Stochastic Volatility Models: Considerations for the Lay Actuary 1. Abstract

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Stochastic Volatility Models: Considerations for the Lay Actuary 1. Abstract"

Transcription

1 Sochasic Volailiy Models: Consideraions for he Lay Acuary 1 Phil Jouber Coomaren Vencaasawmy (Presened o he Finance & Invesmen Conference, 19-1 June 005) Absrac Sochasic models for asse prices processes are now familiar o acuaries. Many of he models used in life office and pension fund valuaion and asse-liabiliy modelling sudies assume deerminisic volailiy parameers. Empirical evidence however, suggess ha volailiy in asse prices varies wih ime. Furher, volailiies implied by raded opion prices show a erm srucure for implied volailiy, as well as an apparen dependence on he "moneyness" of he opion. These observaions seem o be a odds wih a consan volailiy assumpion. In his paper we presen some empirical observaions concerning volailiy, and consider he impac of volailiy on acuarial work. We hen review some of he common models which incorporae sochasic volailiy and consider issues relaed o parameerising such models. 1 We are indebed o Suar Jones for his paience and for providing invaluable echnical assisance. We would also like o hank Dylan Brooks and Carmela Calvosa for providing daa and fruiful discussions. 1

2 1. Inroducion Volailiy is cenral o many applied issues in finance and financial engineering, ranging from asse pricing and asse allocaion o risk managemen. Financial economiss have always been inrigued by he very high precision wih which volailiy can be esimaed under he diffusion assumpion rouinely invoked in heoreical work. The basic insigh follows from he observaion ha precise esimaion of diffusion volailiy does no require a long calendar span of daa; raher, volailiy can be esimaed arbirarily well from an arbirarily shor span of daa, provided ha reurns are sampled sufficienly frequenly. This conrass sharply wih precise esimaion of he drif, which generally requires a long calendar span of daa, regardless of he frequency wih which reurns are sampled. There is also he baffling range of volailiy erms used: Hisorical Volailiy, Implied Volailiy, Forecas Volailiy, ec.. In his paper, he firs wo erms are mos imporan. Hisorical volailiy is a measure of he previous flucuaions in share price (crudely: an indicaor of he share's up/downess). There is much discussion over he bes mehod of calculaing he hisoric volailiy. The mos usual mehod is he sandard deviaion of he log of price reurns - his procedure is fairly sandard and can be found in mos exbooks. While he calculaion iself is sraigh-forward, i is accurae only wihin he parameers of each calculaion (e.g. he specific ime period: 3 monhs, 3 years ec.). There is grea scope for analysing he share price behaviour over differen ime periods, and hereby calculaing differen hisoric volailiies. Insead of inpuing a volailiy parameer ino an opion model (e.g. Black-Scholes) o deermine an opion's fair value, he calculaion can be urned round, where he acual curren opion price is inpu and he volailiy is oupu. The erm implied volailiy is obviously self-explanaory - ha level of volailiy ha will calculae a fair value acually equal o he curren rading opion price. This calculaion can be very useful when comparing differen opions. The implied volailiy can be regarded as a measure of an opion's "expensiveness" in he marke, and is used by raders seing up combinaion sraegies, where hey have o idenify relaively cheap and expensive opions (even hough hese opions have differen erms). I is perhaps useful o noe ha implied volailiy only has any meaning in he conex of a paricular opion model (i is no inrinsic o he opion iself). So, alhough opions have exised for a long ime, implied volailiy has only had any meaning since he opion pricing model of Fisher Black and Myron Scholes (devised in he early 1970's) saed ha he value of an opion was a funcion of he volailiy of he underlying share price. To calculae he fair value, an opion model requires he inpu of volailiy, or, more precisely, he inpu of: forecas volailiy of he share price over he period o expiry of he opion. The big quesion (he ar) of opion heory is how o esimae his forecas volailiy. This single sage provides gainful employmen for a legion of academics, analyss and raders. An esimae of fuure share price flucuaions - pleny of room for "wooliness" here!

3 Below we explore he relaionship beween hese hree conceps, bu firs we moivae our paper: why volailiy, and why boher wih volailiy models? The paper is organised as follows: Secion one gives a brief hisory of he concep of volailiy, and inroduces some of he producs raded in he broader financial markes. Secion wo discusses he definiions of, and differences beween various conceps of volailiy, and considers problems relaed o reliably esimaing heir values. In secion hree we review evidence ha volailiy is random, and in secion four consider several models of asse prices which aemp o capure his. Secion five comprises of he applicaion of hese models o acuarial problems as alluded o above, he problem of marke-consisen valuaion of life insurance business, and effecive risk conrol of he same. We summarise our conclusions in secion six. Noe ha we will ofen refer o opions in his paper. Everyhing applicable o exchange raded opions can be read as being applicable o life conracs wih guaranees. We ake his as dicaed by he regulaors, and do no ener ino he debae as o wheher his is he mos appropriae mehodology Why volailiy? Recenly volailiy has become par of he sandard acuarial jargon. Paricularly for life company acuaries, discussions abou volailiy wha i is, he appropriae value i should ake, how i behaves, and wha i does, are now par of he job. Modern (Life Company) acuarial work consiss broadly of wo asks valuaion of asses and liabiliies, and risk conrol. For hose involved in valuaion, he move o marke-consisen valuaion has mean applying opion pricing echniques o life insurance conracs. A he hear of hese valuaion echniques is he concep of volailiy, so volailiy has been placed direcly on he balance shees of life companies boh on he asse and liabiliy side. As a resul of his audiors are asking increasingly sophisicaed quesions abou volailiy parameers are hey appropriae for he conracs in quesion, will hey reproduce marke prices a a fine level of deail, or only in aggregae, and if so, why? Key conceps o answering hese quesions are: he asses in a given fund, he erms of he liabiliies, and he levels of guaranees of hose liabiliies. Each combinaion will produce a differen marke consisen value for he liabiliies, and requires a differen value of implied volailiy. In [4] Sheldon and Smih provide mehodologies o make he appropriae choice of implied volailiy based on he asses in a fund. In his paper we will address he remaining wo issues. Risk conrol involves calculaing he possible movemens of asses and liabiliies on an insurer s balance shee. Again, volailiy is a key parameer, and in his case i is inernal risk managers, and possibly he regulaor, who should be asking ough quesions abou valuaion echniques and paricularly abou volailiy parameers. A longer erm concern is he increasing number of invesmen banks developing srucured producs aimed for sale o insurers. Many of hese producs will conain complex derivaive insrumens. Undersanding he wo relaed conceps of volailiy volailiy influencing asse price movemens, and implied volailiy he 3

4 governing derivaive values, will allow us make informed decisions on he value and risks of such producs. The fac ha he well-known Black-Scholes model does no fi empirical observaions will be familiar o mos readers. Below we consider a class of models which aemp o more closely explain he daa. In paricular: opion prices given by he Black-Scholes model do no fi hose observed in he marke, and asse price movemens have fa-ails. Sochasic volailiy models may allow he valuaion acuary o achieve a closer mach o he relevan marke-raded asses, and may allow he risk conrol acuary o capure an area of risk which migh oherwise be ignored. 1.. A brief hisory of volailiy I has become radiional recenly in any paper concerning opions o make a hisorical reference o Louis Bachelier. Since mos acuaries will be unfamiliar wih Bachelier, we provide a brief summary. In his 1900 paper Théorie de la Spéculaion [], Bachelier considered a model of sock markes where prices follow wha is now known as a Wiener process Brownian moion. In his paper he derived, amongs oher hings, he price of a barrier opion a full 73 years before Fisher Black and Myron Scholes published heir famous paper [4]. This resul was remarkable, given ha i predaes he birh of modern saisics, and even Einsein s 1905 paper on Brownian moion. So far ahead of is ime, i was los unil fairly recenly, and insead he glory wen o Black and Scholes. Mos readers will recall ha Black and Scholes surprising (a he ime) resul was ha he value of an opion is independen of he expeced reurn of he underlying sock bu explicily dependen on is (expeced) volailiy. Thus he financial-mahemaical concep of volailiy appears o dae back a leas 100 years. No doub he purely financial concep (inuiively he amoun of variaion manifes in sock prices in a given ime period we consider he concep in some deph below) has been recognised ever since men began rading ogeher in markes. Alhough here is some evidence ha opion-ype conracs were used in ancien Greece, Rome and he Arab world, rading in modern Black-Scholes syle opions (wih heir explici dependence on volailiy) began on he Chicago Board of Exchange in 1973 wih calls wrien on 16 socks. Trading in pus sared 4 years laer [7]. Growh in he opions marke has exploded since hen, wih noional principle on ousanding exchange raded derivaives esimaed a $5rn, wih OTC (Over The Couner) conracs perhaps accouning for five imes ha [1]. The growh in rade of opions essenially rading in volailiy has lead o increasingly complicaed and sophisicaed sraegies being underaken. Marke players wishing o gain exposure purely o volailiy movemens can, for example, adop posiions in opions, and hen dela hedge, or adop a posiion in he underlying which negaes he effec of movemens in he price of he underlying 4

5 bu leaves exposure o changes in implied volailiy. Anoher possible rade is o ake posiions in wo opions wih differen srikes known as a srangle a highly risky rade wih pure volailiy exposure. More recenly several sandardised producs have begun rading which offer cheaper and easier access o pure volailiy plays. In 1998 wo new financial producs were launched: volailiy fuures on he Deusche Terminborse and volailiy swaps from Salomon Smih Barney (now par of Ciigroup) [15]. The firs of hese insrumens allowed raders o hedge agains he movemen in he price of opions wrien on he DAX index due o changes in volailiy [6]. The second allowed raders o gain exposure o only he volailiy of underlying insrumens, wihou labour inensive and expensive dela hedging. The VIX is an index of implied volailiy on he S&P100 index. OTC derivaives are available o rade on his index. 5

6 . Definiions and conceps We now consider exacly wha we mean by volailiy. Primarily we mus disinguish beween wo relaed, bu disinc, conceps: he volailiy of a financial insrumen, and he implied volailiy of an opion wrien on such an insrumen..1. Underlying volailiy The price of a financial insrumen can be hough of as a random variable. In order o describe how much ha price migh vary over a paricular ime period we would look for some appropriae saisic. A naural place o sar would be o consider he average of he price movemens, measured a some ime frequency (every 5 minues, every day, a year end). Since we are ineresed in he scale raher han he direcion of changes, we would ake absolue values, giving a saisic 1 N d N d k 1 S k (.1) Where N d is he number of ime periods observed, and S k is he price change in he k h ime period. Sudens of finance will, however, be familiar wih he fac ha i is more ofen reurns ha are of ineres, raher han absolue price changes. We are hen lead o consider log price changes. Then, for reasons relaed o he differeniabiliy of he modulus funcion, we could ake squares of log price changes, normalised by subracing he mean price change over he observaion period. Finally, aking he square roo of he final saisic would yield he familiar Roo Mean Square (RMS) saisic: 1 N S ln S k k 1 (.) The purpose of rehashing wha will be long since forgoen maerial for mos is o emphasise ha (.), despie being he more familiar equaion, is in some respecs he more arificial. In sudying he behaviour of marke prices (.1) can reveal some imporan aspecs see for example [5], chaper.4, where high frequency daa is analysed. Saisics are almos ineviably quoed as RMS or sample variance, which is useful, given ha he laer is an unbiased esimaor for he populaion variance. Black and Scholes proposed he following dynamics for asse prices: ds S d S dw (.3) This equaion says ha he insananeous change in he price of an asse is driven by a deerminisic average componen, and a random componen given by a normal random variable. Calculaing (.) for a large number of observaions of price changes would give an accurae esimae for direcly. 6

7 The siuaion becomes more complicaed when is non-consan however for example if i varies randomly as in he models considered below... Implied volailiy Above in equaion (.3) we recalled he Black-Scholes model for asse prices. The Black-Scholes price of a call opion wrien on an asse is hen given by he well known formula where C d S N( d ) ln ( r )( T ) S K T Ke r ( T ) N( d ) (.4) I was Black and Scholes original belief ha a hisorically esimaed would be used o derive a single, objecive value for an opion. The marke quickly proved hem wrong. Some elemenary calculus will show ha (.4) is a monoonic funcion of, meaning ha we can inver i. Now he price of an opion is no dicaed by he above equaion based on some exernal value of (as Black and Scholes iniially hough), bu is se, as wih all prices, by supply and demand. So if we inver he funcion, we reveal he value of implied by marke prices. This is referred o as he implied volailiy. As saed above, he prices of all financial insrumens are se by supply and demand. There are only a cerain number of shares in a company. The price of hose shares balances supply and demand. Now consider he following argumen: derivaives wrien on he shares of a company are differen here is no limi o he number of derivaive conracs wrien on he shares, hey can be creaed a no cos in infinie amouns (assuming selemen in cash). So excess demand for opion conracs should be immediaely mached by marke makers aemping o make a profi selling such derivaives. Assuming ha he marke is compeiive, margins should be driven o zero, leaving implied volailiy as he value a which here is zero ne supply of opions. This value should be he marke expecaion of fuure volailiy. The above argumen, whils aracive, is wrong. Firsly, he argumen seeks o relae equaion (.3) o (.4) hrough marke prices. However, as discussed below, given ha (.3) is no an adequae model of he marke, his is no necessarily a robus argumen. Furher, opion markes do no have he posulaed perfec elasiciy of supply. Wriing opions requires capial, which is a scarce resource, and will consric supply, as will many oher fricional coss. There are in fac many reasons o suppose ha implied volailiy would be a bes a biased esimaor of fuure volailiy, including he facs ha opion prices will conain loadings for capial coss and possibly profi. See [4] for a more deailed lis. 7

8 We prefer o hink of implied volailiies as normalised opion prices. Jus as we can compare he price of a 3 monh 5% coupon bond wih ha of a five year 10% coupon bond by comparing he yield, so we can compare prices of opions wih differen srikes, mauriies, ec, by comparing implied volailiies..3. Volailiy as risk Noe ha we have no made he common idenificaion of volailiy wih risk. This idenificaion daes back o he CAPM and beyond, and arises from he assumpions of such early finance models essenially ha invesor preferences or asse reurns are adequaely described by wo parameers. In a world of normal reurns all risk measures are equivalen a porfolio seleced using sandard deviaions of reurns (our underlying volailiy above) and one seleced using Value a Risk as opimisaion parameers will be idenical. However, as discussed below, we do no live in a normal world. In he real world no all risk measures give idenical resuls, and in paricular, he sandard deviaion of reurns is no an adequae measure of risk. (or one of is coheren relaives) is preferred. Volailiy is imporan for risk conrol, as we will see, bu as a risk facor, no a measure of risk. We poin he ineresed reader o he excellen reference [5]. 8

9 3. Empirical evidence of Sochasic Volailiy Having defined wha we mean by volailiy, we now moivae he remainder of our discussion by discussing evidence ha volailiy varies sochasically. We (broadly) follow [18] here in presening some empirical observaions of observed price behaviour in boh he cash and opions markes. We consider some economic explanaions, and relae hem o he opic a hand: 3.1 Fa ails I is now generally acceped ha he empirical disribuion of asse reurns is lepokuric meaning (roughly) ha he fourh momen abou he mean is greaer han he same saisic for a normal disribuion wih he same variance. See for example [5] chaper. This means ha more exreme reurns, and fewer midrange reurns are observed, han would be expeced under a Gaussian disribuion. Emperical reurn VS Normal PDF Figure 3.1 Empirical daily S&P log reurn disribuion 1 June December 004 vs. Gaussian PDF. The higher peak and faer ails of he empirical disribuion are eviden. 3. Volailiy clusering & persisence A glance a a financial ime series ofen immediaely reveals periods of high volailiy and periods of low volailiy. Figure 3. S&P daily log reurn absolue value, 004. Period of high volailiy circled in red, period of low volailiy in green. 9

10 In fac, fa ails and volailiy clusering are wo sides of he same coin. I is well known ha a mixure of disribuions, for example price changes disribued according o a normal disribuion, bu wih a random variance, can replicae fa ails. However, boh fa ails and volailiy clusering may be equally well explained by direcly modelling he underlying price disribuion as having fa-ails. Anoher empirical fac is he persisence of volailiy regimes here are periods of high volailiy, and periods of low volailiy, no jus random incidences. This observaion indicaes somehing abou any proposed model of volailiy. See [16] for an ineresing characerisaion of his behaviour, or [17] for furher developmen. 3.3 Leverage effecs The empirical observaion ha volailiy and share prices are negaively correlaed is well known. The erm leverage effec was firs coined in [3]. The argumen is ha falling share prices increase he deb-o-equiy raio of firms. This leads o higher uncerainy or risk, which increases he volailiy of he share price. Hence price movemens and volailiy are negaively correlaed. In marke lore he same phenomenon is ofen explained by he fear and greed effec. When imes are good and prices are rising, raders become lazy, and are happy o si back as heir P&L s increase. Fewer rades means lower volailiy. When prices sar dropping however, raders (and heir cliens) sar panicking, rush o cover posiions, and generally creae more marke aciviy (and hence volailiy). 3.4 Informaion arrivals and marke aciviy Movemens in share prices occur due o he arrival of informaion (his is essenially he efficien markes hypohesis). Clearly news does no arrive as a seady sream; hence he iming of price movemens is random. However, he random process governing he arrival of informaion is such ha i seems o be incompaible wih simple models such as he Black-Scholes geomeric Brownian moion model of share prices. One can consider a rading day: he markes open a 08:00, and here is a flurry of aciviy as raders and heir cliens look o ac on informaion read in he morning newspapers. Afer an hour or wo, he marke calms down a lile, wih raders keeping an eye on he Reuers monior. During he mid-afernoon raders may go ou for a coffee, few big rades will go hrough and he marke will generally be quie. Then here will be anoher flurry of aciviy jus before he marke closes, as people close ou open posiions, companies ry o announce unflaering informaion a he las possible minue, ec. In fac, sudies show precisely hese sors of inraday seasonal aciviy effecs. Similar effecs are observed on longer ime inervals. Furher sudies show ha volailiy and marke aciviy are correlaed. 10

11 3.5 Volailiy co-movemens We observe ha volailiy is dependen across markes. The old adage ha when New York sneezes, London caches a cold holds in he scale of price movemens as well as he direcion. When modelling muliple asse classes i may be imporan o capure his behaviour. 3.6 Implied volailiies A sudy of implied volailiy, as discusses above, is primarily a sudy of normalised opion prices. However, implied volailiies of opions on a given underlying, bu wih differing srikes and mauriies, reveals some ineresing behaviour, which can be direcly linked o he observaions above. Firsly, implied volailiies vary wih erm o expiry srucure of implied volailiy. we call his he erm If we accep ha implied volailiies are, in par, an esimae of fuure volailiy, hen i seems reasonable ha a rader migh have a differen esimae depending on he ime horizon. The volailiy of he share price of a company before i makes is nex earnings repor migh reasonably be expeced o be lower han afer he repor is made. Hence opions mauring before he nex repor dae would have a lower (ime averaged) implied volailiy han hose mauring laer. We could also relae his observaion o 3.: if raders hink we are currenly in a low volailiy environmen, hey would expec volailiies o increase. If hey can make a rough guess (based on pas experience) as o he rae of increase, hey will adjus heir expecaions of fuure volailiies accordingly. The volailiy smile is a well known feaure of opion prices. Essenially he volailiy smile shows ha opions which furher ino or ou of he money are undervalued by he Black-Scholes formula. This shows ha he marke expecs he opions o be exercised wih greaer probabiliy han indicaed by he geomeric Brownian moion assumpion. This is a direc consequence of he fa ails displayed by he price process discussed in Figure 3.3 Implied volailiies of shor daed opions on he FTSE100 showing he implied volailiy smile 11

12 An addiional feaure in some markes is he fac ha his smile is asymmeric he so-called volailiy skew, or someimes volailiy smirk. This shows ha deep ou of he money pus are valued by he marke more highly han similarly deep ou of he money calls. This observaion seems o be linked o 3.3. The skew was apparenly only observed afer he 1987 sock marke crash. Insiuions who had wrien deep ou of he money pus were in he wors posiion a he crash no only had heir own porfolios los money, bu hey had o pay ou claims o ohers as well. Many such players did no survive. Afer he crash anyone wriing pus demanded a premium agains he possibiliy of having o pay ou a he wors possible ime. Economic pricing heories (see he excellen [8] for example) would place a higher relaive price on asses which pay ou well when all oher asses are making losses. The high price he marke pus on hese opions should ell us somehing abou he frequency of such crashes (or of people s aversion o hem)! Implied volailiies % 87% 89% 91% 93% 95% 97% 99% 101% 103% 105% 107% 109% 111% 113% 115% Figure 3.4 Implied volailiies of year FTSE100 opions showing he skew of implied volailies Iems are relevan o he risk conrol acuary, who seeks a realisic model of he markes. Iem 3.6 is of ineres o he valuaion acuary. In placing a marke consisen value on a book of insurance conracs, he is seeking o replicae he prices of marke raded opions, and hence mus capure he smile effec. 1

13 4. Sochasic volailiy models In he previous secion we discussed he reasons why we may wan o model volailiy as a random variable. In his secion we consider firs consider a simple bu effecive exension of he Black-Scholes model, before describing rue sochasic volailiy models, and some of he mahemaics involved. I is no our inenion o reproduce a full derivaion of all relevan resuls. Insead we hope o offer a bluffer s guide o he subjec, wih some qualiaive discussion of he imporan resuls. Readers waning full proofs are poined owards he references Local Volailiy models Local volailiy models as a concep were firs suggesed by Dupire in [14]. These models are no sochasic volailiy models, in ha hey do no add any furher sources of risk (or randomness). Insead hey are an aemp o modify he basic Black-Scholes model o fi observed opion prices. We include hem, no only as a hisorical noe on he developmen of full sochasic volailiy models, bu also because hey are sill in use oday on some rading floors, and may be of some use o acuaries aemping o value life conracs in a marke-consisen way. Essenially Dupire s conribuion was o demonsrae ha, given a se of opion prices (subjec o cerain reasonable consrains), one can find a deerminisic funcion b(s,) of he underlying and ime, such ha he price of he underlying can be wrien as a diffusion-ype equaion: ds rsd b( S, ) SdW and he opion prices implied by his equaion fi he observed prices. To demonsrae how his may be done, we consider a simple example (in fac, his example predaes Dupire s work). Recall in we discussed he erm srucure of implied volailiies. We noe he value of he a-he-money (say) implied volailiy a each ime, and wrie i (). We could hen consruc a (deerminisic) process () such ha ( ) 1 0 ( s) ds i.e. each observed implied volailiy is he average of a volailiy funcion analogous o observed spo ineres raes being he average of (unseen) forward raes. We hen consider he process ds rs d ( ) S dw which has soluion (by applying Io s Lemma) 13

14 S ( ) S0 exp r ( s) dw s 0 he price of a European call on S can be seen o be given by he usual Black- Scholes formula wih volailiy erm ( ) - and hus we rerieve he implied volailiies we sared wih. To see his consider ha he Black-Scholes formula gives he price of an opion assuming ha he erminal price of he underlying is log-normally disribued. Noe ha 0 ( s ) dw lim ( s ) s and recalling ha he sum of normally disribued random variables is again normal, we see ha S is indeed log-normally disribued. Dupire showed ha his procedure can also be used o find consisen funcions o describe o fi he smile and skew of observed implied volailiies. In fac, as he noed, here is lile rouble obaining his fi because he number of possible parameers is large. For example, o capure a skew effec, one could wrie as a linear (affine) funcion of S. The erm srucure of implied volailiies for swapions derived from some ineres rae models will be he implici produc of a similar mechanism. Noe ha, in general, since local volailiy models do no add any randomness, he resuling disribuion of asse prices will be Gaussian so for example here will be no allowance for fa ails in he resuling model. i W s 4.. The general form of sochasic volailiy Coninuous ime financial models are wrien as diffusion processes using sochasic differenial equaions. The general form of he models we are invesigaing is and wih ds S d f (, S ) S dw S d m(, ) d (, ) dw dw dw S d (4.1) These equaions mean ha he insananeous reurn on S is given by some deerminisic erm plus some random noise, he scale of which is given by f( ). iself follows similar (bu more general) random dynamics. 14

15 Before coninuing, we pause o consider some desirable qualiies of a model of volailiy. Drawing on secion 3 and inuiion, we would presume ha a model of volailiy should: Be always posiive Rever o some mean value Display a erm srucure Have some form of negaive relaionship wih price movemens The firs hree qualiies would lead us o consider ineres rae models, which share hese characerisics, as an appropriae saring poin for models of he volailiy process The Heson model The Heson model [1] is he classic model of sochasic volailiy he model which has perhaps come closes o maching he success of Black-Scholes. Mos sochasic volailiy models are benchmarked agains Heson, and Bloomberg offers a Heson implemenaion as sandard. We explain he model in some deail below, in order o examine he workings of a ypical sochasic volailiy model. Heson assumed ha he spo variance process d ( ) d dw obeys he dynamics: This is of course he process proposed by Cox, Ingersoll and Ross in [9] o model he shor ineres rae. The model is mean revering. The parameers may be inerpreed as: - he long-run mean level of volailiy of he asse price. - he mean-reversion rae of he process a higher value means ha volailiy will rever back o is long-erm mean faser from a given perurbaion. - he vol-of-vol, i.e. he scale of changed in he volailiy process iself. This parameer essenially conrols he deph of he implied volailiy smile, ogeher wih. 0 - he iniial level of he volailiy process. This ses he level of he smile. - in he Heson model shocks o he asse price and volailiy process may be correlaed, as empirical evidence suggess. A negaive correlaion will resul in an implied volailiy skew. The soluion o he sysem of equaions (4.1) will have an asse price volailiy erm of he form 0 ds s 15

16 So he volailiy of price changes in separae ime periods will be auocorrelaed a desirable feaure of a volailiy model if we are o capure he marke behaviour described in 3.. Having chosen a process for volailiy, he nex ask is o use he model o price opions. Readers will recall ha he Black-Scholes formula for he price of an opion is obained by solving he Black-Scholes parial differenial equaion (PDE). The PDE is derived by considering a porfolio consising of a derivaive insrumen and offseing posiions in he underlying asse and a risk free asse. Some manipulaion using Io s Lemma gives he PDE: C C 1 C rs S S rc 0 S A similar argumen can allow for oher sources of risk such as sochasic volailiy giving he Heson PDE. To avoid awkward noaion we wrie v for, and he resuling equaion reads C rs C 1 vs C 1 ( v ) C C vs C rc S v v S 0 S v Heson proceeded o solve his equaion, giving a (semi) closed form soluion for he price of a call. Noe ha he firs hree and las erms of his equaion form he Black-Scholes PDE. The remaining erms are due o he volailiy risk, and a cross-risk erm. Each of he parial derivaives in he Black-Scholes equaion has a familiar name dela, gamma, ec, which we associae wih a source of risk movemen in he price of he underlying, risk free rae, ec. In a sochasic volailiy world here are addiional sources of risk and hence addiional parials derivaives. These also have names Volga is he second parial derivaive wih respec o volailiy, Vanna is he second parial derivaive wih respec o boh volailiy and he underlying. The new erm in he Heson PDE arises as a price of volailiy risk. This erm is unknown. The reader will recall ha a complee marke is one in which all derivaive claims can be hedged, in an incomplee marke some risks canno be hedged. So in a complee marke, all risks are priced by he marke, and he marke price of volailiy risk will be known, resuling in a unique soluion o he PDE. In an incomplee marke here will be many, possibly an infinie number, of soluions. Heson assumed ha he price of volailiy risk was proporional o volailiy. Oher researchers have assumed ha he volailiy risk commands no premium (an unlikely scenario). Recen research uses he price of oher raded opions, or of volailiy swaps, o complee he marke and derive a unique price. See [17]. Heson solved he above PDE for he price of a call. This soluion has he form: r( T ) C SP1 Ke P where P1 and P are pseudo- or risk-neural probabiliies. Readers will noe he similariy wih he Black-Scholes formula. In fac he price of a call will always have his form, in any model. The full formula is long, and hence we have relegaed i o an appendix. 16

17 Closed form formulae for he prices of opions are considered essenial for he success of a model by hose working in shor erm finance. Given ha banks mus mark heir books o marke and run risk conrol overnigh, i is easy o see why Oher models Many alernaive models have been proposed o Heson. The Hull-Whie [] model is similar o he Heson model, wih dynamics for he variance process given by a mean-revering geomeric random walk. A recen and popular model is SABR (Sochasic Alpha Bea Rho) [0], which combines some feaures of sochasic volailiy and local volailiy models. Fouque e al. propose a model in [17] where volailiy is driven by an Ornsein- Uhlenbeck (mean-revering arihmeic random walk) process, Y, and posiiviy is achieved by seing = exp(y ) GARCH models Anoher approach for modelling he variabiliy of reurns over ime is o le he condiional variance be a funcion of he squares of previous observaions and pas variances. This leads o he auoregressive condiional heeroscedasiciy (ARCH) models. ARCH processes have proved o be an exremely popular class of nonlinear models for financial ime series. The imporance of ARCH processes in modelling financial ime series is seen mos clearly in models of asse pricing which involve agens, maximising expeced uiliy over uncerain evens. Analogous o Sochasic variance models being discree approximaions o coninuous ime opion valuaion models ha use diffusion processes, ARCH models can also approximae a wide range of sochasic differenial equaions. The ARCH(1) model can be wrien as x 1, x,... ~ NID(0, ) (4.5a) wih where x (4.5b) The ARCH(1) model can be seen as an exension o linear ime series models by adding a componen for he variance ha varies wih pas values of he ime series. New informaion has been added o he model and i is expeced ha he model fis he daa beer. The presence of ARCH can lead o serious model misspecificaion if i ignored: as wih all forms of heeroskedasiciy, analysis assuming is absence will resul in inappropriae parameer sandard errors, and hese will be ypically oo small 17

18 A pracical difficuly wih ARCH models is ha for large lags, unconsrained esimaion will ofen lead o he violaion of he non-negaiviy consrains ha need o ensure ha he condiional variance is always posiive. To obain more flexibiliy he generalised ARCH (GARCH) process was proposed. The GARCH(p,q) process has he condiional variance funcion (replacing equaion 4.5b) 0 p i 1 i i q i 1 i i For posiive variances all he coefficiens mus be posiive. Because ARCH processes are hick ailed he condiions for weak saionariy are ofen more sringen [3]. Many exensions of he simple GARCH model have been developed in he lieraure A brief noe on parameers and parameer esimaion The devil, as always, is in he deails. And he deail of all financial models includes parameer esimaion or calibraion. Hence i is ofen far more difficul o obain informaion abou calibraion echniques han o obain specificaions of models. However, discussions wih praciioners reveal some informaion. To a rading desk quan, he problem is o obain a se of parameers such ha his chosen model mos closely replicaes he marke prices of he calibraion insrumens. A ypical calibraion may use hree opions as calibraion insrumens generally he a-he-money forward, and opions wih srikes a 105% and 95% of his level. The model is hen calibraed by minimising he square difference beween he model prediced prices and he observed prices, using he Levenberg- Marquard algorihm, or a more robus algorihm known as Differenial Evoluion. Minimisaion is complicaed somewha by resricions on some parameer values in he Heson model above for example, obviously mus be beween -1 and +1. This mehodology is inappropriae for a risk conrol environmen, as i will give risk adjused parameers. The real-world parameers may be esimaed from hisorical daa, or a combinaion of hisorical daa and opion daa, wih some assumpion abou marke prices of risk allowing risk-neural parameers o be invered o give real-world ones. Hisorical esimaion generally proceeds ieraively in he Heson model and he mean may be esimaed firs assuming no sochasic volailiy by sandard regression echniques. Then he full model will be assumed and he remaining parameers esimaed. 18

19 4.7. An imporan resul - The 1s heorem of sochasic volailiy We now presen an imporan resul which relaes he reurn on a book of opions (or a book of life conacs wih guaranees) o he reurn prediced by Black- Scholes. Recall ha in he Black-Scholes world a derivaive posiion can be hedged by aking offseing posiions in he underlying and in a risk-free savings accoun. However, in he real world he implied volailiy of he derivaive may change, wihou any change in he value of he underlying. The hedged porfolio would no longer be hedged. Now assume we have hedged our porfolio by using a Black-Scholes model, wih implied volailiy. Suppose ha he acual volailiy beween ime 0 and insead follows a process, for example i may follow one of he processes given above. We will expec our porfolio o have value zero a all imes in he fuure. In fac, wha we find is ha he hedging error evolves as a random process wih dynamics where he reader will recall ha respec o he underlying. r( s) 1 s s s 0 Z e S ( ) ds is he nd derivaive of he opion price wih A dela-hedged porfolio is no hedged a all, alhough i may ake a while before you noice! This is an imporan poin o bear in mind if considering dela-hedging a life insurance fund. From he resul we can deduce several hings: Firsly, if he implied and acual volailiies are close mos of he ime, he hedging error will be small. Secondly if he gamma of he opion (or he overall gamma of a book of opions) is small, hen again, he hedging error will be small. The resul shows how i is possible for a derivaives marke o exis, even in he absence of an exac model for price dynamics. I also explicily demonsraes he effec of volailiy risk, a subjec we reurn o laer. Despie is imporance, his resul is no found in many ex books on opion pricing. [10] is a good source for proof and furher discussion. 19

20 5. Sochasic Volailiy models in acion In his secion we implemen one of he models described above, and use i o demonsrae how sochasic volailiy models may be of use in acuarial work. Finally we discuss some concerns relaed o performance of he class of models. The Heson model is undoubedly he bes known sochasic volailiy models, and was one of he earlies. I could be described as he classic model, and is he one which has come closes o emulaing he success of he Black-Scholes model. Indeed, Bloomberg and oher informaion providers/ brokers offer auomaed Heson valuaion of opions o raders. We have relegaed he acual formulas for he Heson model o appendix A, for reasons which will be eviden o he reader who venures ha far. Readers ineresed in he gus of he model are direced here, or o he original paper [1], or he more accessible maerial in [19] The Heson model resuls Before applying he Heson model o acuarial problems, we presen some graphs which demonsrae he disribuion of asse reurns following a Heson process. A Heson process wih vanishing volailiy of volailiy (vol-vol in he jargon) should reduce o a Black-Scholes syle model. The parameers of he mean-reversion erm in he volailiy process may lead o a volailiy which changes over ime: Figure 5.1 The probabiliy disribuion funcion of he reurns from Black Scholes model 0

21 Figure 5. - The probabiliy disribuion of he reurns from he Heson model wih volailiy of volailiy assumed zero. The vol-vol erm deermines he faness of he ails of he disribuion (he kurosis): Figure The probabiliy disribuion of he reurns from he Heson model wih no skewness 1

22 Finally, he correlaion erm deermines he skewness of he disribuion: Figure 5.4- The probabiliy disribuion of he reurns from he Heson model For comparison we show he daily empirical reurn disribuion, derived from he oal reurn index on he S&P500 over he period 01-June 1988 o 31-December 004: Figure 5.5. The empirical disribuion of he daa All disribuions were generaed by simulaion (he S&P reurns were observed), and normalised o have zero mean and uni variance. We have made no aemp o opimise he PDF shown in Figure 5.4 o mach ha in Figure 5.5. See [13] for a sudy where precisely his was done hey find ha he empirical reurn on he S&P500 is indeed consisen wih a correcly parameerised Heson model.

23 5.. Valuaion We consider a simplified model of a life insurance fund. This fund conains a pool of asses. The liabiliies of he fund are he asses deemed o be owned by he policy-holders (asse shares or value of uni funds), and some promise o pay a minimum guaraneed level of benefis a cerain daes in he fuure. We assume all policies have a single mauriy dae, so as o avoid he influence of facors which are no of ineres o us in his demonsraion. Modern acuarial orhodoxy (now enshrined in he UK by new FSA regulaions) holds ha his fund should be viewed as a long pool of funds, shor he asses shares, and shor a pu opion on hose asse shares. Pu-call pariy ells us ha he liabiliy side is equivalen o a shor bond (face value o guaranee) and a call on some proporion of he value of he asses. Now in mos funds here will be a mix of business wrien a differen imes in he pas, and subjeced o differen levels of declared bonus. Hence here will be a range of differen levels of guaranee in he fund. In he parlance of raded opions, we would say ha here are a range of srikes. As discussed above, a porfolio of call opions wih a range of srikes will show a range of differen implied volailiies. A Black-Scholes implemenaion, eiher analyic or Mone-Carlo, used o value he liabiliies of he fund will no give marke consisen answers, in ha he values obained will no be consisen wih he marke prices of raded opions. However, we could insead use he Heson model. The Heson model has a (semi-) closed formula for he prices of plain-vanilla opions, which we use here. The resuls are shown below. Noe ha we can replicae hese values by assuming ha he asses of he fund follow a Heson process somehing we canno do wih a Black-Scholes model where we have changed he volailiy for each srike level LifeCo ld LifeCo is our mock life insurance company. We will consider one of he company s funds, holding five blocks of business, divided by levels of guaranee, in urn based on pas declared bonuses. The fund is assumed o be invesed 100% in he FTSE100, and currenly has 500m in asses. All business under consideraion will maure in years ime. We assume ha asse shares and guaranees have been aggregaed in a meaningful way: Business block Aggregae Asse shares Aggregae Guaranee Guaranee Presen Value 1 90,000,000 86,968,848 80,057,008 90,000,000 9,734,738 85,364, ,000,000 96,578,665 88,903, ,000,000 10,344,556 94,10, ,000, ,110,446 99,518,380 Table 5.1 LifeCo Asse shares and guaranees by business block 3

24 Guaraneed amouns have been discouned a a risk free rae of 4.615%. A mauriy in years ime he fund will pay ou he greaer of asse shares and he guaraneed level. The liabiliies of he fund are herefore he guaraneed amoun, plus a call opion on he asse shares, sruck a he level of he guaranee. We assume no furher bonuses will be paid. Our ask now is o value he call opions Black-Scholes valuaion The asses of he fund are invesed in he FTSE100, so we mus look o opions wrien on he FTSE100 o deermine wha consiues a marke consisen valuaion. We have used daa on Euronex FTSE100 opions, daed 8/0/005. We will assume his is our valuaion dae. Below we show he implied volailiies for year opions. Implied volailiies % 87% 89% 91% 93% 95% 97% 99% 101% 103% 105% 107% 109% 111% 113% 115% Figure 5.6 Implied volailiies of year FTSE100 opions Noe ha he opions show a disinc skew, bu no smile, i.e., he surface does no urn up a higher srikes. This is a common feaure of long daed opions (in banking parlance, years is a long ime. Only opions wih less han 9 monhs o mauriy rade in a ruly deep and liquid marke). We now have several opions. We can value each of our 5 blocks of business separaely, using differen implied volailiies for each as appropriae. This will give a marke consisen valuaion. However, frequenly in a real-world insurance implemenaion, we would be using a Mone Carlo model of he whole fund. In his case his opion would no be available o us. We could insead use a single implied volailiy value. The a-he-money-forward implied volailiy (he value for opions whose srike is equal o he underlying forward price) is given as 11.34%. Alernaively we could use an average implied volailiy, possibly weighed by asse share or level of guaranee. In a Mone Carlo implemenaion using he Black-Scholes (or similar) model his is cerainly wha we would have o do. 4

25 Audiors and oher ineresed paries are likely o quesion his mehod. Primarily hey will wan o know if a valuaion mehod which is marke consisen on average, bu no a a deailed level, can be said o be marke consisen a all. In he able below we show he Black-Scholes value of call opions under each of he possible mehods. Business block Srike volailiy ATM volailiy 11.34% Asse share weighed volailiy 11.5% 1 1,339,077 11,558,65 11,67,379 8,341,349 7,96,537 8,016,34 3 6,054,191 5,96,919 6,059, ,35,006 3,70,605 3,798,70 5 1,576,051,170,670,53,473 Toal 31,635,674 31,31,356 31,755,154 Error 0.4% -1.0% Table 5. Value of call componen of liabiliies using Black-Scholes and several possible choices for volailiy parameers The srike volailiy values in Table 5. are he rue marke consisen values in ha hey are consisen wih he values of raded opions wih he same characerisics. The alernaive mehods will produce answers which are wrong 0.4% oo large and 1% oo small respecively. The business block level errors are in many cases greaer Heson valuaion We now consider valuing he same block of business using a Heson model insead. Again, given he complicaed naure of real insurance business, in pracise his would probably involve a Mone Carlo model of he fund. We will simply use he analyical formula. We calibraed he Heson model o he prices of opions wih srikes a (approximaely) 95%, 100% and 105% of forward, as we believe is cusomary on derivaives desks. Our calibraion involved minimising he square disance of marke from model prices, and used he Levenberg-Marquard algorihm. The resuling parameers are: mean Table 5.3 Heson parameers 5

26 Below we show he values place on he business by he Heson model. We have shown he rue marke consisen values as before. Business block Srike volailiy Heson value 1 1,339,077 1,300,595 8,341,349 8,341,3 3 6,054,191 6,054, ,35,006 3,34, ,576,051 1,570,63 Toal 31,635,674 31,591,361 Error -0.1% Table 5.4 Heson valuaion resuls As can be seen, he Heson model provides a good fi o marke values across a range of srikes. Noe ha for LifeCo we had he freedom o choose our levels of guaranee. In fac, we seleced hem such ha we could mark he business blocks o marke using our daa. Of course, we hen calibraed o ha daa, so i is unsurprising ha we obain a close fi. Noe however, ha blocks 1 and 5 were no calibraed o, and ye a close fi is sill obained. When using such a model in he real world i is unlikely ha he guaranees in your book of life conracs will align so well wih he srikes of marke raded opions! However, despie his, he abiliy of he Heson model o fi a range of srikes is an advanage over he alernaives of using differen implied volailiies for each level of guaranee, or using an implied volailiy which is correc in aggregae, bu incorrec a each level of guaranee. 6

27 5.3. Risk Conrol We coninue o consider he same fund in LifeCo, bu we now consider risk conrol. To be precise, we consider how he realisic valuaion (i.e. he marke-consisen valuaion) of he fund migh vary over a single year. We use Value a Risk mehodology o compare he level of risk calculaed assuming he asses in he fund follow: a) A Black-Scholes process b) A Heson process The fund is solven a ime zero, wih 500million in asses Simulaions We generaed 10,000 simulaions from a Black-Scholes process, assuming a fla riskfree rae of 4.615%, and a % equiy risk premium. For volailiy we used he ATM forward volailiy of 11.34%. For he Heson process we use he same parameers as above. However, we mus move from he esimaed pricing or risk-neural parameers o real world parameers. This can be done in an analogous fashion o he way i is done in he Black-Scholes model by adding back a risk premium. Firs we increase he asse price drif parameer, as in he Black-Scholes model. We use a % risk premium again o be consisen. We now need o include a premium for volailiy risk. As invesors are risk-averse and dislike volailiy, when pricing hey assume ha he long-run average volailiy will be higher han i is, and ha i will reurn o his value a a faser rae. As discussed briefly above, we could aemp o derive an exac value for he adjusmen by considering he prices of volailiy relaed asses (e.g. dela-hedged opions) or by considering he average reurn on opions as compared o heir prices. However, we use an ad-hoc adjusmen of = -0.04, applied in he following formulae: RN RN RN Where he subscrip RN denoes risk-neural values and variables wih no subscrip are real-world parameers. We considered he realisic balance shee a he end of one year, wih liabiliies valued using a Black-Scholes volailiy of 11.34%, and he same Heson parameers as in secion We calculaed he 0.5 h percenile of realisic ne asses. 7

28 5.3.. Resuls The resuls are as follows: Model Black-Scholes 76,967,075 Heson 159,7,058 Table 5.5: A comparison of he oupus from he Black-Scholes model and he Heson Model Clearly he greaer value given by he Heson model indicaes ha he Black-Scholes model is severely undersaing he risk of he fund. A priori one would expec his, he shor comings of Black-Scholes as a risk measuremen ool are well known. We would expec a greaer level of risk from he Heson model o arise from wo facors: he fa-ailed and skewed asse reurns affecing he asse side of he balance shee, and he shif in volailiy affecing he liabiliy side of he balance shee. However, we presen separae asse and liabiliy resuls below: Model Asses 0.5 h percenile Opion liabiliies 99.5 h percenile Black-Scholes 39,47,16 179,777,63 Heson 310,3,04 146,895,098 Table 5.6: A comparison of he resuls from he Black-Scholes model and he Heson Model As can be seen, he Heson model shows a significan asse-side risk. However, on he liabiliy side, he risk is greaer from he Black-Scholes model. This seems o be because he opions are mos valuable when asse prices have risen a he same ime as spo volailiies. However, our calibraion of he model reurned a large negaive correlaion in volailiies and asse prices, so he perfec sorm scenario seems o happen wih negligible probabiliy. This issue illusraes a problem wih he Heson model he parameers which replicae opion prices do no seem o generae realisic projecions. In paricular, some alernaive mechanism o generae an appropriae mach o he observed skew of implied volailiies is suggesed. An alernaive parameerisaion may be used which will resul in simulaions which more closely mach hisorical observaion. [13] perform such an exercise, wih good resuls. However, his ype of calibraion is unlikely o give a good mach o marke prices a ime zero. This is a siuaion which will be familiar o acuaries who have been involved in similar risk measuremen projecs. 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

More information

Chapter 1.6 Financial Management

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

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

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

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

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

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

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More 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

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

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

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

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

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

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

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

More information

Options and Volatility

Options and Volatility Opions and Volailiy Peer A. Abken and Saika Nandi Abken and Nandi are senior economiss in he financial secion of he Alana Fed s research deparmen. V olailiy is a measure of he dispersion of an asse price

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

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

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

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

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

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

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

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

The Real Business Cycle paradigm. The RBC model emphasizes supply (technology) disturbances as the main source of

The Real Business Cycle paradigm. The RBC model emphasizes supply (technology) disturbances as the main source of Prof. Harris Dellas Advanced Macroeconomics Winer 2001/01 The Real Business Cycle paradigm The RBC model emphasizes supply (echnology) disurbances as he main source of macroeconomic flucuaions in a world

More information

DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR

DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios

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

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

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

SPEC model selection algorithm for ARCH models: an options pricing evaluation framework

SPEC model selection algorithm for ARCH models: an options pricing evaluation framework Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,

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

GOOD NEWS, BAD NEWS AND GARCH EFFECTS IN STOCK RETURN DATA

GOOD NEWS, BAD NEWS AND GARCH EFFECTS IN STOCK RETURN DATA Journal of Applied Economics, Vol. IV, No. (Nov 001), 313-37 GOOD NEWS, BAD NEWS AND GARCH EFFECTS 313 GOOD NEWS, BAD NEWS AND GARCH EFFECTS IN STOCK RETURN DATA CRAIG A. DEPKEN II * The Universiy of Texas

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

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

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

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

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

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,

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

Distributing Human Resources among Software Development Projects 1

Distributing Human Resources among Software Development Projects 1 Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources

More information

House Price Index (HPI)

House Price Index (HPI) House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales

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

Can Individual Investors Use Technical Trading Rules to Beat the Asian Markets?

Can Individual Investors Use Technical Trading Rules to Beat the Asian Markets? Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weak-form of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien

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

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry

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

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

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 Soc Prices? * Neil D. Pearson Universiy of Illinois a Urbana-Champaign Allen M. Poeshman Universiy of Illinois a Urbana-Champaign Joshua Whie Universiy

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

GUIDE GOVERNING SMI RISK CONTROL INDICES

GUIDE GOVERNING SMI RISK CONTROL INDICES GUIDE GOVERNING SMI RISK CONTROL IND ICES SIX Swiss Exchange Ld 04/2012 i C O N T E N T S 1. Index srucure... 1 1.1 Concep... 1 1.2 General principles... 1 1.3 Index Commission... 1 1.4 Review of index

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

Appendix D Flexibility Factor/Margin of Choice Desktop Research

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

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

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

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.

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

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

I. Basic Concepts (Ch. 1-4)

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

More information

One dictionary: Native language - English/English - native language or English - English

One dictionary: Native language - English/English - native language or English - English Faculy of Social Sciences School of Business Corporae Finance Examinaion December 03 English Dae: Monday 09 December, 03 Time: 4 hours/ 9:00-3:00 Toal number of pages including he cover page: 5 Toal number

More information

INTRODUCTION TO FORECASTING

INTRODUCTION TO FORECASTING INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren

More information

Performance Center Overview. Performance Center Overview 1

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

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

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

INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE

INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE INVESMEN UARANEES IN UNI-LINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN

More information

Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions to Nonfarm Payroll Employment: 1964 to 2011 Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

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

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

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

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

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

ARCH 2013.1 Proceedings

ARCH 2013.1 Proceedings Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

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

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

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

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

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were

More information

Stochastic Calculus and Option Pricing

Stochastic Calculus and Option Pricing Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes

More information

Vector Autoregressions (VARs): Operational Perspectives

Vector Autoregressions (VARs): Operational Perspectives Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Volatility Forecasting Techniques and Volatility Trading: the case of currency options

Volatility Forecasting Techniques and Volatility Trading: the case of currency options Volailiy Forecasing Techniques and Volailiy Trading: he case of currency opions by Lampros Kalivas PhD Candidae, Universiy of Macedonia, MSc in Inernaional Banking and Financial Sudies, Universiy of Souhampon,

More information

A Further Examination of Insurance Pricing and Underwriting Cycles

A Further Examination of Insurance Pricing and Underwriting Cycles A Furher Examinaion of Insurance ricing and Underwriing Cycles AFIR Conference, Sepember 2005, Zurich, Swizerland Chris K. Madsen, GE Insurance Soluions, Copenhagen, Denmark Svend Haasrup, GE Insurance

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

Forecasting Sales: A Model and Some Evidence from the Retail Industry. Russell Lundholm Sarah McVay Taylor Randall

Forecasting Sales: A Model and Some Evidence from the Retail Industry. Russell Lundholm Sarah McVay Taylor Randall Forecasing Sales: A odel and Some Evidence from he eail Indusry ussell Lundholm Sarah cvay aylor andall Why forecas financial saemens? Seems obvious, bu wo common criicisms: Who cares, can we can look

More information

Agnes Joseph, Dirk de Jong and Antoon Pelsser. Policy Improvement via Inverse ALM. Discussion Paper 06/2010-085

Agnes Joseph, Dirk de Jong and Antoon Pelsser. Policy Improvement via Inverse ALM. Discussion Paper 06/2010-085 Agnes Joseph, Dirk de Jong and Anoon Pelsser Policy Improvemen via Inverse ALM Discussion Paper 06/2010-085 Policy Improvemen via Inverse ALM AGNES JOSEPH 1 Universiy of Amserdam, Synrus Achmea Asse Managemen

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

LECTURE: SOCIAL SECURITY HILARY HOYNES UC DAVIS EC230 OUTLINE OF LECTURE:

LECTURE: SOCIAL SECURITY HILARY HOYNES UC DAVIS EC230 OUTLINE OF LECTURE: LECTURE: SOCIAL SECURITY HILARY HOYNES UC DAVIS EC230 OUTLINE OF LECTURE: 1. Inroducion and definiions 2. Insiuional Deails in Social Securiy 3. Social Securiy and Redisribuion 4. Jusificaion for Governmen

More information

Measuring the Downside Risk of the Exchange-Traded Funds: Do the Volatility Estimators Matter?

Measuring the Downside Risk of the Exchange-Traded Funds: Do the Volatility Estimators Matter? Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper

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

Research Article Optimal Geometric Mean Returns of Stocks and Their Options

Research Article Optimal Geometric Mean Returns of Stocks and Their Options Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics

More information