Definiion 7.1 (Forward-rae dynamics in he HJM model). In he Heah Jarrow Moron model, brieflyhjm model, he insananeous forward ineres rae
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 informationBrownian 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 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 informationSystemic 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 informationOn multicurve models for the term structure.
On mulicurve models for he erm srucure. Wolfgang Runggaldier Diparimeno di Maemaica, Universià di Padova WQMIF, Zagreb 2014 Inroducion and preliminary remarks Preliminary remarks In he wake of he big crisis
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 informationA 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 informationFINAL 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 informationPricing 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 informationMacroeconomics 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 informationOption 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 informationVALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION
Aca Universiais Mahiae Belii ser. Mahemaics, 16 21, 17 23. Received: 15 June 29, Acceped: 2 February 21. VALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION TOMÁŠ BOKES
More informationIJRSS 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 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 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 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 informationFIXED 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 informationMean Field Games and Systemic Risk
Mean Field Games and Sysemic Risk Jean-Pierre Fouque Universiy of California Sana Barbara Join work wih René Carmona and Li-Hsien Sun Mahemaics for New Economic Thinking INET Workshop a he Fields Insiue
More informationLecture 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 informationAdvanced 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 informationPDE APPROACH TO VALUATION AND HEDGING OF CREDIT DERIVATIVES
PDE APPROACH TO VALUATION AND HEDGING OF CREDIT DERIVATIVES Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6066, USA Monique Jeanblanc Déparemen de Mahémaiques
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 informationCoupling Smiles. November 18, 2006
Coupling Smiles Valdo Durrleman Deparmen of Mahemaics Sanford Universiy Sanford, CA 94305, USA Nicole El Karoui Cenre de Mahémaiques Appliquées Ecole Polyechnique 91128 Palaiseau, France November 18, 2006
More informationInterest 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 informationMulti Currency Credit Default Swaps
Muli Currency Credi Defaul Swaps Quano effecs and FX devaluaion jumps Damiano Brigo Nicola Pede Andrea Perelli arxiv:1512.07256v2 [q-fin.pr] 21 Jan 2018 Firs posed on SSRN and arxiv on December 2015 Second
More informationCredit 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 informationModeling of Tradeable Securities with Dividends
Modeling of Tradeable Securiies wih Dividends Michel Vellekoop 1 & Hans Nieuwenhuis 2 June 15, 26 Absrac We propose a generalized framework for he modeling of radeable securiies wih dividends which are
More informationFAIR 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 informationValuation and Hedging of Correlation Swaps. Mats Draijer
Valuaion and Hedging of Correlaion Swaps Mas Draijer 4298829 Sepember 27, 2017 Absrac The aim of his hesis is o provide a formula for he value of a correlaion swap. To ge o his formula, a model from an
More informationPricing 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 informationFinancial Markets And Empirical Regularities An Introduction to Financial Econometrics
Financial Markes And Empirical Regulariies An Inroducion o Financial Economerics SAMSI Workshop 11/18/05 Mike Aguilar UNC a Chapel Hill www.unc.edu/~maguilar 1 Ouline I. Hisorical Perspecive on Asse Prices
More informationOptimal Consumption and Investment with Habit Formation and Hyperbolic discounting. Mihail Zervos Department of Mathematics London School of Economics
Oimal Consumion and Invesmen wih Habi Formaion and Hyerbolic discouning Mihail Zervos Dearmen of Mahemaics London School of Economics Join work wih Alonso Pérez-Kakabadse and Dimiris Melas 1 The Sandard
More informationExotic FX Swap. Analytics. ver 1.0. Exotics Pricing Methodology Trading Credit Risk Pricing
Exoic FX Swap Analyics ver 1. Exoics Pricing Mehodology Trading Credi Risk Pricing Exoic FX Swap Version: ver 1. Deails abou he documen Projec Exoics Pricing Version ver 1. Dae January 24, 22 Auhors Deparmen
More informationVaR 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 informationModeling of Tradeable Securities with Dividends
Modeling of Tradeable Securiies wih Dividends Michel Vellekoop 1 & Hans Nieuwenhuis 2 April 7, 26 Absrac We propose a generalized framework for he modeling of radeable securiies wih dividends which are
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 informationDrift conditions on a HJM model with stochastic basis spreads. Teresa Martínez Quantitative Product Group. Santander Quantitative Product Group
Drif condiions on a HJM model wih sochasic basis spreads eresa Marínez Quaniaive Produc Group Sanander Quaniaive Produc Group Conens 1 Inroducion 1 2 Seing of he problem. Noaion 3 2.1 Bonds and curves...................................
More informationR e. Y R, X R, u e, and. Use the attached excel spreadsheets to
HW # Saisical Financial Modeling ( P Theodossiou) 1 The following are annual reurns for US finance socks (F) and he S&P500 socks index (M) Year Reurn Finance Socks Reurn S&P500 Year Reurn Finance Socks
More informationUCLA 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 informationFinal 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 informationMA 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 informationThe Investigation of the Mean Reversion Model Containing the G-Brownian Motion
Applied Mahemaical Sciences, Vol. 13, 219, no. 3, 125-133 HIKARI Ld, www.m-hikari.com hps://doi.org/1.12988/ams.219.918 he Invesigaion of he Mean Reversion Model Conaining he G-Brownian Moion Zixin Yuan
More informationPricing options on defaultable stocks
U.U.D.M. Projec Repor 2012:9 Pricing opions on defaulable socks Khayyam Tayibov Examensarbee i maemaik, 30 hp Handledare och examinaor: Johan Tysk Juni 2012 Deparmen of Mahemaics Uppsala Universiy Pricing
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 informationThe Cox-Ingersoll-Ross Model
The Cox-Ingersoll-Ross Model Mahias Thul, Ally Quan Zhang June 2, 2010 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 1 References Cox, John C.; Ingersoll, Jonahan E.; Ross, Sephen A. An Ineremporal
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 informationConstructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes. Johnny Li
1 / 43 Consrucing Ou-of-he-Money Longeviy Hedges Using Parameric Moraliy Indexes Johnny Li Join-work wih Jackie Li, Udiha Balasooriya, and Kenneh Zhou Deparmen of Economics, The Universiy of Melbourne
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 informationA 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 informationQuanto Options. Uwe Wystup. MathFinance AG Waldems, Germany 19 September 2008
Quano Opions Uwe Wysup MahFinance AG Waldems, Germany www.mahfinance.com 19 Sepember 2008 Conens 1 Quano Opions 2 1.1 FX Quano Drif Adjusmen.......................... 2 1.1.1 Exensions o oher Models.......................
More informationAN 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 informationFinancial 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 informationOptimal 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 informationOn 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 informationAgenda. What is an ESG? GIRO Convention September 2008 Hilton Sorrento Palace
GIRO Convenion 23-26 Sepember 2008 Hilon Sorreno Palace A Pracical Sudy of Economic Scenario Generaors For General Insurers Gareh Haslip Benfield Group Agenda Inroducion o economic scenario generaors Building
More informationBond 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 informationContagion models with interacting default intensity processes
ICCM 2007 Vol. II 1 4 Conagion models wih ineracing defaul inensiy processes Kwai Sun Leung Yue Kuen Kwok Absrac Credi risk is quanified by he loss disribuion due o unexpeced changes in he credi qualiy
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 informationA Two-Asset Jump Diffusion Model with Correlation
A Two-Asse Jump Diffusion Model wih Correlaion Mahew Sephen Marin Exeer College Universiy of Oxford A hesis submied for he degree of MSc Mahemaical Modelling and Scienific Compuing Michaelmas 007 Acknowledgemens
More informationThe 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 informationCredit Spread Option Valuation under GARCH. Working Paper July 2000 ISSN :
Credi Spread Opion Valuaion under GARCH by Nabil ahani Working Paper -7 July ISSN : 6-334 Financial suppor by he Risk Managemen Chair is acknowledged. he auhor would like o hank his professors Peer Chrisoffersen
More informationProceedings 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 informationHEDGING OF CREDIT DERIVATIVES IN MODELS WITH TOTALLY UNEXPECTED DEFAULT
HEDGING OF CREDIT DERIVATIVES IN MODELS WITH TOTALLY UNEXPECTED DEFAULT Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Monique Jeanblanc Déparemen
More informationCompleteness of a General Semimartingale Market under Constrained Trading
Compleeness of a General Semimaringale Marke under Consrained Trading Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 666, USA Monique Jeanblanc Déparemen de
More informationChanges of Numeraire for Pricing Futures, Forwards, and Options
Changes of Numeraire for Pricing Fuures, Forwards, and Opions Mark Schroder Michigan Sae Universiy A change of numeraire argumen is used o derive a general opion pariy, or equivalence, resul relaing American
More informationCalibrating and pricing with embedded local volatility models
CUING EDGE. OPION PRICING Calibraing and pricing wih embedded local volailiy models Consisenly fiing vanilla opion surfaces when pricing volailiy derivaives such as Vix opions or ineres rae/ equiy hybrids
More information(c) Suppose X UF (2, 2), with density f(x) = 1/(1 + x) 2 for x 0 and 0 otherwise. Then. 0 (1 + x) 2 dx (5) { 1, if t = 0,
:46 /6/ TOPIC Momen generaing funcions The n h momen of a random variable X is EX n if his quaniy exiss; he momen generaing funcion MGF of X is he funcion defined by M := Ee X for R; he expecaion in exiss
More informationAvailable 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 informationOnline 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 informationINFORMATION 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 informationLIDSTONE 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 informationCompleteness of a General Semimartingale Market under Constrained Trading
1 Compleeness of a General Semimaringale Marke under Consrained Trading Tomasz R. Bielecki, Monique Jeanblanc, and Marek Rukowski 1 Deparmen of Applied Mahemaics, Illinois Insiue of Technology, Chicago,
More informationVolatility and Hedging Errors
Volailiy and Hedging Errors Jim Gaheral Sepember, 5 1999 Background Derivaive porfolio bookrunners ofen complain ha hedging a marke-implied volailiies is sub-opimal relaive o hedging a heir bes guess of
More informationEquity-credit modeling under affine jump-diffusion models with jump-to-default
Equiy-credi modeling under affine jump-diffusion models wih jump-o-defaul Tsz Kin Chung Deparmen of Mahemaics, Hong Kong Universiy of Science and Technology E-mail: kchung@us.hk Yue Kuen Kwok Deparmen
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 information