# Calculation of variable annuity market sensitivities using a pathwise methodology

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 cuing edge Variable annuiie Calculaion of variable annuiy marke eniiviie uing a pahwie mehodology Under radiional finie difference mehod, he calculaion of variable annuiy eniiviie can involve muliple Mone Carlo imulaion, leading o high compuaional co A pahwie approach reduce hi dramaically, while providing an unbiaed eimae By Carey Hobb, Bala Krihnaraj, Ying Liu and Jay Muelman 1 Inroducion Variable annuiie (VA) wih guaraneed minimum benefi (deah, accumulaion, income, or wihdrawal) repreen produc wih complex embedded derivaive rucure Deermining he marke value and eniiviie ( greek ) of hee produc i imporan in a number of conex However, generaion of hee value can be expenive o compue uing he mo prevalen mehod in he indury In order o evaluae an in-force book in erie, buinee mu employ ignifican compuing reource a coniderable co o generae pricing and eniiviy run For inance, grid enabled compuing farm or high peed compuing faciliie generae ochaic cenario for each conrac Furher, any geography or axonomy employed by he buine o raify he in-force book will need o be modelled, uch a indice or fund group, and hee will ofen be hocked and imulaed o obain componen greek ued for hedging purpoe Addiionally, increaed ale, he growing complexiy of produc, muli-rik facor modelling, and heerogeneiy in rike and ime-o-mauriy pace of aging in-force book can imply addiional imulaed pah o enure convergence and abiliy A uch, any innovaion in he compuaional apec of hee Mone Carlo imulaion could have ignifican impac hi paper will ouline how o develop and apply an alernaive o finie difference baed on he pahwie differeniaion mehod (or infinieimal perurbaion) o calculae marke value eniiviie hi moly em from exenion of Broadie & Glaerman, 1996 for pecific applicaion o VA conrac We begin by reviewing hee reul applied o vanilla equiy opion and exend he mehod o a complex guaraneed minimum wihdrawal benefi (GMWB) conrac Finally, we conclude wih a urvey of applicaion of he mehod in calculaing a wide variey of greek for differen model including hoe wih dynamic behaviour 2 Review of he finie difference mehod A common mehod ued o price VA guaranee i he Mone Carlo imulaion framework, where a number of differen economic cenario, ofen rik neural, are projeced and he value i equal o he dicouned expeced value of he payoff acro all cenario Greek, or marke value derivaive uch a dela, rho, vega, or gamma, are ofen eimaed wih a Mone Carlo imulaion approach uing finie difference approximaion, which are a andard indirec mehod of eimaing a derivaive value uing re-imulaion of pah and perurbing iniial value Addiionally, i i common ha he higher he order of derivaive, he more pah are required o guaranee convergence, creaing coniderable pracical iue involving co and runime he finie difference approximaion for dela i defined a: + h V ( S 0 ) h 2 V ( S 0 ) +K V ( S 0 ) + h V ( S 0 ) 1 2 h 2 V ( S 0 ) K = V S h 2 V ( S 0 ) V S 0 One of he key challenge ariing wih he finie difference cheme, a hown above, i he exience of an error erm over V (S 0 ) ha creae a biaed eimae of he greek hi erm will end o zero a h end o zero, bu here i a rade-off beween elecing a mall enough meh ize o obain a negligible bia and mainaining an accepable level of variance (ee Glaerman, 2003) 2h 40 Life & Penion

2 cuing edge Variable annuiie 3 Pahwie mehod he pahwie mehod i an aracive alernaive o he finie difference approximaion of deermining greek eniiviie of a financial derivaive; i i a direc mehod ha no only offer reduced compuaional effor, bu i alo unbiaed in i eimae of he derivaive he pahwie mehod leverage he relaionhip beween he ecuriy payoff and he iniial parameer of inere and i obained by differeniaing he expreion for he payoff of he derivaive wih repec o he parameer A uch, higher order eniiviie can be expreed a a funcion of he iniial parameer and eimaed from he iniial pah, perhap generaed during a bae marke valuaion 31 Overview of pahwie mehod o demonrae he pahwie mehod conider he price of a European call opion defined o be: C = E e r max S K where E i he rik-neural expecaion operaor, i he ime o mauriy of he opion, r i he conan inere rae from ime 0 o, K i he rike price, and S i he price of he ock a ime he pahwie dela of he opion i defined o be: C = ( ) = { S S E e r S K E e r I S max S K} 0 0 S 0 where I {S i he indicaor funcion which equal uniy when he K} brackeed condiion i aified and zero oherwie We have inerchanged he expecaion and he derivaive operaor, which i permied for Lipchiz coninuou payoff Auming ha under he rik-neural meaure S evolve wih lognormal dynamic, a in he claic Black-Schole-Meron model, we have So ha S = S 0 e σ 2 ( Z + ( r σ 2 ) ) = S I i clear from he above formula ha in order o obain he dela of he opion uing he pahwie mehod one need o a mo know he value of S on each cenario he value of S i available from he bae marke value Mone Carlo imulaion and, herefore, addiional imulaion i no required o obain he opion dela Broadie & Glaerman how ha hi mehod can be exended o opion which are pah-dependen, ubjec o cerain coninuiy condiion We have uccefully applied hi o a number of more complex rucure However, we alo recommend ha he reader conul he reference cied herein o beer appreciae he limiaion of hi mehod, which involve real analyi and are ouide he cope of hi paper S 0 In he following ecion we demonrae how he mehod can be exended o complex pah dependen financial derivaive uch a a GMWB conrac 4 Pahwie dela for GMWB conrac 41 GMWB conrac pecificaion In wha follow we will chooe a paricularly rich wihdrawal benefi o demonrae he mehod Conider a GMWB conrac where he policy holder can elec o ar aking wihdrawal a any poin unil conrac mauriy Addiionally, he wihdrawal amoun are defined a a percenage of a benefi bae by he iniial wihdrawal he benefi bae will depend on he underlying accoun value he price of hi conrac no only depend on he underlying ae value mechanic, bu i alo dependen on he policy holder behaviour However, a hi poin in he model developmen, we will aume aic behaviour In ecion 44 we how ha hi mehodology work in he preence of dynamic behaviour We define he value of he GMWB conrac a: V = E D p c max( WD AV ) = 0 where E i he rik-neural expecaion of he dicouned payoff of he GMWB policy, D i he dicoun facor from ime 0 o, p c i he probabiliy of urvival o ime given a aic lape aumpion, WD i he wihdrawal a ime, and AV i defined a he accoun value of he conrac a ime he AV will be equal o he accoun value a ime -1 le wihdrawal accumulaed by an equiy reurn r a given by: AV = ( AV 1 WD 1 )e r where WD, i defined a a percenage of benefi bae, BB, a, and AV 0 = P S 0 where P i ued o denoe he iniial premium, and AV 0 can herefore be inerpreed a a noional amoun In many GMWB rucure markeed oday he benefi bae roll up hrough ime and can rache if AV > BB -1 In addiion, many annuiy provider offer feaure where he benefi bae i he high waer mark of he accoun value and he rolled up previou benefi bae We define: BB = max AV, BB 1 WD = w BB where w i he rae of wihdrawal and i aumed o be conan and reaonably bounded We aume wo differen wihdrawal cohor: Cohor 1 i aumed o ar aking wihdrawal a he end of he 1 year and Cohor 2 a he end of he 10h year he marke valuaion model developed exend unil =30 year In general he pecific of he cohor approach or variaion of modelling wihdrawal behaviour are a he dicreion of he modeller wwwlife-penioncom Sepember

3 cuing edge Variable annuiie 42 Pahwie dela for he GMWB conrac he pahwie dela for he GMWB conrac pecified i defined a: PW = V = E D p c max WD AV S 0 = 0 Auming aic behaviour he derivaive will ac excluively on he hird erm reuling in he following expreion max WD AV = I WD > AV { } WD AV Secion 44 deal wih derivaive on he behaviour funcion when i i dynamic he reulan PW of he GMWB conrac i he dicouned (ime 0) rik neural expecaion acro he variou cenario wih he appropriae derivaive for WD and AV numerically calculaed in he imulaion I i worh noing ha derivaive involving he guaranee ake on recurive form ha lend hemelve o imple coding: AV = AV 1 S 0 which erminae wih he condiion WD WD 1 er = P Furher, = BB w = AV max( AV, BB 1 ) w = I BB S R + I 1 0 S NR 0 w he indicaor funcion in hi la expreion exi o rack wheher or no he accoun value racheed a ime, R, or no, NR 43 Pahwie key rae duraion for he GMWB conrac o exend he mehod o a more complex problem we will apply he pahwie mehod o compuing key rae duraion (rho), or imply KRD Rho eniiviie are of paricular relevance o hee produc deign ince i become imporan o recognie ae liabiliy managemen need and poibly hedge he long daed inere rae rik of he produc KRD are preferable in hee cae ince hey permi effecive hedging a numerou enor of, for inance, he wap curve, and ake ino accoun poible non-parallel yield curve dynamic (ee, for inance, uckman, 2002) We will begin wih a review of key rae eniiviie he KRD i defined o be he vecor of n fir order inere rae eniiviie a each enor uch ha he reulan change in liabiliy ariing from he inere rae impac i: KRD 1 δr 1 δ [ Liabiliy] = KRD δr = KRD 2 M KRD n δr 2 M δr n where δ[] indicae a change over ome period, and r i he po rae a ime he liabiliy arie due o he embedded opion from he guaranee in he GMWB conrac I i change in hee liabiliie ha he praciioner would eek o hedge, perhap wih wap radiionally, he vecor KRD involve performing finie differencing a each Clearly finie differencing can conume coniderable runime when numerou hocked key rae are required Wih he pahwie mehod one only require he bae valuaion run o ge inere rae eniiviie a every enor Le he pahwie KRD be defined a: KRD PW = V r = V E D p c max( WD AV ) r = 0 We are now concerned wih wo derivaive wihin he expecaion, pecifically, he dicoun facor and he guaranee I i clear ha any change in he po rae r will affec he adjacen forward a ime and +1 Hence, D r = D r + D r where we have now inroduced he forward rae, f I i worh noing ha derivaive involving po and forward rae ake he following form: D = D d δ where d = 025 year in our model and δ i he Kronecker dela equal o one if = Furher, = f r r = r r d d r ( d ) + 1 Derivaive of he guaranee ake he form: r max WD AV = I WD > AV { } WD r d AV r I i imporan o noe ha for < he above expreion i zero Similar o he cae of dela, recurive relaionhip arie when derivaive on he guaranee are compued Specifically, AV = + AV f AV + r f r r + 1 where for k=, +1 he accoun value varie in forward rae by AV = AV k k WD 1 1 and wihdrawal vary in forward rae by WD k = BB k k 1 rk e + AV rk ( 1 WD 1) e d MAW = AV max( AV, BB 1 ) w = I BB R + I k NR k k w Where again he indicaor funcion rack wheher or no he accoun value racheed a ime, R, or no, NR 42 Life & Penion

4 cuing edge Variable annuiie A hi poin we have idenified all of he perinen expreion required o generae pahwie dela and KRD eniiviie for he pecified GMWB conrac wih aic behaviour 44 Dynamic behaviour Dynamic behaviour i a complex funcion of moneyne above aic bae lape aumpion (imilar o a morgage prepaymen funcion) Ofen, buinee and academic reearcher employ complicaed proprieary funcion derived for exiing produc or from experience udie he augmened GMWB conrac dela in he preence of dynamic behaviour i: V = V + p E D p S = S S V he augmened GMWB conrac KRD in he preence of behaviour i: V = D E + + V p p V p V r r r r D = 0 V where we have inroduced he (now dynamic) p Le he probabiliy of urvival be defined a p = p p 1 uch ha p depend on ome (o be defined) funcion φ(s 0 ) p = 1 φ( S 0 ) Derivaive in he behaviour funcion will involve differeniable approximaion of he indicaor funcion, ince i conain expreion for AV and lape will occur when he benefi i ou-of-he-money For he implemenaion of hi cheme we employed he following approximaion: anh x ε H x able 1: Comparion of ime 0 dela a a percenage of iniial premium for a GMWB conrac wih no policyholder behaviour Cohor 1 Cohor 2 Finie difference dela -7589% -2526% Pahwie dela -7592% -2550% able 2: Comparion of ime 0 dela a a percenage of iniial premium for a GMWB conrac wih policyholder behaviour Cohor 1 Cohor 2 Finie difference dela -6302% -3390% Pahwie dela -6333% -3399% 45 Implemenaion Implemenaion wa performed wih excel/vba calling malab funcion and run were performed on everal ordinary PC One of he main challenge in hi implemenaion i o keep rack of he variou indicaor hrough ime in he imulaion hee indicaor would evenually be ued o calculae greek a he end of he imulaion Addiionally, higher order or more complex greek, uch a vanna or key rae vega, rely on careful coding and derivaion of he perinen derivaive he reducion in runime due o no re-imulaion valy ouweighed he increaed complexiy of racking hee indicaor While exac number are highly dependen on coding language, produc pecific, yem, and any opimizaion, we oberved run-ime reducion of 44% for he inrumen modelled for hi paper In all imulaion 5000 rik-neural pah were ued for each cohor 46 Dicuion of reul able 1 and 2 how comparion of dela uing he pahwie mehod o ha of he radiional finie difference mehod We find ha he pahwie mehod of obaining dela give u aifacory value wih and wihou dynamic behaviour In addiion, Figure 1 how ha he dela uing boh mehod are roughly idenical a he GMWB conrac age In addiion, he pahwie mehod i een o be robu when re eed In Figure 2, we how he impac of changing moneyne on he dela uing boh he mehod We find ha he dela calculaed uing pahwie coincide wih ha of he finie difference mehod he andard error i impaced wih hi approach a hown in able 3 In general he pahwie mehod will have le andard error relaive o finie difference for differen h and N hi relaionhip beween h and N can be beneficial wih higher order greek where convergence may be dependen on hock ize able 4 how a imilar comparion Figure 1 Dynamic behaviour dela Dela 5% 0% -5% -10% -15% -20% -25% -30% Valuaion year Finie difference dela Pahwie dela Comparion of dela for a GMWB conrac ample pah during he fir 10 year wih dynamic behaviour Figure 2 Variable moneyne dela Dela 5% 0% -5% -10% -15% -20% Moneyne Finie difference dela Pahwie dela Comparion of dela for a GMWB conrac (Cohor 1) when moneyne (AV 0 /BB 0 ) i varied 00% 080% -20% 070% 060% -40% 050% -60% 040% -80% 030% -100% 020% -120% 010% -140% 000% Valuaion year Finie difference rho error Pahwie rho error Finie difference rho Pahwie rho Rho Figure 3 KRD projecion Comparion of KRD projecion for a GMWB conrac (Cohor 1) Sandard error wwwlife-penioncom Sepember

5 cuing edge Variable annuiie for KRD for = 1, 3, 5, 7, 10 uing he pahwie mehod and he radiional finie difference mehod he projecion in Figure 3 of one KRD10 pah alo how conien agreemen wih finie difference along a 10 year projecion I i worh noing ha he periodic piking in he error, while mall in abolue magniude, i he reul of he annual rache feaure rendering he opion AM hi being only one conrac, he porfolio effec will end o mooh hi feaure, bu imilar o dela of a book of pah dependen opion, here may be non-rivial convexiy co o hedging a porfolio of hee conrac if here i any ype of heerogeneiy in conrac anniverarie due o pike in ale or eaonaliy hi undercore he value of he pahwie mehod in ha he abiliy o perform projecion and need hedge imulaion are very imporan for proper rik managemen and produc developmen of long daed produc wih hee complex feaure Rache and wihdrawal end o diplay complex behaviour and impac greek ued in hedging hi lend ielf o he noion ha i i imporan and good rik managemen o be able o calculae a va number of complex greek, and projec hem for behaviour and imulaion purpoe o beer underand hedge impac and raegy deign We have ofen found ha i i compuing ime and reource ha make hi challenging o manage 5 Exenion of mehodology he power of he pahwie mehodology increae a more greek are calculaed uing he bae marke valuaion imulaion We have exended hi framework o calculae oher greek, uch a key rae vega, gamma, and componen dela and gamma By exenion he mahemaic become more cumberome emming from manipulaing he variable in he fir order expreion, uch a he indicaor funcion Performing ucceful differeniaion may involve numerical or funcional approximaion ha have cerain limi of appropriae ue or involve more complex mahemaic, like generalied funcion, which are beyond he cope of hi paper We have uccefully calculaed complex GMIB and GMWB greek wih hi mehod, however, i i imporan o noe ha many payoff funcion generally encounered may no be C 2 and oher mehod, like imporance ampling, may need o be employed Laly, hi mehod can admi more ochaic rik facor, hould hey be warraned, uch a ochaic rae or volailiy in he bae pricing run 6 Concluion In hi paper we have derived and implemened a pahwie greek mehodology for a complex GMWB conrac he moivaion emmed mainly from he deire o reduce he expenive compuing ime for compuing marke value eniiviie and he reulan addiional reource ha may be employed for hedging, raegy developmen, ae-liabiliy managemen and reearch hi i of pracical ue for many marke paricipan, epecially hedging or raegy R&D eam he mehod produce an unbiaed eimae of greek We have uccefully exended hi mehod o oher produc and oher greek However, we would like o end on a cauionary noe o fully underand he limiaion of hi mehod, when i i appropriae o ue, and o alway have anoher, more brue force mehod of validaing and checking reul readily available, uch a finie difference L&P he auhor are member of Derivaive Sraegie eam wihin ING USFS Marke Rik Managemen Addiionally, we wih o hank he reviewer for heir helpful commen able 3: Comparion of dela andard error for a GMWB conrac (Cohor 1) for variou h and N Sandard error (h=0001) Sandard error (h=00001) Finie difference dela (N = 500) Finie difference dela (N = 5000) Pahwie dela (N = 500) Pahwie dela (N = 5000) able 4: Comparion of KRD value a a percenage of iniial premium for a GMWB conrac wihou policyholder behaviour Key rae Finie difference KRD Pahwie KRD Difference 1 805% 804% 001% % 1035% 003% 5 879% 878% 001% 7 739% 742% -003% % -400% 000% he view expreed herein are hoe of he auhor and no of ING Group or any of i global buine uni Reference 1 Broadie, Mark & Glaerman, Paul Eimaing Securiy Price Derivaive Uing Simulaion Managemen Science, V42, No 2, , February Glaerman, Paul Mone Carlo Mehod in Financial Engineering Springer, uckman, Bruce Fixed Income Securiie Wiley Finance, 2nd Ed 2002, Page Life & Penion welcome ubmiion o i peer-reviewed Cuing Edge ecion Aricle hould be en o Submiion guideline are available a hp://wwwlife-penioncom/ public/howpagehml?page= Life & Penion

### 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

### How Much Can Taxes Help Selfish Routing?

How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU) Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from

### 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

### How has globalisation affected inflation dynamics in the United Kingdom?

292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he

### A Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting

A Comparaive Sudy of Linear and Nonlinear Model for Aggregae Reail Sale Forecaing G. Peer Zhang Deparmen of Managemen Georgia Sae Univeriy Alana GA 30066 (404) 651-4065 Abrac: The purpoe of hi paper i

### Topic: Applications of Network Flow Date: 9/14/2007

CS787: Advanced Algorihm Scribe: Daniel Wong and Priyananda Shenoy Lecurer: Shuchi Chawla Topic: Applicaion of Nework Flow Dae: 9/4/2007 5. Inroducion and Recap In he la lecure, we analyzed he problem

### A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *

A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone:

### Stock option grants have become an. Final Approval Copy. Valuation of Stock Option Grants Under Multiple Severance Risks GURUPDESH S.

Valuaion of Sock Opion Gran Under Muliple Severance Rik GURUPDESH S. PANDHER i an aian profeor in he deparmen of finance a DePaul Univeriy in Chicago, IL. gpandher@depaul.edu GURUPDESH S. PANDHER Execuive

### CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton

CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA R. L. Chamber Deparmen of Social Saiic Univeriy of Souhampon A.H. Dorfman Office of Survey Mehod Reearch Bureau of Labor Saiic M.Yu. Sverchkov

### The International Investment Position of Jamaica: An Estimation Approach

WP/04 The Inernaional Invemen Poiion of Jamaica: An Eimaion Approach Dane Docor* Economic Informaion & Publicaion Deparmen Bank of Jamaica Ocober 2004 Abrac Thi paper eek o inroduce he inernaional invemen

### 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

### 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

### 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

### Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities

Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for

### 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

### Modeling Energy American Options in the Non-Markovian Approach

Modeling Energy American Opion in he Non-Markovian Approach Valery Kholodnyi Vienna Auria 06.05.015 VERBUND AG www.verbund.com Ouline Ouline Inroducion Mehodology he Non-Markovian Approach Modeling Energy

### 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

### 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

### Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical

### New Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and

New Evidence on Muual Fund Performance: A Comparion of Alernaive Boorap Mehod David Blake* Trian Caulfield** Chrio Ioannidi*** and Ian Tonk**** June 2014 Abrac Thi paper compare he wo boorap mehod of Koowki

### Chapter 1.6 Financial Management

Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

### 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

### Fortified financial forecasting models: non-linear searching approaches

0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: non-linear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,

### Heat demand forecasting for concrete district heating system

Hea demand forecaing for concree diric heaing yem Bronilav Chramcov Abrac Thi paper preen he reul of an inveigaion of a model for hor-erm hea demand forecaing. Foreca of hi hea demand coure i ignifican

### 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

### Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear

### Equity Valuation Using Multiples. Jing Liu. Anderson Graduate School of Management. University of California at Los Angeles (310) 206-5861

Equiy Valuaion Uing Muliple Jing Liu Anderon Graduae School of Managemen Univeriy of California a Lo Angele (310) 206-5861 jing.liu@anderon.ucla.edu Doron Niim Columbia Univeriy Graduae School of Buine

### ARTICLE IN PRESS Journal of Computational and Applied Mathematics ( )

Journal of Compuaional and Applied Mahemaics ( ) Conens liss available a ScienceDirec Journal of Compuaional and Applied Mahemaics journal homepage: www.elsevier.com/locae/cam Pricing life insurance conracs

### 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:

### Explore the Application of Financial Engineering in the Management of Exchange Rate Risk

SHS Web o Conerence 17, 01006 (015) DOI: 10.1051/ hcon/01517 01006 C Owned by he auhor, publihed by EDP Science, 015 Explore he Applicaion o Financial Engineering in he Managemen o Exchange Rae Rik Liu

### 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

### 2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics

.4 Nework flow Problem involving he diribuion of a given produc (e.g., waer, ga, daa, ) from a e of producion locaion o a e of uer o a o opimize a given objecive funcion (e.g., amoun of produc, co,...).

### SOLVENCY II: QIS5 FOR NORWEGIAN LIFE AND PENSION INSURANCE

SOLVENCY II: QIS5 FOR NORWEGIAN LIFE AND PENSION INSURANCE BY KEVIN DALBY THESIS for he degree of MASTER OF SCIENCE (Modeling and Daa Anali) Facul of Mahemaic and Naural Science UNIVERSITY OF OSLO Ma 2011

### The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies

1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz- und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany

### 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

### 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

### 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

### The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.

The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081

### 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:

### Trading Strategies for Sliding, Rolling-horizon, and Consol Bonds

Trading Sraegie for Sliding, Rolling-horizon, and Conol Bond MAREK RUTKOWSKI Iniue of Mahemaic, Poliechnika Warzawka, -661 Warzawa, Poland Abrac The ime evoluion of a liding bond i udied in dicree- and

### Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs

Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edge-dijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.

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

### CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly

### Optimal Longevity Hedging Strategy for Insurance. Companies Considering Basis Risk. Draft Submission to Longevity 10 Conference

Opimal Longeviy Hedging Sraegy for Insurance Companies Considering Basis Risk Draf Submission o Longeviy 10 Conference Sharon S. Yang Professor, Deparmen of Finance, Naional Cenral Universiy, Taiwan. E-mail:

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

Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics Friedrich-Alexander-Universiy

### 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.

### 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

### On the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes

On he Managemen of Life Insurance Company Risk by raegic Choice of Produc Mix, Invesmen raegy and urplus Appropriaion chemes Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen Working Paper Deparmen

### The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees

1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081

### 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

### 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

### 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

### 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

### 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

### Robust Bandwidth Allocation Strategies

Robu Bandwidh Allocaion Sraegie Oliver Heckmann, Jen Schmi, Ralf Seinmez Mulimedia Communicaion Lab (KOM), Darmad Univeriy of Technology Merckr. 25 D-64283 Darmad Germany {Heckmann, Schmi, Seinmez}@kom.u-darmad.de

### Empirical heuristics for improving Intermittent Demand Forecasting

Empirical heuriic for improving Inermien Demand Forecaing Foio Peropoulo 1,*, Konanino Nikolopoulo 2, Georgio P. Spihouraki 1, Vailio Aimakopoulo 1 1 Forecaing & Sraegy Uni, School of Elecrical and Compuer

### 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

### The Twin Agency Problems in Corporate Finance - On the basis of Stulz s theory -

The Twin Agency Problem in Corporae Finance - On he bai of Sulz heory - Von der Fakulä für Machinenbau, Elekroechnik und Wirchafingenieurween der Brandenburgichen Technichen Univeriä Cobu zur Erlangung

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

### 6.003 Homework #4 Solutions

6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d

### 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

### Performance Center Overview. Performance Center Overview 1

Performance Cener Overview Performance Cener Overview 1 ODJFS Performance Cener ce Cener New Performance Cener Model Performance Cener Projec Meeings Performance Cener Execuive Meeings Performance Cener

### 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

### Chapter Four: Methodology

Chaper Four: Mehodology 1 Assessmen of isk Managemen Sraegy Comparing Is Cos of isks 1.1 Inroducion If we wan o choose a appropriae risk managemen sraegy, no only we should idenify he influence ha risks

### Subsistence Consumption and Rising Saving Rate

Subience Conumpion and Riing Saving Rae Kenneh S. Lin a, Hiu-Yun Lee b * a Deparmen of Economic, Naional Taiwan Univeriy, Taipei, 00, Taiwan. b Deparmen of Economic, Naional Chung Cheng Univeriy, Chia-Yi,

### Market Liquidity and the Impacts of the Computerized Trading System: Evidence from the Stock Exchange of Thailand

36 Invesmen Managemen and Financial Innovaions, 4/4 Marke Liquidiy and he Impacs of he Compuerized Trading Sysem: Evidence from he Sock Exchange of Thailand Sorasar Sukcharoensin 1, Pariyada Srisopisawa,

### This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.

This documen is downloaded from DR-NTU, Nanyang Technological Universiy Library, Singapore. Tile A Bayesian mulivariae risk-neural mehod for pricing reverse morgages Auhor(s) Kogure, Asuyuki; Li, Jackie;

### Two-Group Designs Independent samples t-test & paired samples t-test. Chapter 10

Two-Group Deign Independen ample -e & paired ample -e Chaper 0 Previou e (Ch 7 and 8) Z-e z M N -e (one-ample) M N M = andard error of he mean p. 98-9 Remember: = variance M = eimaed andard error p. -

### GMWB For Life An Analysis of Lifelong Withdrawal Guarantees

GMWB For Life An Analysis of Lifelong Wihdrawal Guaranees Daniela Holz Ulm Universiy, Germany daniela.holz@gmx.de Alexander Kling *) Insiu für Finanz- und Akuarwissenschafen Helmholzsr. 22, 8981 Ulm, Germany

### 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,

### Money-Back Guarantees in Individual Pension Accounts: Evidence from the German Pension Reform

No. 22/3 Money-Back Guaranees in Individual Pension Accouns: Evidence from he German Pension Reform Raimond Maurer / Chrisian Schlag Cener for Financial Sudies an der Johann Wolfgang Goehe-Universiä Taunusanlage

### Longevity 11 Lyon 7-9 September 2015

Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univ-lyon1.fr

### Life insurance cash flows with policyholder behaviour

Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,

### I. Basic Concepts (Ch. 1-4)

(Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

### 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

### On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction

On he Connecion Beween Muliple-Unica ework Coding and Single-Source Single-Sink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg ework Error Correcion Problem: Adverary

### 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

### Lecture III: Finish Discounted Value Formulation

Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

### Cross-sectional and longitudinal weighting in a rotational household panel: applications to EU-SILC. Vijay Verma, Gianni Betti, Giulio Ghellini

Cro-ecional and longiudinal eighing in a roaional houehold panel: applicaion o EU-SILC Viay Verma, Gianni Bei, Giulio Ghellini Working Paper n. 67, December 006 CROSS-SECTIONAL AND LONGITUDINAL WEIGHTING

This page inenionally lef blank Marke-Valuaion Mehods in Life and Pension Insurance In classical life insurance mahemaics, he obligaions of he insurance company owards he policy holders were calculaed

### 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

### Long Term Spread Option Valuation and Hedging

Long Term Spread Opion Valuaion and Hedging M.A.H. Demper, Elena Medova and Ke Tang Cenre for Financial Reearch, Judge Buine School, Univeriy of Cambridge, Trumpingon Sree, Cambridge CB 1AG & Cambridge

### 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

### ON A FAIR VALUE MODEL FOR PARTICIPATING LIFE INSURANCE POLICIES

Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 05 ON A FAIR VALUE MODEL FOR PARTICIPATING LIFE INSURANCE POLICIES Fabio Baione, Paolo De Angeli, Andrea Forunai Abrac The aim of hi aer

### 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

### OPTIMAL BATCH QUANTITY MODELS FOR A LEAN PRODUCTION SYSTEM WITH REWORK AND SCRAP. A Thesis

OTIMAL BATH UANTITY MOELS FOR A LEAN ROUTION SYSTEM WITH REWORK AN SRA A Thei Submied o he Graduae Faculy of he Louiiana Sae Univeriy and Agriculural and Mechanical ollege in parial fulfillmen of he requiremen

### 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,

### 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.

### Three Dimensional Grounding Grid Design

Three Dimenional Grounding Grid Deign Fikri Bari Uzunlar 1, Özcan Kalenderli 2 1 Schneider Elecric Turkey, Ianbul, Turkey bari.uzunlar@r.chneider-elecric.com 2 Ianbul Technical Univeriy, Elecrical-Elecronic

### Appendix D Flexibility Factor/Margin of Choice Desktop Research

Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\223489-00\4

### 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

### 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

### IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include so-called implici or embedded opions.

### 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

### 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

### Variance Swap. by Fabrice Douglas Rouah

Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach

### THE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS

THE IPACT OF THE ECONDARY ARKET ON LIFE INURER URRENDER PROFIT Nadine Gazer, Gudrun Hoermann, Hao chmeiser Insiue of Insurance Economics, Universiy of. Gallen (wizerland), Email: nadine.gazer@unisg.ch,

### Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a