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 (V@R) 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. V@R (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 (V@R) 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