Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities

Size: px
Start display at page:

Download "Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities"

Transcription

1 Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for pricing pass-hrough morgage backed securiies (MBS) and provide a novel hedging ool for invesors in his marke. To calculae he price of an MBS, raders use wha is known as opion-adjused spread (OAS). The resuling OAS value represens he required basis poins adjusmen o reference curve discouning raes needed o mach an observed marke price. The OAS suffers from some drawbacks. For example, i remains consan unil he mauriy of he bond (hiry years in morgage-backed securiies), and does no incorporae ineres rae volailiy. We sugges insead wha we call dynamic opion adjused spread (DOAS). The laer allows invesors in he morgage marke o accoun for boh prepaymen risk and changes of he yield curve. Keywords: Asse pricing, Morgage Backed Securiies, Term Srucure. JEL Classificaion: C23, G34 We wish o hank john Crosby for consrucive commen. The usual disclaimer applies. Corresponding auhor: Mario Cerrao, Universiy of Glasgow, Deparmen of Economics, 1

2 1. Inroducion Morgage Backed Securiies (MBS) are securiies collaeralised by residenial morgage loans. The MBS marke has grown o become he larges fixed income marke in he Unied Saes. The reason of his enormous growh is probably due o he higher reurn and lower risk profile compared o oher fixed income securiies. However, alhough he marke is growing very quickly, neverheless here are sill quie a few issues concerning he pricing and risk managemen of hese securiies. Because of he borrowers prepaymen opion in he underlying morgage loans, morgage-backed securiies have characerisics similar o hose of callable bonds. Unlike callable bonds, however, for which he issuers refinancing sraegies are assumed o be close o opimal, morgage borrowers may be slow o refinance when i would financially favourable and someimes prepay when i is financially unfavourable. Invesors in morgage-backed securiies hold long posiions in noncallable bonds and shor posiions in call (prepaymen) opions. The noncallable bond is a effecively a porfolio of zero coupon bonds, and he call opion gives he borrower he righ o prepay he morgage a any ime prior o he mauriy of he loan. Therefore, he value of he MBS is he difference beween he value of he noncallable bond and he value of he call (prepaymen) opion. In he marke place, dealers generally price he morgage by pricing hese wo componens separaely. To evaluae he call opion, he Opion-Adjused Spread mehodology uses opion pricing echniques. When he opion componen is quanified and aken away from he oal yield spread, he yield o mauriy of a non-benchmark bond can be compared o a risk-free of a benchmark securiy 1. Any model employed o value a MBS should be able o value he noncallable componen of a morgage and he call opion componen. Ceeris paribus, given ha ineres rae and prepaymen risks have been accouned for, and incorporaed in he heoreical model, one would expec he heoreical price of an MBS o be equal o is marke price. If hese values are no equal, hen marke paricipans demand compensaion for he unmodeled risks. The difference in values migh be due o unmodeled risks which are aribuable o he srucure and liquidiy of he bond. One of hese unmodeled risks is 2

3 he forecas error associaed wih he prepaymen model. For example, he acual prepaymen may be faser or slower han wha he model predics. In his case, he OAS is he marke price for he unmodeled risks. Because here is no agreemen on how o model prepaymens among morgage holders, and many differen ineres rae models exiss, opion-adjused spread calculaion suffers from he lack of a sandard erm. The academic lieraure in his area has mainly focused on modelling OAS dynamics such ha he embedded morgage call opion price can be esimaed and consequenly he morgage priced (see for example, Dunn and Spa (1986), Liu and Xu (1998), Schwarz and Torous (1992) amongs ohers). However, hese models alhough helping o clarify a number of issues concerning he pricing of MBS, are no used in pracice. On he oher hand, many researchers working in financial insiuions, and amongs hem op academics, have insead oped for economeric models o esimae he parameers of ineres o calibrae reduced form models and price MBS (see for example Chen (2004)). Therefore, from a praciioner s poin of view reduced form models seem o be he ideal way of pricing MBS. However, since mos of hese models are proprieary models heir funcional form is no known in he marke. This paper is organised as follows: we discuss he MBS model used in his sudy in Secion 2, Secion 3 discusses he ineres rae model and is calibraion, Secion 4 presens a numerical example, Secion 5 he dynamic opion adjused spread, Secion 6 presens he empirical resuls finally Secion 7 concludes. 1 See our applicaion of opion adjused spread in his paper. 3

4 2. The Morgage Backed Securiy Model Consider he following probabiliy space ( Ω, F, P), and suppose he process ψ (, D, C, z), represening he price process of a morgage backed securiy, is adaped o he filraion F. The price process depends on he risk neural vecor of discoun bond price D i 0 < i < N, wih Q being he risk neural probabiliy measure, and he sae variable z. Also denoe wih C he cash-flow paid by he morgage a. Define he price process for a morgage a imet when z = 0 as he expeced value of he discouned fuure cash-flows: T Q i i = 0 E(ψ ) = E [ C D ] (1) i The main problem when deermining he price of his securiy is ha i is no simply deermined by discouningc, since he borrower can a each ime consider a i prepaymen acion. In he inroducion we have already menioned differen ways of modelling he prepaymen opion when pricing MBS. In his paper we shall follow Chen (2004) and implemen a reduced form model 2. In general, when pricing MBS one has o, firs, generae he morgage cash flows C( D, z) using, for example, a reduced form model. Once cash-flows have been generaed, he value of he morgage can be obained by discouning he simulaed cash flows beween1 < i < N : ~ ~ T ~ Q i i = 0 E(ψ ) = E [ C D ] (2) i If we use Mone Carlo o generae m pahs for m, we have ha ~ C E Q ~ 1 T M ~ ~ (ψ ) = [ C i Di ], limm C i m C m i = 0 m= 1 and he soluion of (2) gives he value of he morgage. Using Equaion (2) one can also esimae he opion adjused 2 Refer o he Appendix for a descripion of he model. 4

5 spread z in he following way. Define wih P he observed marke price of he morgage. We can compue z using a roo finding mehod o solve (3) below: ~ ~ ψ (, C, D, z) = Ρ (3) 0 3. The Term Srucure Model To solve Equaion (2) one has o simulae he erm srucure of ineres raes ou of he mauriy of he morgage. We exend he above model by using a wo facor Heah, Jarrow, and Moron (1992) model (HJM). The HJM model is a class of models, and herefore one needs o specify he iniial forward raes and volailiies o specify he model iself. Below we explain he way we have deal wih his problem. The HJM model aemps o consruc a model of he erm srucure of ineres raes ha is consisen wih he observed erm srucure. The sae variable in his model is he forward rae in ime for insananeous borrowing a a laer imet, F (, T ). In differenial form he model can be wrien as: df (, T ) m(, T ) d + (, T ) dw ( ) N = σ for 0 T (4) k = 1 k k Or also in inegral form Here ( 0, T ) F ( T ) = F( 0, T ) + m( v, T ) dv + ( v, T ) 0 N, σ (5) 0 k = 1 k dwk ( v) F is he fixed iniial forward rae curve, ( T ) m, is he insananeous forward rae drif, σ (,T ) is he insananeous volailiy process of he forward rae curve, and W is a sandard Brownian moion process. The model above is very general and encompasses all he shor rae models such as, for example, he Hull and Whie (1993) model. 5

6 The drif process is specified as: N T m (, T ) = k (, T ) σ (, s) ds k = 1 σ (6) The hardes problem when using he HJM model o simulae F(, T ) is ha he model is specified in erms of insananeous forward raes and he laer are no observable in he marke. To overcome he problem we use he following deerminisic specificaion for he volailiies, and he Musiela parameerizaion: σ k (, T ) = σ k (, T ) Tha means ha our model belongs o he Gaussian class of models and mauriy is specified as ime o mauriy. Therefore, if we se τ = T i follows ha: d F (, τ ) m(, τ ) d + σ (, τ ) dw ( ) = (7) Wih he drif specified as: τ m(, τ ) = σ (, τ ) σ (, s) ds + F(, τ ) (8) τ 0 We use he above parameerisaion when simulaing he forward raes. The spo rae z () used o discoun he cash flows can be deermined from (7) as follows: z( ) lim df (, τ ) τ 6

7 To use he wo facor model above, one has o specify he iniial forward raes and volailiies. In his applicaion we have used Bloomberg o obain he forward raes necessary o iniiae he process. Also, we have used Bloomberg o obain implied volailiies on ineres rae caps necessary for he calibraion of our model. Two volailiies are used. The firs is se fixed for all he mauriies and equal o he implied volailiy of a hiry year ineres rae cap opion. The second refers o implied volailiies of ineres raes cap wih mauriies 1 o 30 years. An Euler discreizaion scheme, wih 360 ime seps and 5000 simulaions, is used. 4. Numerical Example Table 1 shows a sample of simulaed prices for he morgage backed securiy using he model described above wih heir sandard errors. Table 1: Morgage Backed Securiy Valuaion wih 5% coupon rae ~ ψ 0 % SE Consider, for example, he morgage wih value equal o %. Suppose he size of he underlying morgage pool is $1,000,000.00, he price of a morgage-backed securiy issued from he underlying pool will be $1,021, The observed marke price is assumed o be 100% of he par value. One can herefore compue, using a roo finding mehod, he opion adjused spread ha in his example is 46 basis poins. 7

8 1 MBS Cash Flow 5% Coupon 0.9 Presen Value of Fiure Cash Flows Time o Mauriy (monhs) Figure 1. Figure 1 above shows simulaed pahs of he monhly cash flows of he morgage. As he bond approaches mauriy he value of he prepaymen opion decreases and consequenly he morgage cash flow becomes less uncerain. 5. Dynamic Opion Adjused Spread The opion-adjused spread (OAS) above can be viewed as a measure of he yield spread. I is consan over he benchmark curve chosen for he valuaion process. The reason why his spread is referred o as opion-adjused is because he cash flows of he underlying securiy are adjused o reflec he embedded opion. Mos marke paricipans find i more convenien o hink abou yield spread han price differences. One issue wih he opion spread is ha i assumes he yield spread o say unchanged over he mauriy of he bond. Therefore, if fuure ineres raes become volaile, he OAS remains unchanged. This implies ha raders will have o compue i and recalibrae heir models frequenly. In his secion we propose a modificaion of he OAS ha we call Dynamic Opion Adjused Spread (DOAS). The DOAS allows one o capure prepaymen risk as well as changes in he yield curve. A poenial invesor holding a morgage can use he DOAS as a hedging ool. Figure 2 below shows he condiional prepaymen rae (CPR) funcion, he refinancing incenive (RI) and he porfolio value (PV). A he beginning of he morgage here is a posiive spread (i.e. he difference beween he value of he porfolio and he cash flow of he morgage). The difference would compensae he invesor if he opion is exercised by he borrower. The spread is paricularly relevan in he firs one hundred monhs which, in general, corresponds o he ime when he 8

9 prepaymen risk is higher. As he prepaymen risk becomes less accenuae, he spread decreases. Presen Value of Fuure cash Flows CPI CPR PV RI Time o Mauriy Figure 2. From an invesor poin of view he DOAS can be viewed as an invesmen 3. The value of his porfolio can be posiive or negaive depending on he spread adjusmen. A bond having a posiive OAS has a posiive porfolio value. On he oher hand, a bond wih a negaive OAS will have a negaive porfolio value. 4 To compue he dynamic opion adjused spread, we used he following procedure. Use simulaions o simulae he cash-flows, a each, over he lifeime of he morgage. Compue he opion adjused spread (i.e z ) a 0 and use i o adjus he cash-flows of he bond a each.you have compued he adjused cash-flows. The difference, a each, beween he plain vanilla bond cash flows and he morgage cash flows, is he dynamic opion adjused spread in. The summaion of hese up o 0 is he porfolio value n ~ Q 0 = E [( ψ ) E( ψ i= 0 PV )] i (9) Equaion (9) describes he way we compued he porfolio value. Therefore he porfolio value is jus he difference beween a non-callable bond and a callable bond. 3 We call his invesmen a porfolio value (PV). 4 OAS can be negaive when he morgage coupon is low bu ineres rae volailiy is relaively high. In his case invesors in his marke migh no be very concerned wih he MBS opionaliy, a leas no in he shor run. 9

10 I migh be worh noicing ha, by buying a MBS and invesing in he above porfolio, he invesor has indeed creaed a synheic non-callable bond bu wih he difference ha he is also hedging agains ineres rae risk. 5.1 Numerical Example Table 2 shows esimaed porfolio values using Mone Carlo simulaions. We also repor sandard errors. Table 2: Porfolio Values (5% Coupon) and Their Sandard Errors. PV % SE The DOAS we use in our example is % par value. If we assume ha he pool size of he morgage is $1,000,000.00, he porfolio value will be $ 20, The invesor can buy his opion o hedge ineres rae risk. In he nex secion, we show his wih an example Porfolio Value 5% Coupon Time o Mauriy (monhs) Figure 3. 10

11 5.2 Numerical Example The invesor can use he porfolio described above as a hedging insrumen agains prepaymen risk in general and changes of he yield curve. The examples below show exacly his. Example1: 5% Coupon rae: Invesor A buys a ime 0 a 30-year morgage-backed securiy wih he price of he MBS being 100% of he face value. The invesor receives Treasury rae plus 46 basis poin (OAS). We assume he pool size o be $1,000,000. Anoher invesor, say Invesor B, buys a ime 0 he same morgage and also buys a DOAS opion. The DOAS opion is % of he par value. Therefore he value of his invesmen will be %. Suppose a ime 1 he ineres rae volailiy increases from 13bp o 26bp. Wha is he impac of his increase on he MBS price, and he invesor`s porfolio? A ime 1, he price of he morgage drops o % or $ 998, Therefore ha implies a $1,466 loss on he morgage for Invesor A. On he oher hand, he value of he invesmen for he Invesor B, is given by: Pay-off = bond value a ime 1 - bond value a ime 0 + (porfolio value a ime 1 - porfolio value a ime 0 ) Pay-off = ( ) = or $1,337 Example2: 6% coupon rae: We repor below anoher example choosing a coupon rae ha is above he iniial ineres rae used in he simulaion. Invesor A buys a ime 0 he morgage and receives ineress plus basis poins. 5 Invesor B buys he same morgage bu also invess ino a DOAS opion whose price is % for a oal of %. 5 OAS has been calculaed as in (5). 11

12 Suppose ha a ime 1 he ineres raes volailiy increases, as before, from o Wha is he impac of his increase on he bond price, and he invesor`s porfolio? A ime 1 he price of he morgage drops o % or $ 999, The loss for he Invesor A is herefore $ As a consequence of he increase in ineres rae volailiy he value of he DOAS opion increases o %. The payoff for he Invesor B is herefore given by: Pay-off = bond value a ime 1 - bond value a ime 0 + (porfolio value a ime 1 - porfolio value a ime 0 ) Pay-off = ( ) = % or $ Empirical Resuls Table 3 shows MBS prices wih differen coupons and also he opion adjused spread. We noe ha he price of he morgage increases as he coupon rae increases. Table 3: Morgage-Backed Securiy Values and Dynamic Opion Adjused Spreads Coupon Rae % MBS Price SE OAS bp DOAS % SE The highes price is reached when he coupon is 7% and i is Such a high premium clearly canno be explained jus by par plus a number of refinancing poins. These high prices are consisen wih wha generally is observed in he marke where morgage prices can easily reach hese levels (see also Longsaff, 2004, for a discussion on his issue). 12

13 Condiionally on he ineres rae level used in our simulaion, we noe ha higher coupon raes will increase he incenive for he borrower o repay he morgage and his clearly will affec he spread ha an evenual invesor would require as a compensaion for he prepaymen opion. In fac our model suggess a spread on he Treasury curve of more han 400bp when a 7% coupon is considered. We have also compued sandard errors from he simulaion by using 100 independen rials of he model in secion 2. A he boom of Table 3, we repor he simulaed dynamic opions adjused values. As we see, given he ineres rae level used in he simulaion, he value of he opion increases as he coupon increases. This is consisen wih a higher prepaymen risk implici wih higher coupons. As we showed above an invesor migh decide o buy his opion, and pay a higher price for he morgage, if he wishes o be hedge agains prepaymen risk and changes in he slope of he yield curve. 13

14 Conclusions The Morgage Backed Securiies marke is he larges fixed income marke in he Unied Saes. These asses are collaeralised by a pool of morgages and allow invesors o gain higher ineres raes wih a relaively lower risk compared o oher fixed income insrumens. Given he imporance of hese securiies, in he las decade, here has been a proliferaion of models rying o explain he opimal prepaymen behaviour of he borrower. The main problem wih mos of hese models is ha hey canno always explain, wihin a raional model, how borrowers decide o refinance heir loans. Therefore, some of hese models have ried o model he prepaymen acion as an endogenous problem (see Sanon and Wallace, 1998 amongs he ohers), bu MBS prices obained by using hese models canno generally mach marke prices. If on one hand various differen models have been proposed in he lieraure o price MBS. On he oher hand here has been very lile done in erms of he hedging and risk managemen of hese securiies. In his paper we have ried o fill his gap. We exend a reduced form model o price MBS and propose a novel approach o managing ineres raes risk. We show ha an invesor in his marke, by aking a long posiion on an opion (DOAS), can hedge ou ineres rae risk. The DOAS is simply he difference beween he cash flows of a non-callable bond and a callable bond over he mauriy of he morgage. The concep of DOAS can be easily exended o oher fixed income securiies such as callable bonds and a variey of exoic swaps. 14

15 References Brace, A., D., Gaarek, and M., Musiela, 1997, The Marke Model of Ineres Rae Dynamics, mahemaical Finance, 7, Chen Jian, 2004, Simulaion Based Pricing of Morgage Backed Securiies, Proceeding of he 2004 Winer Simulaion Conference. Dunn, Kenneh B., and Cheser S., Spa, 1986, The Effec of Refinancing Coss and Marke Imperfecions on he Opimal Call Sraegy and he Pricing of Deb Consracs, Working Paper, Carnegie-Mellon Universiy. Heah, D., R., A., Jarrow, and A., Moron, 1992, Bond Pricing and he Term Srucure of Ineres Raes: A New Mehodology for Coningen Claims Valuaion, Economerica 60 (1), Hull, J., and A., Whie, 1993, One Facor Ineres rae Models and he Valuaion of Ineres Rae Derivaive Securiies, Journal of Financial and Quaniaive Analysis (28), Liu Jian Guao and Eugene Xu, 1998, Pricing of Morgage Backed Securiies wih Opion Adjused Spread, Managerial Finance, vol. 24, No. 9/10. Longsaff, F., 2004, Borrower Credi and he Valuaion of Morgage Backed Securiies, UCLA Anderson School, Working Papers in Finance. Schwars, Eduardo, S., and Waler N., Torous, 1992, Prepaymen, Defaul, and he Valuaion of Morgage Pass-Through Securiies, Journal of Business, 65, Sanon, Richard and N., Wallace, 1998, Morgage Choice: Wha`s he Poin?, Real Esae Economics, 26,

16 Appendix 1 The model assumes ha four facors (i.e. refinancing incenive, burnou, seasoning, and seasonaliy) explain 95% of he variaion in prepaymen raes. These facors are hen combined ino one model o projec prepaymens: CPR = RI AGE MM BM where, RI represens he refinancing incenive; AGE represens he seasoning muliplier; MM represens he monhly muliplier; BM represens he burnou muliplier. Therefore, he prepaymen model is: CPR = RI AGE MM BM where: 1 RI = an 10 AGE = min 1, 30 B 1 BM = B0 [ ( WAC r ( ) )] MM akes he following values, which sar from January and end in December: (0.94, 0.76, 0.74, 0.95, 0.98, 0.92, 0.98, 1.1, 1.18, 1.22, 1.23, 0.98), r is 10-year Treasury rae, and WAC is he weighed average coupon rae Refinancing Incenive 5% Coupon Refinancing Incenive Time o Mauriy (monhs) Figure 3. 16

17 0.12 Refinancing Incenive 7% Coupon Refinancing Incenive Figure Time o Mauriy (monhs) Figure 3 and 4 above show he refinancing incenive funcion for 5% and 7% coupon raes. Borrowers have a higher incenive o exercise he prepaymen opion and refinance he morgage when he coupon rae is higher han ineres raes. This is shown in Figure 4. 17

18 18

Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities

Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao and Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics Previous Draf: 27 January 28 This Draf: 27 April 29

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

CVA calculation for CDS on super senior ABS CDO

CVA calculation for CDS on super senior ABS CDO MPRA Munich Personal RePEc Archive CVA calculaion for CDS on super senior AS CDO Hui Li Augus 28 Online a hp://mpra.ub.uni-muenchen.de/17945/ MPRA Paper No. 17945, posed 19. Ocober 29 13:33 UC CVA calculaion

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya. Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

More information

Newton's second law in action

Newton's second law in action Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

More information

THE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS

THE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models

More information

Chapter 9 Bond Prices and Yield

Chapter 9 Bond Prices and Yield Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

More information

Pricing Single Name Credit Derivatives

Pricing Single Name Credit Derivatives Pricing Single Name Credi Derivaives Vladimir Finkelsein 7h Annual CAP Workshop on Mahemaical Finance Columbia Universiy, New York December 1, 2 Ouline Realiies of he CDS marke Pricing Credi Defaul Swaps

More information

ABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION

ABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.

More information

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

Price elasticity of demand for crude oil: estimates for 23 countries

Price elasticity of demand for crude oil: estimates for 23 countries Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh

More information

Markit Excess Return Credit Indices Guide for price based indices

Markit Excess Return Credit Indices Guide for price based indices Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semi-annual

More information

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance Finance Leers, 003, (5), 6- Skewness and Kurosis Adjused Black-Scholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Pricing Black-Scholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension

Pricing Black-Scholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension Pricing Black-choles Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zu-lang Liao a, and Hsing-Hua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

Supplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect Risk-Taking?

Supplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect Risk-Taking? Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec Risk-Taking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF

More information

Tanaka formula and Levy process

Tanaka formula and Levy process Tanaka formula and Levy process Simply speaking he Tanaka formula is an exension of he Iô formula while Lévy process is an exension of Brownian moion. Because he Tanaka formula and Lévy process are wo

More information

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

Estimating Time-Varying Equity Risk Premium The Japanese Stock Market 1980-2012

Estimating Time-Varying Equity Risk Premium The Japanese Stock Market 1980-2012 Norhfield Asia Research Seminar Hong Kong, November 19, 2013 Esimaing Time-Varying Equiy Risk Premium The Japanese Sock Marke 1980-2012 Ibboson Associaes Japan Presiden Kasunari Yamaguchi, PhD/CFA/CMA

More information

The Interest Rate Risk of Mortgage Loan Portfolio of Banks

The Interest Rate Risk of Mortgage Loan Portfolio of Banks The Ineres Rae Risk of Morgage Loan Porfolio of Banks A Case Sudy of he Hong Kong Marke Jim Wong Hong Kong Moneary Auhoriy Paper presened a he Exper Forum on Advanced Techniques on Sress Tesing: Applicaions

More information

CBOE VIX PREMIUM STRATEGY INDEX (VPD SM ) CAPPED VIX PREMIUM STRATEGY INDEX (VPN SM )

CBOE VIX PREMIUM STRATEGY INDEX (VPD SM ) CAPPED VIX PREMIUM STRATEGY INDEX (VPN SM ) CBOE VIX PREIU STRATEGY INDEX (VPD S ) CAPPED VIX PREIU STRATEGY INDEX (VPN S ) The seady growh of CBOE s volailiy complex provides a unique opporuniy for invesors inen on capuring he volailiy premium.

More information

Chapter 6 Interest Rates and Bond Valuation

Chapter 6 Interest Rates and Bond Valuation Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Long-erm deb-loosely, bonds wih a mauriy of one year or more Shor-erm deb-less han a year o mauriy, also called unfunded deb Bond-sricly

More information

Equities: Positions and Portfolio Returns

Equities: Positions and Portfolio Returns Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi

More information

Fair games, and the Martingale (or "Random walk") model of stock prices

Fair games, and the Martingale (or Random walk) model of stock prices Economics 236 Spring 2000 Professor Craine Problem Se 2: Fair games, and he Maringale (or "Random walk") model of sock prices Sephen F LeRoy, 989. Efficien Capial Markes and Maringales, J of Economic Lieraure,27,

More information

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of Erlangen-Nuremberg Lange Gasse

More information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return Calculation of U.S. Treasury Constant Maturity Indices Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

More information

Nikkei Stock Average Volatility Index Real-time Version Index Guidebook

Nikkei Stock Average Volatility Index Real-time Version Index Guidebook Nikkei Sock Average Volailiy Index Real-ime Version Index Guidebook Nikkei Inc. Wih he modificaion of he mehodology of he Nikkei Sock Average Volailiy Index as Nikkei Inc. (Nikkei) sars calculaing and

More information

Investor sentiment of lottery stock evidence from the Taiwan stock market

Investor sentiment of lottery stock evidence from the Taiwan stock market Invesmen Managemen and Financial Innovaions Volume 9 Issue 1 Yu-Min Wang (Taiwan) Chun-An Li (Taiwan) Chia-Fei Lin (Taiwan) Invesor senimen of loery sock evidence from he Taiwan sock marke Absrac This

More information

FX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS

FX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS FX OPTION PRICING: REULT FROM BLACK CHOLE, LOCAL VOL, QUAI Q-PHI AND TOCHATIC Q-PHI MODEL Absrac Krishnamurhy Vaidyanahan 1 The paper suggess a new class of models (Q-Phi) o capure he informaion ha he

More information

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100... Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The Black-Scholes Shl Ml Moel... pricing opions an calculaing

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

LEASING VERSUSBUYING

LEASING VERSUSBUYING LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

CONVEXITY ADJUSTMENT AND DELIVERY OPTION IN AUSTRALIAN DOLLAR 90 DAY BILLS FUTURES

CONVEXITY ADJUSTMENT AND DELIVERY OPTION IN AUSTRALIAN DOLLAR 90 DAY BILLS FUTURES CONVEXITY ADJUSTMENT AND DELIVERY OPTION IN AUSTRALIAN DOLLAR 90 DAY BILLS FUTURES MARC HENRARD Absrac. Ausralian dollar bills fuures are very paricular, no only on he valuaion a epiry bu also for he mauriy

More information

A general decomposition formula for derivative prices in stochastic volatility models

A general decomposition formula for derivative prices in stochastic volatility models A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

Default Risk in Equity Returns

Default Risk in Equity Returns Defaul Risk in Equiy Reurns MRI VSSLOU and YUHNG XING * BSTRCT This is he firs sudy ha uses Meron s (1974) opion pricing model o compue defaul measures for individual firms and assess he effec of defaul

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.

UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment. UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL

More information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot College Physics Lab RC Circuits Scott Hildreth Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

More information

Foreign Exchange and Quantos

Foreign Exchange and Quantos IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in

More information

Optimal Time to Sell in Real Estate Portfolio Management

Optimal Time to Sell in Real Estate Portfolio Management Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr

More information

Annuity Decisions with Systematic Longevity Risk

Annuity Decisions with Systematic Longevity Risk Annuiy Decisions wih Sysemaic Longeviy Risk Ralph Sevens This draf: November, 2009 ABSTRACT In his paper we invesigae he effec of sysemaic longeviy risk, i.e., he risk arising from uncerain fuure survival

More information

Why Did the Demand for Cash Decrease Recently in Korea?

Why Did the Demand for Cash Decrease Recently in Korea? Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in

More information

The performance of popular stochastic volatility option pricing models during the Subprime crisis

The performance of popular stochastic volatility option pricing models during the Subprime crisis The performance of popular sochasic volailiy opion pricing models during he Subprime crisis Thibau Moyaer 1 Mikael Peijean 2 Absrac We assess he performance of he Heson (1993), Baes (1996), and Heson and

More information

Impact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences

Impact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process, Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ

More information

Rationales of Mortgage Insurance Premium Structures

Rationales of Mortgage Insurance Premium Structures JOURNAL OF REAL ESTATE RESEARCH Raionales of Morgage Insurance Premium Srucures Barry Dennis* Chionglong Kuo* Tyler T. Yang* Absrac. This sudy examines he raionales for he design of morgage insurance premium

More information

The yield curve, and spot and forward interest rates Moorad Choudhry

The yield curve, and spot and forward interest rates Moorad Choudhry he yield curve, and spo and forward ineres raes Moorad Choudhry In his primer we consider he zero-coupon or spo ineres rae and he forward rae. We also look a he yield curve. Invesors consider a bond yield

More information

NASDAQ-100 Futures Index SM Methodology

NASDAQ-100 Futures Index SM Methodology NASDAQ-100 Fuures Index SM Mehodology Index Descripion The NASDAQ-100 Fuures Index (The Fuures Index ) is designed o rack he performance of a hypoheical porfolio holding he CME NASDAQ-100 E-mini Index

More information

The Grantor Retained Annuity Trust (GRAT)

The Grantor Retained Annuity Trust (GRAT) WEALTH ADVISORY Esae Planning Sraegies for closely-held, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business

More information

FORWARD AND FUTURES CONTRACTS

FORWARD AND FUTURES CONTRACTS Page1 C H A P T E R 2 FORWARD AND FUTURES CONTRACTS 2.1 INTRODUCTION The main purpose of forward and fuures conracs is he managemen of risk. The exposure o risk as a resul of ransacing in he spo marke

More information

Derivatives. Forwards and Futures. Forward. Futures. Options. Initial Cost

Derivatives. Forwards and Futures. Forward. Futures. Options. Initial Cost Derivaives Forwards and Fuures A derivaive securiy is a securiy whose value depends on he values of oher more basic underlying variables. Forward The mos common derivaive securiies are forward, fuures

More information

Fifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance

Fifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of bes-esimae provisions... 3 2.1

More information

Does Option Trading Have a Pervasive Impact on Underlying Stock Prices? *

Does Option Trading Have a Pervasive Impact on Underlying Stock Prices? * Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? * Neil D. Pearson Universiy of Illinois a Urbana-Champaign Allen M. Poeshman Universiy of Illinois a Urbana-Champaign Joshua Whie Universiy

More information

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.

More information

Pricing Futures and Futures Options with Basis Risk

Pricing Futures and Futures Options with Basis Risk Pricing uures and uures Opions wih Basis Risk Chou-Wen ang Assisan professor in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan Ting-Yi Wu PhD candidae in

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

Technical Description of S&P 500 Buy-Write Monthly Index Composition

Technical Description of S&P 500 Buy-Write Monthly Index Composition Technical Descripion of S&P 500 Buy-Wrie Monhly Index Composiion The S&P 500 Buy-Wrie Monhly (BWM) index is a oal reurn index based on wriing he nearby a-he-money S&P 500 call opion agains he S&P 500 index

More information

Efficient Pricing of Energy Derivatives

Efficient Pricing of Energy Derivatives Efficien Pricing of Energy Derivaives Anders B. Trolle EPFL and Swiss Finance Insiue March 1, 2014 Absrac I presen a racable framework, firs developed in Trolle and Schwarz (2009), for pricing energy derivaives

More information

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely

More information

Modeling VXX. First Version: June 2014 This Version: 13 September 2014

Modeling VXX. First Version: June 2014 This Version: 13 September 2014 Modeling VXX Sebasian A. Gehricke Deparmen of Accounancy and Finance Oago Business School, Universiy of Oago Dunedin 9054, New Zealand Email: sebasian.gehricke@posgrad.oago.ac.nz Jin E. Zhang Deparmen

More information

APPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY. January, 2005

APPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY. January, 2005 APPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY Somnah Chaeree* Deparmen of Economics Universiy of Glasgow January, 2005 Absrac The purpose

More information

FIN 472 Fixed-Income Securities Approximating Price Changes: From Duration to Convexity Professor Robert B.H. Hauswald Kogod School of Business, AU

FIN 472 Fixed-Income Securities Approximating Price Changes: From Duration to Convexity Professor Robert B.H. Hauswald Kogod School of Business, AU FIN 47 Fixed-Income Securiies Approximaing rice Changes: From Duraion o Convexiy rofessor Rober B.H. Hauswald Kogod School of Business, AU Bond rice Volailiy Consider only IR as a risk facor Longer M means

More information

Fixed Income Analysis: Securities, Pricing, and Risk Management

Fixed Income Analysis: Securities, Pricing, and Risk Management Fixed Income Analysis: Securiies, Pricing, and Risk Managemen Claus Munk This version: January 23, 2003 Deparmen of Accouning and Finance, Universiy of Souhern Denmark, Campusvej 55, DK-5230 Odense M,

More information

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul universiy of copenhagen Universiy of Copenhagen A Two-Accoun Life Insurance Model for Scenario-Based Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:

More information