Term 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
|
|
- Judith Farmer
- 5 years ago
- Views:
Transcription
1 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 ineres raes. The so-called marke models ha were developed 1 in he lae 90 s overcome his problem by direcly modelling observable marke raes such as LIBOR and swap raes. These models are sraighforward o calibrae and have quickly gained widespread accepance from praciioners. The firs marke models were acually developed in he HJM framework where he dynamics of insananeous forward raes are used via Iô s Lemma o deermine he dynamics of zero-coupon bonds. The dynamics of zero coupon bond prices were hen used, again via Iô s Lemma, o deermine he dynamics of LIBOR. Marke models are herefore no inconsisen wih HJM models. In hese lecure noes, however, we will prefer o specify he marke models direcly raher han derive hem in he HJM framework. In he process, we will derive Black s formulae for caples and swapions hereby demonsraing he consisency of hese formulae wih maringale pricing heory. Throughou hese noes, we will ignore he possibiliy of defaul or counerpary risk and rea LIBOR ineres raes as he fundamenal raes in he marke. Zero-coupon bond prices are hen compued using LIBOR raher han he defaul-free raes implied by he prices of governmen securiies. This does resul in a minor inconsisency in ha we price derivaive securiies assuming no possibiliy of defaul ye he ineres raes hemselves ha play he role of underlying securiy, i.e. LIBOR and swap raes, implicily incorporae he possibiliy of defaul. This inconsisency acually occurs in pracice when banks rade caps, swaps and oher insrumens wih each oher, and ignore he possibiliy of defaul when quoing prices. Insead, he associaed credi risks are kep o a minimum hrough he use of neing agreemens and by counerparies limiing he oal size of rades hey conduc wih one anoher. This approach can also be jusified when counerparies have a similar credi raing and similar exposures o one anoher. Finally, we should menion ha i is indeed possible 2, and someimes necessary, o explicily model credi risk even when we are pricing sandard securiies such as caps and swaps. I goes wihou saying of course, ha defaul risk needs o be modelled explicily when pricing credi derivaives and relaed securiies. 1 LIBOR, Swap Raes and Black s Formulae for Caps and Swapions We now describe wo paricularly imporan marke ineres raes, namely LIBOR and swap raes. We firs define LIBOR and forward LIBOR, and hen describe Black s formula for caples. Afer defining LIBOR we hen proceed o discuss swap raes and forward swap raes as well as describing Black s formula for swapions. In pracice, he underlying securiy for caps and swapions are LIBOR and LIBOR-based swap raes. Therefore by modelling he dynamics of hese raes direcly we succeed in obaining more realisic models han hose developed in he shor-rae or HJM framework. LIBOR The forward rae a ime based on simple ineres for lending in he inerval [T 1, T 2 ] is given by 3 F, T 1, T 2 ) = 1 T 2 T 1 ) Z T 1 Z T 2 Z T2 1) 1 See Milersen, Sandmann and Sondermann 1997), Brace, Gaarek and Musiela 1997), Jamshidian 1997) and Musiela and Rukowski 1997). 2 See chaper 11 of Cairns for a model where swaps are priced aking he possibiliy of defaul explicly ino accoun. 3 This follows from a simple arbirage argumen.
2 Marke Models 2 where, as before, Z T is he ime price of a zero-coupon bond mauring a ime T. Noe also ha if we measure ime in years, hen 1) is consisen wih F, T 1, T 2 ) being quoed as an annual rae. LIBOR raes are quoed as simply-compounded ineres raes, and are quoed on an annual basis. The accrual period or enor, T 2 T 1, is usually fixed a δ = 1/4 or δ = 1/2 corresponding o 3 monhs and 6 monhs, respecively. Wih a fixed value of δ in mind we can define he δ-year forward rae a ime wih mauriy T as L, T ) := F, T, T + δ) = 1 δ Noe ha he δ-year spo LIBOR rae a ime is hen given by L, ). Z T Z T +δ Z T +δ ). 2) Remark 1 LIBOR or he London Iner-Bank Offered Rae, is deermined on a daily basis when he Briish Bankers Associaion BBA) polls a pre-defined lis of banks wih srong credi raings for heir ineres raes. The highes and lowes responses are dropped and hen he average of he remainder is aken o be he LIBOR rae. Because here is some credi risk associaed wih hese banks, LIBOR will be higher han he corresponding raes on governmen reasuries. However, because he banks ha are polled have srong credi raings he spread beween LIBOR and reasury raes is generally no very large and is ofen less han 100 basis poins. Moreover, he pre-defined lis of banks is regularly updaed so ha banks whose credi raings have deerioraed are replaced on he lis wih banks wih superior credi raings. This has he pracical impac of ensuring ha forward LIBOR raes will sill only have a very modes degree of credi risk associaed wih hem. Black s Formula for Caples Consider now a caple wih payoff δlt, T ) K) + a ime T + δ. The ime price, C, is given by C = [ ] B E Q δlt, T ) K) + B T +δ = δz T +δ E P [ T +δ LT, T ) K) + ]. where B, Q) is an arbirary numeraire-emm pair and Z T +δ, P T +δ ) is he forward measure-numeraire pair. The marke convenion is o quoe caple prices using Black s formula which equaes C o a Black-Scholes like formula so ha C = δz T +δ [ L, T )Φ logl, T )/K) + σ 2 ) T )/2 logl, T )/K) σ 2 )] σ T )/2 KΦ T σ T where Φ ) is he CDF of a sandard normal random variable. Noe ha 3) is wha you would ge for C if you assumed ha dl, T ) = σl, T ) dw T +δ ) where W T +δ ) is a P T +δ -Brownian moion and σ is an implied volailiy ha is used o quoe prices. Black s formula for caps is o equae he cap price wih he sum of caple prices given by 3) bu where a common σ is assumed. Similar formulae exis for floorles and floors. Swap Raes Consider a payer forward sar swap where he swap begins a some fixed ime in he fuure and expires a ime T M. We assume he accrual period is of lengh δ. Since paymens are made in arrears, he firs paymen occurs a +1 = + δ and he final paymen a T = T M + δ. Then maringale pricing implies ha he ime < value, SW, of his forward sar swap is M SW = E Q B δ LT j, T j ) R) B Tj+1 j=n 3)
3 Marke Models 3 where R is he fixed rae annualized) specified in he conrac. A sandard argumen using he properies of floaing-rae bond prices implies ha SW Tn = 1 Z T This in urn easily implies why?) ha for < we have SW = Z Z T Rδ Rδ Definiion 1 The forward swap rae is he value R = R,, T M ) for which SW = 0. In paricular, we obain The swap rae is hen obained by aking = in 4)... R = R,, T M ) = ZTn Z T δ. 4) ZT j Now consider he ime price 4 of a payer-swapion ha expires a ime > and wih paymens of he underlying swap aking place a imes +1,..., T. Assuming a fixed rae of R annualized) and a noional principle of $1, he value of he opion a expiraion is given by C Tn = 1 Z T Rδ Z Tj Subsiuing 4) a = ino 5) we find ha [ C Tn = δ R,, T M ) R ] = δ Z Tj +. 5) + [ R,, T M ) R] +. 6) Therefore we see ha he swapion is like a call opion on he swap rae. The ime value of he swapion, C, is hen given by he Q-expecaion of he righ-hand-side of 6), suiably deflaed by he numeraire. Black s Formula for Swapions Marke convenion, however, is o quoe swapion prices via Black s formula which equaes C o a Black-Scholes-like formula so ha [ C = δ logr,, T M )/ R,, T M )Φ R) ) + σ 2 )/2 σ RΦ logr,, T M )/ R) )] σ 2 )/2 σ 7) where again σ is an implied volailiy ha is used o quoe prices. Noe ha he expression in 7) is wha we would obain for he expecaion of δ [ R,, T M ) R 4 Noe ha in 5) we have implicily assumed ha he srike is k = 0. ] +
4 Marke Models 4 if dr,, T M ) = σr,, T M ) dw. I should be saed ha Black s formulae for caps and swapions did no originally correspond o prices ha arise from he applicaion of maringale pricing heory o some paricular model. As originally conceived, hey merely provided a framework for quoing marke prices. The marke models of hese lecure noes will provide a belaed jusificaion for hese formulae. We shall see ha he jusificaions are muually inconsisen, however, in ha i is impossible for boh formulae o hold simulaneously wihin he one model. 2 The Term Srucure of Volailiy The erm srucure of volailiy 5 is a graph of volailiy ploed agains ime o mauriy, τ. There are of course many definiions of volailiy and care is needed in specifying which definiion is inended. Some commonly used definiions of he erm srucure of volailiy a ime include: 1. The volailiy of spo raes Y +τ as a funcion of τ. Depending on he model under consideraion, his volailiy may be available in closed form and he model calibraed o hisorical or implied raes. 2. The volaily, σ, + τ), of insananeous forward raes, f, + τ). 3. The implied volailiy, σ, given by Black s formula for caples. This will vary wih ime o mauriy and can be compued a any ime from marke prices for caples. 4. The implied volailiy, σ, given by Black s formula for caps. Again his will vary wih ime o mauriy and can be compued a any ime from marke prices for caps. When calibraing erm srucure models i is common o calibrae using boh marke prices and he erm srucure of volailiy. As a resul we ofen wan o work wih models ha allow for a rich variey of erm srucures of volailiy as well of course, as a rich variey of erm srucures of ineres raes. 3 Numeraires and Zero-Coupon Bond Prices While he cash accoun wih B := exp ) 0 r s ds has been he defaul numeraire o dae, we will no work wih he cash accoun as our numeraire in he conex of marke models. The reason is clear: in marke models we ake LIBOR raes or swap raes) wih a fixed enor, δ, in mind, as our fundamenal ineres raes. I would herefore be very inconvenien as well as defeaing he purpose) if we had o deermine he insananeous shor rae a each poin in ime. As a resul we will generally work wih oher numeraire-emm pairs as described below. Bu firs we will fix he mauriies or enor daes o which our marke models will apply. A ime we could in principal have LIBOR raes, L, T ), available for all T >. This is unnecessary, however, as he prices of mos imporan securiies, e.g. caps, floors, swaps, swapions, Bermudan swapions, ec., are deermined by he raes LIBOR or swap) applying o only a finie se of mauriies. We herefore fix in advance a se of enor daes 6 0 := T 0 < T 1 < T 2 <... < T M < T wih δ i := T i+1 T i, i = 0, 1,..., M. While he δ i s are usually nominally equal, e.g. 1/4 or 1/2, day-coun convenions will resuls in slighly differen values for each δ i. We le Z n denoe he ime price of a zero-coupon bond mauring a ime > for 5 Quans in he fixed-income indusry commonly refer o he erm-srucure of volailiy when discussing fixed-income derivaives and models. In his secion we briefly give some possible definiions of he erm-srucure of volailiy bu we will no need hese definiions elsewhere in he course. 6 The noaion and seup in his secion and he nex will borrow heavily from Secion 3.7 in Mone Carlo Mehods in Financial Engineering by Glasserman.
5 Marke Models 5 n = 1,..., M. Similarly, we use L n ) o denoe he ime forward rae applying o he period [, +1 ] for n = 0, 1,..., M. In paricular, a simple arbirage argumen hen implies L n ) = Zn Z n+1 δ n Z n+1, for 0, n = 0, 1,..., M. 8) Wih some work we can inver 8) o obain an expression for bond prices in erms of LIBOR raes. We find Z n T i = j=i δ j L j T i ) for n = i + 1,..., M ) Equaion 9) only deermines he bonds prices a he fixed mauriy daes. However, for an arbirary dae we can easily check ha Z n = Z φ) j=φ) δ j L j ) where we define φ) o be nex enor dae afer ime. Tha is, for 0. 10) φ) := min {i : < T i}. i=1,..., Remark 2 The presence of Z φ) in 10) suggess ha i may no be sufficien o model only he dynamics of he forward LIBOR raes, L n ), when we specify a marke model since hey are no sufficien o deermine a an arbirary ime. However, as we shall see below, his will no prove o be a problem as he φ) facor vanishes upon deflaing by he numeraire. Z φ) Exercise 1 Prove equaions 9) and 10). Numeraire-EMM Pairs The following numeraire-emm pairs are commonly used in marke models: 1. The spo measure, Q, assumes ha B is he numeraire where B is defined as follows. sar wih $1 a = 0 and hen purchase 1/Z 1 0 of he zero-coupon bonds mauring a ime T 1 a ime T 1 reinves he funds in he zero-coupon bond mauring a ime T 2 by coninuing in his way, we see ha a ime he spo numeraire will be worh B = Z φ) φ) 1 j=0 [1 + δ j L j T j )]. 11) 2. The forward measure, P T, akes he zero-coupon bond mauring a ime T as numeraire. We have seen his numeraire-emm pair already. 3. The swap measure, P X, is useful for pricing swapions analyically. I akes he numeraire o be X = δ M k=1 Zk, which is indeed a posiive securiy price process. Deflaing Zero-Coupon Bond Prices by he Spo Numeraire Equaions 10) and 11) show ha deflaed 7 zero-coupon bond prices, D n, saisfy φ) 1 D n ) = 1 1 for 0. 12) 1 + δ j L j T j ) 1 + δ j L j ) j=0 In paricular, we see ha he facor, Z φ), has vanished. 7 We will ake he spo numeraire o be he defaul numeraire. j=φ)
6 Marke Models 6 4 Arbirage-Free LIBOR Dynamics Dynamics under he Spo Measure We assume ha he dynamics of he LIBOR raes saisfy dl n ) = µ n )L n ) d + L n )σ n ) T dw ), 0, n = 1,..., M 13) where W ) is a d-dimensional Brownian moion, and µ n ) and σ n ) are adaped processes ha may depend on he curren vecor of ineres raes L) := L 1 ),..., L M )). The assumpion of no arbirage and he posiiviy of deflaed bond prices implies he exisence of an R d -valued process ν n ) such ha dd n ) = D n )ν ) dw ). 14) We could apply Iô s Lemma direcly o our expression for D n ) in 12) bu his would be awkward. Insead we will apply Iô s Lemma o Y n ) := log D n ). We see from 14) ha dy n ) = 1 2 ν n) 2 d + ν ) dw ) 15) We can also find an alernaive expression for dy n ) using 12). In paricular, noing ha he firs facor in 12) is consan beween mauriies, we obain via Iô s Lemma dy n ) = = j=φ) j=φ) d log 1 + δ j L j )) δ j µ j )L j ) 1 + δ j L j ) δ2 j L j) 2 σj T )σ ) j) δ j L j )) 2 d j=φ) δ j L j )σj T ) dw ).16) 1 + δ j L j ) Comparing he volailiy erms in 15) and 16) hen gives us ν n ) = j=φ) δ j L j )σ j ) 1 + δ j L j ). 17) We would now like o find an expression for he µ j s. Towards his end, we could compare he drif erms in 15) and 16), and his is easy o do when n = 2 and φ) = 1. Afer some sraighforward algebra, we easily find 8 More generally, we obain µ 1 ) = σ T 1 )ν 2 ), 0 T 1. µ n ) = σ ) ν n+1 ) = n j=φ) δ j L j )σn T )σ j ). 18) 1 + δ j L j ) We could have obained 18) by again comparing he drif erms in 15) and 16) bu his appears o be very cumbersome. Exercise 2 insead provides a more elegan approach. Exercise 2 Use inducion o esablish ha he drifs, µ n ), mus saisfy 18) under he no-arbirage assumpion. In paricular, firs assume µ 1,..., µ have been chosen consisen wih he Q-maringale assumpion on D 1,..., D n. Show ha D n+1 is a maringale if and only if L n D n+1 and hen apply Iô s Lemma o obain 9 18). 8 Noe ha L 1 ), and herefore µ 1 ), do no have any meaning for > T 1. 9 See Glasserman, page 170.
7 Marke Models 7 We herefore obain ha he arbirage free Q-dynamics of he forward LIBOR raes are given by n dl n ) = δ j L j )σn T )σ j ) L n ) d + L n )σ n ) T dw ), 0, n = 1,..., M. 19) 1 + δ j L j ) j=φ) Dynamics under he Forward Measure Consider now he case where we use he forward measure, P, and he associaed numeraire, Z. We now find ha deflaed zero-coupon bond prices are given by D n ) = M 1 + δ j L j )). 20) j=n We would like o find he marke-price-of-risk process, η ) R d, ha relaes he Q-Brownian moion W ) o he he P Brownian moion, W ), so ha dw ) = dw ) η) d. 21) There are a number of ways o do his bu perhaps he easies is he approach we followed wih he Vasicek model when we swiched o he forward measure. Equaion 20) implies D M ) = 1 + δ M L M ) so ha dd M ) = δ M dl M ). 22) We now subsiue for dl M ) in 22) using 19) evaluaed a n = M, and hen subsiue for W ) using 21). Since D M ) is a P -maringale we find ha η) = M j=φ) δ j L j )σ j ) 1 + δ j L j ). In paricular, we obain he arbirage-free P -dynamics of he forward LIBOR raes are given by dl n ) = M δ j L j )σn T )σ j ) L n ) d + L n )σ n ) T dw ), 1 + δ j L j ) 0, n = 1,..., M. Black s Formula for Caples We are now in a posiion o derive Black s formula see 3)) for caple prices. If we ake n = M in 23), hen we obain dl M ) = L M )σ M ) T dw ) 24) implying in paricular 10 ha L M ) is a P -maringale. If we assume ha σ M ) is a deerminisic funcion, hen we easily see ha L M ) is lognormally disribued. In paricular, we obain log L M ) N logl M 0)) σ M s) 2 ds, We can now obain 3) if we le T M = T and reinerpre σ appropriaely. 0 ) σ M s) 2 ds. Noe also ha here is no problem when we ake σ M ) o be deerminisic in 24) which conrass wih he HJM framework. This is because while he numeraors in he drif of 19) are quadraic in L j ), he 1 + δ j L j ) 10 Subjec, as usual, o echnical condiions. 23)
8 Marke Models 8 erm in he denominaor ensures ha here is no possibiliy of explosion in he SDE. This is a furher advanage of he marke model framework where we model simple LIBOR raes raher han insananeous forward raes. BGM s Approximaion for Swapion Prices In heir original paper, Brace, Gaarek and Musiela BGM) succeeded in deriving Black s formula for caples and hereby demonsraed is consisency wih maringale pricing. Their framework did no enable hem o derive Black s formula for swapions, however. Insead hey provided an analyic approximaion for swapion prices ha we will no describe 11 here. I is worh menioning, however, ha heir approximaion works well in pracice and provides swapion prices ha are very close o hose obained via Mone Carlo simulaion. 5 A Swap Marke Model for Pricing Swapions Consider a payer-swapion ha expires a ime > and wih paymens of he underlying swap aking place a imes +1,..., T. Assuming a fixed rae of R annualized) and a noional principle of $1, we showed in 6) ha he ime price of he swapion is given by C Tn = δ This implies ha he ime price of he swapion, C, saisfies C = X E P x [ R,, T M ) R] +. 25) δ ZT j ) [R,, T M ) R X Tn ] + 26) where X is he ime price of he chosen numeraire securiy and P x is he corresponding EMM. A paricularly convenien choice of numeraire ha we will adop is he porfolio 12 consising of δ unis of each of he zero-coupon bonds mauring a imes +1,..., T. Then X = δ ZT j and we find [ [ C = δ R] ] + E P x R,, T M ) 27) Jamshidian 1997) developed a erm srucure framework where a any ime he curren erm srucure was given in erms of he forward swap raes, R, T i, T M ) for i = φ),..., M. In paricular, he showed ha i was possible o assume ha he P x -dynamics of R,, T M ) saisfy dr,, T M ) = R,, T M )σ) T dw x ) 28) where σ) is a deerminisic vecor of volailiies. This implies ha he forward swap rae is lognormally disribued so we can obain 13 Black s formula for swapion prices 7). Remark 3 When we model swap raes direcly as in 28) we say ha we have a swap marke model. This conrass wih he LIBOR marke models of Secion See Chaper 9 of Cairns for a derivaion. 12 There is no difficuly aking a porfolio of securiies raher han a fixed individual securiy as he numeraire. More generally in fac, we could ake a dynamic self-financed porfolio as he numeraire securiy, assuming of course ha i has sricly posiive value a all imes. 13 Of course we need o reinerpre σ in 7) in erms of he deerminisic funcion σ) in 28).
9 Marke Models 9 Remark 4 The advanage of Black s swapion formula is ha i is elegan and exac, whereas he BGM formula is cumbersome and only an approximaion. However, he BGM approximaion is consisen wih Black s formulae for caples and caps whereas Black s swapion formula is no. Indeed, i may be shown 14 ha if forward LIBOR raes have deerminisic volailiies hen i i is no possible for swap raes o also have deerminisic volailiies. Therefore Black s formulae for caples and swapions canno boh hold wihin he same model. Tha said, wihin he LIBOR marke framework wih deerminisic volailiies, i can be argued ha forward swap raes are approximaely lognormally disribued. 6 Mone-Carlo Simulaion While i is possible o price many commonly raded derivaive securiies such as caps, floors and swapions in he marke model framework, i is in general necessary o use Mone Carlo mehods o price oher securiies. Indeed, if our marke model has sochasic volailiy funcions hen i will ypically be necessary o also use Mone Carlo mehods o price even caps, floors and swapions. The ypical approach is o use some discreizaion scheme such as he Euler scheme when performing he Mone Carlo simulaion. This does no creae oo much of a compuaional burden as we will only need o simulae he SDE s describing he forward LIBOR dynamics for a finie number of mauriies. This conrass wih he HJM framework where we had infiniely many mauriies which mean i was pracically infeasible o use a very fine discreizaion. This in urn promped he developmen of he discree-ime HJM framework wih he resuling discree-ime arbirage-free resricion on he drif. I is also possible o develop discree-ime arbirage-free marke models in a manner ha is analogous o our discree-ime HJM developmen. As described above, however, he need o do so is no as urgen as i is pracically feasible o simulae he marke model SDE s on a sufficienly fine grid and his is wha is ypically done in pracice. Noneheless, Glasserman s Mone Carlo Mehods for Financial Engineering describes how o build discree-ime arbirage-free marke models. I urns ou o be inconvenien o choose he LIBOR raes as he fundamenal variables ha we choose o discreize. Insead i is more convenien o direcly model deflaed bond prices as discree-ime Q-maringales 15 and o define LIBOR raes in erms of hese bond prices. Oher choices of discreizaion variable are also possible. As usual, we can choose o simulae under any EMM ha we prefer and all of he usual variance reducion echniques may be employed. 14 This is done by applying Iô s Lemma o he forward swap rae given in 4). 15 This ensures he discree-ime model is arbirage-free.
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 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 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 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 informationForeign Exchange, ADR s and Quanto-Securities
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Foreign Exchange, ADR s and Quano-Securiies These noes consider foreign exchange markes and he pricing of derivaive
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 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 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 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 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 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 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 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 information(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 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 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 informationOrigins of currency swaps
Origins of currency swaps Currency swaps originally were developed by banks in he UK o help large cliens circumven UK exchange conrols in he 1970s. UK companies were required o pay an exchange equalizaion
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 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 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 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 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 informationMORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quaniaive Finance and Invesmen Core Exam QFICORE MORNING SESSION Dae: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion
More 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 informationDYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Krzysztof Jajuga Wrocław University of Economics
DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 2006 Krzyszof Jajuga Wrocław Universiy of Economics Ineres Rae Modeling and Tools of Financial Economerics 1. Financial Economerics
More informationProblem Set 1 Answers. a. The computer is a final good produced and sold in Hence, 2006 GDP increases by $2,000.
Social Analysis 10 Spring 2006 Problem Se 1 Answers Quesion 1 a. The compuer is a final good produced and sold in 2006. Hence, 2006 GDP increases by $2,000. b. The bread is a final good sold in 2006. 2006
More informationFundamental 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 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 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 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 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 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 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 informationAlexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009
lexander L. Baranovski, Carsen von Lieres and ndré Wilch 8. May 2009/ Defaul inensiy model Pricing equaion for CDS conracs Defaul inensiy as soluion of a Volerra equaion of 2nd kind Comparison o common
More informationMartingale Methods in Financial Modelling
Sochasic Modelling and Applied Probabiliy 36 Maringale Mehods in Financial Modelling Bearbeie von Marek Musiela, Marek Rukowski 2nd ed. 25. Corr. 3rd prining 28. Buch. xvi, 638 S. Hardcover ISBN 978 3
More informationt=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 informationLIBOR MARKET MODEL AND GAUSSIAN HJM EXPLICIT APPROACHES TO OPTION ON COMPOSITION
LIBOR MARKET MODEL AND GAUSSIAN HJM EXPLICIT APPROACHES TO OPTION ON COMPOSITION MARC HENRARD Absrac. The win brohers Libor Marke and Gaussian HJM models are invesigaed. A simple exoic opion, floor on
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 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 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 informationwhere lnp(, ) f(, ) = P(, ) = exp { f(, u)du} = exp{q(, )} Q(, ) = f(, u)du Heah, Jarrow, and Moron (1992) claimed ha under risk-neural measure, he dr
HJM Model HJM model is no a ransiional model ha bridges popular LIBOR marke model wih once popular shor rae models, bu an imporan framework ha encompasses mos of he ineres rae models in he marke. As he
More 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 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 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 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 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 informationSwaps & Swaptions. by Ying Ni
Swaps & Swapions by Ying i Ineres rae swaps. Valuaion echniques Relaion beween swaps an bons Boosrapping from swap curve Swapions Value swapion by he Black 76 moel . Inroucion Swaps- agreemens beween wo
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 informationBlack-Scholes and the Volatility Surface
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Black-Scholes and he Volailiy Surface When we sudied discree-ime models we used maringale pricing o derive he Black-Scholes
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 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 informationAMS 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 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 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 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 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 informationVALUATION OF CREDIT DEFAULT SWAPTIONS AND CREDIT DEFAULT INDEX SWAPTIONS
VALATION OF CREDIT DEFALT SWAPTIONS AND CREDIT DEFALT INDEX SWAPTIONS Marek Rukowski School of Mahemaics and Saisics niversiy of New Souh Wales Sydney, NSW 2052, Ausralia Anhony Armsrong School of Mahemaics
More informationContinuous-time term structure models: Forward measure approach
Finance Sochas. 1, 261 291 (1997 c Springer-Verlag 1997 Coninuous-ime erm srucure models: Forward measure approach Marek Musiela 1, Marek Rukowski 2 1 School of Mahemaics, Universiy of New Souh Wales,
More informationExtended MAD for Real Option Valuation
Exended MAD for Real Opion Valuaion A Case Sudy of Abandonmen Opion Carol Alexander Xi Chen Charles Ward Absrac This paper exends he markeed asse disclaimer approach for real opion valuaion. In sharp conras
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 information2. Quantity and price measures in macroeconomic statistics 2.1. Long-run deflation? As typical price indexes, Figure 2-1 depicts the GDP deflator,
1 2. Quaniy and price measures in macroeconomic saisics 2.1. Long-run deflaion? As ypical price indexes, Figure 2-1 depics he GD deflaor, he Consumer rice ndex (C), and he Corporae Goods rice ndex (CG)
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 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 informationdb t = r t B t dt (no Itô-correction term, as B has finite variation (FV), so ordinary Newton- Leibniz calculus applies). Now (again as B is FV)
ullin4b.ex pm Wed 21.2.2018 5. The change-of-numeraire formula Here we follow [BM, 2.2]. For more deail, see he paper Brigo & Mercurio (2001c) cied here, and H. GEMAN, N. El KAROUI and J. C. ROCHET, Changes
More 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 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 informationCARF Working Paper CARF-F-196. A Market Model of Interest Rates with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies
CARF Working Paper CARF-F-196 A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he presence of Collaeral and Muliple Currencies Masaaki Fujii The Universiy of Tokyo Yasufumi Shimada Shinsei Bank,
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 informationDual Valuation and Hedging of Bermudan Options
SIAM J. FINANCIAL MAH. Vol. 1, pp. 604 608 c 2010 Sociey for Indusrial and Applied Mahemaics Dual Valuaion and Hedging of Bermudan Opions L. C. G. Rogers Absrac. Some years ago, a differen characerizaion
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 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 informationFunding beyond discounting: collateral agreements and derivatives pricing
cuing edge. DERIVAIVES PRICING Funding beyond discouning: collaeral agreemens and derivaives pricing Sandard heory assumes raders can lend and borrow a a risk-free rae, ignoring he inricacies of he repo
More informationAppendix B: DETAILS ABOUT THE SIMULATION MODEL. contained in lookup tables that are all calculated on an auxiliary spreadsheet.
Appendix B: DETAILS ABOUT THE SIMULATION MODEL The simulaion model is carried ou on one spreadshee and has five modules, four of which are conained in lookup ables ha are all calculaed on an auxiliary
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 informationOutput: The Demand for Goods and Services
IN CHAPTER 15 how o incorporae dynamics ino he AD-AS model we previously sudied how o use he dynamic AD-AS model o illusrae long-run economic growh how o use he dynamic AD-AS model o race ou he effecs
More informationApplications of Interest Rate Models
WDS'07 Proceedings of Conribued Papers, Par I, 198 204, 2007. ISBN 978-80-7378-023-4 MATFYZPRESS Applicaions of Ineres Rae Models P. Myška Charles Universiy, Faculy of Mahemaics and Physics, Prague, Czech
More informationStandard derivatives pricing theory (see, for example, Hull,
Cuing edge Derivaives pricing Funding beyond discouning: collaeral agreemens and derivaives pricing Sandard heory assumes raders can lend and borrow a a risk-free rae, ignoring he inricacies of he repo
More informationAMS Computational Finance
AMS 54 - Compuaional Finance European Opions Rober J. Frey Research Professor Sony Brook Universiy, Applied Mahemaics and Saisics frey@ams.sunysb.edu Feb 2006. Pu-Call Pariy for European Opions A ime T
More informationA Note on Construction of Multiple Swap Curves with and without Collateral
Financial Research and Training Cener Discussion Paper Series A Noe on Consrucion of Muliple Swap Curves wih and wihou Collaeral Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi DP 2009-6 February, 2010
More informationINTEREST RATES AND FX MODELS
INTEREST RATES AND FX MODELS 5. Shor Rae Models Andrew Lesniewski Couran Insiue of Mahemaics New York Universiy New York March 3, 211 2 Ineres Raes & FX Models Conens 1 Term srucure modeling 2 2 Vasicek
More informationBond Implied CDS Spread and CDS-Bond Basis. Abstract
ond Implied Spread and -ond asis Richard Zhou Augus 5, 8 Absrac We derive a simple formula for calculaing he spread implied by he bond mare price. Using no-arbirage argumen, he formula expresses he bond
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 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 informationIntroduction. Enterprises and background. chapter
NACE: High-Growh Inroducion Enerprises and background 18 chaper High-Growh Enerprises 8 8.1 Definiion A variey of approaches can be considered as providing he basis for defining high-growh enerprises.
More informationHULL-WHITE ONE FACTOR MODEL: RESULTS AND IMPLEMENTATION
HULL-WHITE ONE FACTOR MODEL: RESULTS AND IMPLEMENTATION QUANTITATIVE RESEARCH Absrac. Deails regarding he implemenaion of he Hull-Whie one facor model are provided. The deails concern he model descripion
More informationANSWER ALL QUESTIONS. CHAPTERS 6-9; (Blanchard)
ANSWER ALL QUESTIONS CHAPTERS 6-9; 18-20 (Blanchard) Quesion 1 Discuss in deail he following: a) The sacrifice raio b) Okun s law c) The neuraliy of money d) Bargaining power e) NAIRU f) Wage indexaion
More informationPARAMETER ESTIMATION IN A BLACK SCHOLES
PARAMETER ESTIMATIO I A BLACK SCHOLES Musafa BAYRAM *, Gulsen ORUCOVA BUYUKOZ, Tugcem PARTAL * Gelisim Universiy Deparmen of Compuer Engineering, 3435 Isanbul, Turkey Yildiz Technical Universiy Deparmen
More informationCHAPTER 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 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 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 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 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 informationModeling of Interest Rate Term Structures under Collateralization and its Implications
Modeling of Ineres Rae Term Srucures under Collaeralizaion and is Implicaions Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi Firs version: 22 Sepember 2010 Curren version: 24 Sepember 2010 Absrac In
More informationUniversity of Cape Town
The copyrigh of his hesis vess in he auhor. No quoaion from i or informaion derived from i is o be published wihou full acknowledgemen of he source. The hesis is o be used for privae sudy or noncommercial
More informationPrinciples 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 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 informationSOCIETY OF ACTUARIES Quantitative Finance and Investment Advanced Exam Exam QFIADV MORNING SESSION
SOCIETY OF ACTUARIES Exam Exam QFIADV MORNING SESSION Dae: Thursday, Ocober 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion has a oal of 100 poins.
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 informationPricing Inflation-Indexed Derivatives Using the Extended Vasicek Model of Hull and White
Pricing Inflaion-Indexed Derivaives Using he Exended Vasicek Model of Hull and Whie Alan Sewar Exeer College Universiy of Oxford A hesis submied in parial fulfillmen of he MSc in Mahemaical Finance April
More information