Mathematics in finance and economics: importance of teaching higher order mathematical thinking skills in finance

Transcription

1 Tularam Volume 7, Issue 1 (13) e-journal of Business Educaion & Scholarship of Teaching Vol. 7, No. 1, 13, pp: hp:// Mahemaics in finance and economics: imporance of eaching higher order mahemaical hinking skills in finance Gurudeo Anand Tularam Faculy of Science Environmen Engineering and Technology [ENV], Griffih Universiy, Brisbane Ausralia a.ularam@griffih.edu.au ABSTRACT This paper addresses he imporance of eaching mahemaics in business and finance schools of eriary insiuions of Ausralia. The paper explores he naure of hinking and reasoning required for advancemen financial or economic sudies involves he use of higher order hinking and creaiviy skills (HOTS) for eaching in mahemaics classes. The paper demonsraes how various skills required in finance are relaed o he mahemaical way of hinking and reasoning. The 7 global crisis has focused aenion on financial hinking generally and mahemaics eaching of finance in paricular for he lack of skills by finance personnel. This could be addressed hrough suden learning and eaching by reshaping business schools o include well designed financial mahemaics courses ha are compulsory, in degree programs. The mahemaical reasoning paerns, hinking, explanaions, simplificaions required in ransfer of knowledge o sudens is effecively augh if expers in he field are explicaing or presening proofs. The essence of he concepual developmen of models and heir assumpions and soluions requires raining beyond hose of he nonmahemaics personnel, mainly due o heir lack of years of ime spen on such mahemaically work. If he mehodology described is adhered o our fuure business and finance sudens could be afforded a beer educaional experience in mahemaical hinking and reasoning as well as higher order hinking and creaive skills in heir courses and degree programs. Keywords: Financial mahemaics; applied mahemaics, higher order hinking (HOTS), mahemaical reasoning, finance educaion JEL Classificaion: A, C PsycINFO Classificaion: 353 FoR Code: 13; 15 e-jbest Vol.7, Iss.1 (13) 43

2 Tularam Volume 7, Issue 1 (13) Inroducion Years of lecuring mahemaics and saisics and researching suden learning allow lecurers a universiy o gain some knowledge abou he many difficulies sudens freshly coming ino universiies and adul learners coming in laer years, experience while learning mahemaics. The non-rouine ask of lecuring firs year sudens o supervising o heir docorae compleions especially from widely differen disciplines allows lecurers o comprehend more deeply suden learning and undersanding of procedures and conceps in heir respecive fields. In his way, universiy lecurers are researchers on he job and hus gain a deeper undersanding of suden prior knowledge as well as heir ongoing universiy learning experiences. I seems ha mahemaics lecurers see greaer numbers of sudens from a wide variey of course programs in universiies for mahemaics is a compulsory course for many programs, alhough his poin may be debaable. In he auhor s experience his large pool of sudens allow mah lecurers opporuniies o observe more criically differen suden learning behaviours as many clearly find mahemaics difficul and in many cases sudens believe mahemaics o be a useless field of sudy and no applicable o heir immediae life or heir careers more generally. The difficuly feaure of mahemaics provides lecurers opporuniies o learn abou suden prior mah experiences, heir undersanding of conceps, heir self-beliefs; he auhor claims perhaps more so han oher disciplines since sudens complain more abou heir mahemaics learning han many oher courses (Tularam, 13). The many ypes and expression of concerns wih mahemaics ofen exposes sudens inadequacies. The auhor believes ha from such a rich daa base of experiences, mahemaics lecurers may be able o provide significan insighs in he progress of eriary sudens in differen disciplines especially hose ha require mahemaical insighs in heir applicaions (Roca and Tularam, 1; Tularam, 1997, 1998,, 11, 13; Tularam e al, 1). Alhough a mahemaician, he auhor s case is special in ha he has had added opporuniies o each mahemaics o finance sudens for a number of years; eaching courses such as risk managemen, ime series, forecasing and financial modelling and sochasics and Io calculus. In addiion, he supervising of higher degree posgraduae sudens in finance, including hose graduaing wih docoraes in mahemaical finance have allowed he auhor significan opporuniies o observe suden learning hroughou heir universiy life ill achievemen of heir highes educaion. The rich experiences gained over ime has allowed he auhor access o deeper undersaing of sudens of finance in erms of heir mahemaical needs; ha is, in erms of heir capabiliies and as well as heir gaps or misconcepions in learning and undersanding. This research paper is a reflecive analysis from a rich daa base of research experiences ha may provide possible ways in which gaps in suden learning may be filled in finance disciplines so ha sudens may wih undersanding apply various mahemaically based financial ools. Alhough such an analysis can be underaken for oher disciplines, his paper is focused on finance, and he analysis and research are qualiaive, and based on a reflecive analysis of suden inerviews and documened commens, A criical analysis is also presened regarding higher order mahemaical hinking and is imporance in financial sudies. The firs par of his paper presens incoming suden reflecions from hose hinking of aking up finance as a degree course followed by analysis of suden experiences regarding heir learning a he undergraduae and higher degree levels. A brief hisorical perspecive is hen presened on he influence of mahemaics boh in procedural and he more-deeper criical finance issues ha iself has iniiaed a live and producive research area for mahemaicians and in some cases propagaed real advancemens in higher mahemaics iself. In addiion o ohers, he case of Black- Scholes is presened as an example for he use of higher order criical mahemaical hinking in finance. In he end, he in deph analysis shows ha mahemaics is such e-jbest Vol.7, Iss.1 (13) 44

3 Tularam Volume 7, Issue 1 (13) in grained in finance ha finance sudens ough o learn he mahemaics in a manner ha shows deeper undersanding of mahemaics including assumpions and simplificaion issues considered in hem. Mahemaics in finance should be undersood by he sudens raher han be augh in an ad hoc pach up rouine exercise manner ha is highlighed in he reflecions of many acive workers in he field even when hey do make a valid poin. The analyses show ha he manner of eaching and approach generally does no lead o he learning of he higher order hinking skills (HOTS) so desperaely needed in advanced financial hinking paricularly in he imes of global crises. Using mahemaical procedures in finance, his paper shows he naure and levels of hinking skills sudens require o cope wih higher financial learning and decision making in financial sudies; all of which can be exposed and augh in a faciliaing manner o sudens only in well-designed financial mahemaics courses ha is augh by mahemaicians who are rained in boh pure mahemaics and applicaions raher han by solely finance personnel who may have aken some aspecs of mahemaics as applicaion ools in some courses and in higher sudies in finance or relaed sudies. Har s (11) definiion is a good sar o undersanding he naure of mahemaical hinking - A good criical hinker (a) raises vial quesions and problems, formulaes hem clearly and precisely; (b) gahers and assesses relevan informaion, using absrac ideas o inerpre i effecively comes o well-reasoned conclusions and soluions, esing hem agains relevan crieria and sandards; (c) hinks openmindedly wihin alernaive sysems of hough, recognizing and assessing, as need be, heir assumpions, implicaions, and pracical consequences; and (d) communicaes effecively figuring ou soluions o complex problems wih supporing evidence and proofs (Har, 11; Tularam, 1997, 1998; Tularam and Kelson, 1998) The quesion of finance educaion The imporance of financial knowledge in everyday living le alone advanced complex applicaions in he real world of business and accouning canno be doubed. As a suden a he Universiy of Wisconsin saed ha he mos useful major ha a young person can choose oday for a career in he fuure is a finance degree. A finance degree will allow a person o have he knowledge on how o effecively conrol heir finances, give hem vas employmen opporuniies, and allow hem o conribue o individuals and he economy. One may argue ha he siuaion is now changed because of he Global Financial Crisis (GFC), ye he auhor would argue oherwise since crises are he poins a which revoluions in hinking ake place and new way of hinking is ofen generaed or proposed. The knowledge o make decisions abou deb, invesmen, and fuure reiremen plans are all imporan basic ools necessarily in real life. Finance degree sudens will be able o make criical decisions abou sock marke, morgages, and invesmen opporuniies. Clearly, he financial knowledge and ools acquired a universiies will help bring real life examples o workforce making Ausralian workplace a smarer one. Alhough he new finance majors may be worried abou employmen i is sill noed ha he field of finance is one of he fases growing occupaions. There is a demand for he bes wih higher saring salary and excellen benefis as noed in he CEO salaries around he world. Sock markes will no go away any ime soon and people wih money will wan o improve heir wealh posiions so finance majors will have an edge because hey work wih differen aspecs of he economy on a daily basis. I is imporan o realize he responsibiliy placed on finance majors in real life is grea when hey are given he opporuniy o guide individuals and businesses o possible invesmen decisions wih goal o provide for heir bes possible wealh posiions. The recen crisis is he impeus and an opporuniy for financial deparmens o reshape he eaching of incoming sudens by making cerain hey leave universiies wih he bes ools in finance. There is a need for our sudens o forge a new way of e-jbest Vol.7, Iss.1 (13) 45

4 Tularam Volume 7, Issue 1 (13) hinking in mahemaical finance ha is desperaely needed raher han being amused by debaes abou blame. The main aim of his paper is o highligh he ingrained naure of mahemaics in finance. A second aim is o show how various skills needed in finance are relaed o mahemaical way of hinking and as such have o be augh by hose in wih a high level of mahemaics raining such as mahemaicians who have undergone years of problem solving sudies. Finally, i is proposed ha suden learning and eaching needs o be reshaped o include well designed financial mahemaics courses ha are compulsory yearly in finance and business degree programs. The firs secion of his paper presens an analysis of prospecive sudens in search of informaion regarding finance degrees. The prospecive sudens inquiring abou possible fields of sudy hey may underake a universiies. Ou of many only some examples are presened. The replies by members of finance communiy who are acually working in he field are hen analysed. Such an analysis provides a perspecive ha is ofen missed in oher quaniaive based research. This secion is followed by a brief view on mahemaicians and hisory of mahemaical finance hus providing a background o he imporance of mahemaics o finance. The definiion and examples of various higher order mahemaical hinking and heir link o mahemaics and financial sudies is hen presened. The paper concludes by demonsraing he imporance of mahemaical higher order hinking skills wih examples followed by a discussion of he fuure of mahemaical hinking in finance. A real life qualiaive analysis Case Sudy 1: Informaion gahering by a prospecive finance suden asking for advice from field workers in finance I am saring school as early as his Fall. I'd like o make a decision on a major before I even sign up. I am rying o hink of hings I'm good a and enjoy. I'm basically on he fence wih majoring in Social Work, where I would be working eiher in a VA, a general hospial, communiy clinic or counseling cener, ec. Pay would be around $4k a year afer abou wo years (on average in his area), a which poin I could ge my Maser in jus a year. Then here's Finance major. I enjoy working wih my finances. I hink I'm prey good a i. I'm prey sure I could do i all day, five days a week, for he nex 3 years. However...I dropped ou of High School, having only compleed Algebra 1.5. I immediaely go my GED afer dropping ou, and scored a 54 of 8 on my mah. The lowes of all my scores I checked Boise Sae Universiy's regisrar, and i appears ha only wo mah courses are required for he Bachelors (no sure abou Masers), and you have a choice of wo for each of he wo slos. Mah 143 College Algebra and Mah 16 look like I may be able o grasp hem well. However, i sill scares me a bi. Mah is and always has been my weakes subjec. However, I wasn' one o sudy in High School. I didn' apply myself well. And I even ook my GED wihou sudying. So maybe I'm no as weak wih mah as I hink? Maybe i's simply one of he wo subjecs ha challenge me (he oher being science, mainly because I don' believe a lo of i). Wha do you guys hink? Oher han speaking wih my universiy's advisors, how else could I figure ou if his is a good field for me o pursue? Thanks so much everyone! Ou of many responses, one of he answers by a member of he finance field: You really ough o be good a mah o do finance. Being srong a mah made finance prey easy for me when I go my MBA. However, i really depends on wha you are doing wih your finance degree. A lo of he jobs are prey rouine. Run he same spreadshee, plug in new numbers and analyze he oupu. Morgages are like ha. Almos everyhing is auomaed and here really isn' much new. Also anyhing o do wih invesing oher people's money doesn' require you o be Einsein. There are pleny of quaniaive people in he background ha crunch he numbers. Los of finance posiions are really jus sales. If i is only wo course, I'd look ino wha hey cover and jus brush up on hem before you ake he classes. Having said all of his, if e-jbest Vol.7, Iss.1 (13) 46

5 Tularam Volume 7, Issue 1 (13) you are really good a mah, i's easy o do many differen jobs. You never know migh have been a bad high school eacher. Good luck. Case Sudy : I am a high school graduae. I am hinking of majoring in finance bu do I need o do well in mah, mainly calculus? and do I need o know economics as well?...accouning skills is a mus in finance I presume? and does i pay good? One of he many answers chosen as he bes by he candidae was: I've been where you've been so I wan you o ake his o hear and know ha i's he bes answer you will receive! The only kinds of Mah you will ever need in your lifeime no maer wha career you go ino will be jus he basics! PERIOD! Unless you're going ino some kind of design field, hen you'll need a lo more Mah han ha. Oher han ha, you'll only need o know how o add, subrac, muliply, and divide! Tha's i! The res of he Mah classes ha oher insiuions of higher learning decide o give you for a degree are irrevelan and hey will be once you graduae! For a career in Finance, you'll need o ake a Corporae Finance course. I akes you over all of he ypes of Mah and equaions ha you'll need for a career in Finance. Since his ype of Mah poins owards your career, i's good o ake. You'll also need Accouning, which will be grea for you once you maser (or pass) i. You'll be augh abou differen ypes of accouning for he hree ypes of business offered: sole proprieorships, parnerships, and corporaions. Do no worry abou classes like calculus, physics, or rigonomery. They won' maer o you years from now and you'll never use hem in he real world. These classes are bound o give you headaches and pleny of fis, especially if you don' know wha you're doing and you're forced o find someone o help you who doesn' know wha hey're doing eiher! I never needed calculus, bu rigonomery gave me so much hell I was forced o ake i a anoher college ha was real lenien on me when i came o his ype of Mah. Tha's wha you wan o always look for: leniency. No everyone is a genius when i comes o Mah. Some need a lo more help han ohers. If you are forced o ake any high-end Mah classes like he ones I menioned, hen his is somehing you wan o keep in mind: You don' need o know everyhing abou ha ype of Mah, all you need is a moderae amoun, jus enough o ge you by! A deep qualiaive analysis of he wo replies can be underaken using inerview analysis echniques ha involve caegorising of commens ha can be done eiher in a word by word or senence-wise manner. In he end however such an analysis can be revealing and can help generae hypoheses ha can be esed. Essenially, in boh cases, here is a search for knowledge by he inquirer. The firs reply provides sound informaion of how imporan an in deph undersanding of mahemaics is in finance generally when he person says, Having said all of his, if you are really good a mah, i's easy o do many differen jobs.. Clearly mahemaics is criically imporan when he goal is o have an overall undersanding in he finance area. Ye in he second case, he replier places he mahemaics in a differen conex. The reply makes a lile mockery of much of mahemaics o be learned in general. An imporan commen was: The res of he Mah classes ha oher insiuions of higher learning decide o give you for a degree are irrevelan and hey will be once you graduae! ha shows ha he advice is indeed incorrec even when his person perceives i o be rue in real life in he finance field. Anoher ineresing commen is also imporan and one o be considered seriously by finance deparmens is: You don' need o know everyhing abou ha ype of Mah, all you need is a moderae amoun, jus enough o ge you by! This is a raher disurbing even hough here are many more similar commens made in he reply. Jus enough o ge you by! is imporan and here is no doub ha his is rue in many real cases bu he danger is also very real. The inquirer or replier are persons who may indeed one day become a CEO; he manager or direcor or a head of a major company in charge of making criical finance decisions. This is no an unlikely scenario as he readers can provide examples of similar individuals in he field during he crises. Ofen such individuals feel e-jbest Vol.7, Iss.1 (13) 47

6 Tularam Volume 7, Issue 1 (13) he same way abou he mahemaics as he commens made by he replier quoed. As o how much mahemaics he CEO s or managers really do undersand one canno say bu he auhor would be fairly cerain o say many would be lacking in heir undersanding of criically imporan aspecs of models ec. Wha is more imporan is such prospecive direcor or CEO will hen go ahead and make he criical decisions abou he aciviies of large fund managemen insiuions and heir invesmen possibiliies (see Wall Sree Greas or high flyers of he Pas or read any risk managemen books on crashes of public companies). The more imporan issue ha is no highlighed in he above exchange is he processes involved in he learning of mahemaics. In he cases quoed, he acquisiion of mahemaical procedures or low order skills has been highlighed bu more criically imporan mahemaics hinking skills has been ignored. The acquiring of higher order hinking skills can be readily gained hrough a proper mahemaics learning program. The pursui of mahemaical knowledge forces individuals o acquire wha is ermed higher order hinking creaive skills ha are largely reflecive and absrac hinking based. The processes of learning such skills will be by passed if a rouine learning or eaching approach is underaken; ha is, if Case guidance quoed earlier is aken hen while suden being proficien in procedural work he or she will generally lack applicaion and general problem solving abiliy. This can aribued o he learning and eaching syle underaken; his is mainly because of criical mahemaical skills referred o as he higher order hinking skills are no acquired in he conex of mahemaical finance learning. If such skills are no praciced rouinely hen sudens will show a lack of heir use while problem solving. The imporan lesson o be learn is ha when financial mahemaics is augh by hose who have no underaken rigorous mahemaical learning over long periods of ime, hey will hemselves lack such criical conex based skills base and during heir eaching processes such criical pracicaliies will no be passed on or highlighed. The criical mahemaical problem solving skills will no be passed ono hose who are in he learning phase in financial sudies. Clearly, if Case ill advice is aken, hen he suden will indeed lack he learning or pracice of such skills mainly because heir mahemaics learning will be only o pass he course raher han underaken seriously o become a financial problem solver. A quesion posed by a suden a few years ago was o explain he imporance of Financial Mahemaics and he bes voed answer was: he mos imporan use of financial mah can be described in words: minimum and maximum minimize cos and maximize profis. Also, financial mah will give you concree numbers for invenory, bes use of sorage coss, compound ineres and fuure values. I'm sure ohers could add o his lis. While he answer appears appropriae is focus is mainly on rouine conceps augh from sraigh financial mahemaics exs ha explains maximums and minimums and mahemaics of forward and fuures pricing including sorage coss ec. The mahemaics of risk analysis, criical assumpions in he models used or modeling of financial sock pricing are no noed; and neiher is sochasical naure of price changes over ime. I is however ineresing ha ohers could add o his lis was menioned. Essenially, he eaching of mahemaics in finance appears o be focused on some specific ools raher han concepual mahemaical hinking of finance and modeling or financial problem solving. The criical concern is ha financial mahemaics eaching may be focused on rouine procedural mahemaical hinking raher he higher order creaive hinking ha is required for decision making and in complex analyses. Mahemaicians working in finance I is ineresing o noe ha ofen mahemaicians are seen o be differen from normal persons. I is believed ha mahemaicians are raher serious individuals wih possibly glasses siing a heir compuers day long working on absrac and raher incomprehensible or impracical mah. This is rue o a grea exen abou pure e-jbest Vol.7, Iss.1 (13) 48

7 Tularam Volume 7, Issue 1 (13) mahemaicians as indeed hey do sudy prime numbers, zea funcions and so on bu if pure mahemaicians did no exis much of applicaions ha are in his world would no be possible; such as undersanding aomic behaviour, explaining complex weaher sysems, space and iner-plane ravel and so on. The applied mahemaicians place a higher focus on he applicaion of heir pure mahemaical knowledge as ools and models o solve real life problems. In he field of finance one of he firs major applicaions of mahemaics was shown by Black and Scholes. This applicaion of mahemaical knowledge in finance may have los some favour over ime due o various reasons bu he leaders in he field realize he some major concepual leaps were made in finance due o his new way of hinking and approach. In he pas, he calculaions such as he ne value or a heoreical sudy of inflaion and money supply and applicaion of probabiliy and saisics in risk analysis, as well as Io calculus in sochasics of marke pricing have all conribued majorly in he finance indusry. New or exoic producs have been developed based on he advancemen and more sysemaic and accurae risk-managemen opions. The advancemens made by mahemaicians have led o high levels of profis over he years for he business and finance indusry generally. In he recen global crisis (GFC) many have dispued he use of models and mahemaics relaed o hem. There are a number of analogies we can make bu he earhquake example appears useful here. The Richer magniude scale was developed in 1935 by Charles F. Richer of he California Insiue of Technology as a mahemaical device o compare he size of earhquakes. One mus noe ha he occurrence of devasaing earhquakes do no make seismic mahemaics or he Richer scale useless bu raher much more learning mus occur concerning ouliers or exreme incidens so ha more knowledge abou earhquake sysems can be gained. Only hen he new knowledge can be uilized by engineers o build sronger and earhquake resisan buildings and bridges; or even render areas uninhabiable; ha is, o build away from high risk areas. This analogy is no o far-feched and clearly may be applied o finance because he GFC may have been caused by financiers no making appropriae decisions abou risks relaed o exoic opions and allowing ranches o be given credi raing for invesors o use before he recen Global Financial Crisis (GFC). Therefore, hisory shows ha mahemaicians have been working on financial problems over a long period of ime providing much help o finance personnel paricularly in providing proofs for heir insighs or hunches so ha he same ools may be applied wih confidence in he field. The ingrained naure of mahemaics in finance Loius Bachelier was he firs o place applied mahemaics ino he forefron of finance. In his disseraion he used some sophisicaed mahemaics in finance relaing o he heory of speculaion (19). I seems ha Bachelier was one of he firs o sudy coninuous-ime sochasic processes discussing Brownian moion and is applicaion o opion-pricing. Opion rading is now an immense financial aciviy and some exoica opions in recen imes ha led us o he pah of he GFC. Bachelier also developed he soluion o a paricular diffusion equaion, some five years before Einsein ha shows he exen and naure of his mahemaical knowledge he had; The diffusion equaions changed physics and chemisry learning forever and of course opion pricing did as well bu afer he work done much laer by Black and Scholes. Bachelier's work was no noed in he finance area ill he mid 195 s. Up o his ime i would seem he mos sophisicaed mahemaics had been elemenary discouning o calculae presen values of fuure cash-flows in finance. In he mid 195 s, Markowiz sudied porfolio selecion o maximize reurn by holding cerain amouns of desirable socks. Harry Markowiz sudied he momens of socks and devised a mean-variance opimisaion problem in which he expeced reurn of he porfolio was sudied subjec o variabiliy of ha reurn. Thus grew he modern porfolio heory - MPT. Linear regression was used o sudy risk and reurn of an e-jbest Vol.7, Iss.1 (13) 49

8 Tularam Volume 7, Issue 1 (13) enire porfolio. An opimizaion sraegy was used o choose a porfolio wih larges mean reurn subjec o accepable levels of variance in he reurn. William Sharpe a he same ime used mahemaics in deermining he correlaion beween each sock and he marke. Evenually Harry received a Nobel Prize wih William Sharpe for he developmen of he famous capial asse pricing model in 199. This has been he basis of much insiuional invesing bu many have criicized he descripive aspecs of he model in recen imes. Laer, Rober Meron and Paul Samuelson replaced he one-period models by coninuous ime, Brownian-moion models, and he quadraic uiliy funcion implici in mean variance opimizaion was replaced by more general increasing, concave uiliy funcions. However, he grea leap in sophisicaion came wih he developmen of opion pricing he Fisher Black and Myron Scholes who applied sochasic calculus o opions. The high leverage of an opion allows a buyer he righ, bu no he obligaion, o buy somehing in he fuure a a price which is se now. In fac he CBOE (Chicago Board Opions Exchange) began rading lised opions in April a monh before he publicaion of he Black and Scholes paper. By 1975, virually all raders were valuing and hedging heir opion porfolios, using he Black and Scholes model and he res is hisory; alhough oday highly criicized by hose who do no undersand he assumpions of he mahemaical model i is sill used widely. In fac he auhors hemselves raher ha aemping o make he model more real life like, desired o make money ou of heir success a he firs level of developmen and as such were doomed o failure. By he 198s, he developmen of academic heory and is applicaion was a an advanced sage wih applicaions. Now i seems ha much of he leading edge work is occurring mosly in commercial organizaions. This is no unusual as links wih indusry ofen lead o research funds and hen of course he dealings wih he funding companies are enhanced. Harry had dealings wih DAIWA, while Sharpe became a consulan. A US invesmen Bank employed Black while Scholes worked for a compeing firm. There is presenly many PhDs in number heory, quanum relaiviy, conrol heory (bes from Cambridge o Sanford) working in full ime research posiions in major financial insiuions. More sophisicaed mahemaical models and derivaive pricing sraegies have been developing alhough credibiliy of some has been damaged. A number of auhors of books such as he Black Swan (Taleb, 1) have had much o do wih his nowihsanding he crashing of many financial companies. However he mos elling blow has been caused by he fall ou caused by he major financial crisis of 7 (GFC). Noneheless here are now bodies similar o INET (Insiue of New Economic Thinking) examining and helping esablish more effecive heories and mehods o move forward wih. A brief background including he hisorical analysis shows ha finance appears o be ingrained wih mahemaics. Even from he beginnings of his cenury mahemaicians have made major conribuions o he developmen of finance more generally. Imporan concepual leaps and advances can be aribued o he use of mahemaics. Given his high saus of mahemaics in finance, i is appropriae o commen on he eaching of he mahemaics in finance and such a commen can be well explained by someone well versed in all of he above. Clearly, mahemaics needs o be augh in finance degree courses using a well developmen siuaed program if only o undersand he hisorical growh of financial ideas and he relaed calculaions. Judging from he brief hisorical analysis i is appropriae o sae ha a sandard o advanced level of pure and applied mahemaics is required and ough o be augh in he finance degree programs. e-jbest Vol.7, Iss.1 (13) 5

9 Tularam Volume 7, Issue 1 (13) Wha wen wrong if he bes mahemaicians developed he models? While he accuracy of model developmen and applicaion is a lenghy process in oday s compuer response ime framework he demands on fas delivery are being imposed on he bes workers in he field. There are many examples bu one of he recen raher simple bu imporan example is examined here. In paricular, one can examine case of he person who developed and designed he ranches (exoic opions) in US. A highly rained young mahemaician developed he so called ranches for a major company. However, he assumed he risk of defaul in he model o be very low ha led o he assumpion of independence of individual households in suburbs of USA. Clearly, his assumpion is a firs aemp o model risk and fauly as he suburbs are usually inhabied by hose who are alike in a number of ways; such as raher similar income levels, socio-economic levels or class of work ec. The assumpion of independence beween individuals could no be applied seriously o he valuing he real life ranches bu i was done. The criical facor was no aken ino accoun in he final modeling of risk as such he ranches were given a high credi raing. Now one would hink a mahemaician could idenify his error bu he process of modeling and subsequen calculaions relaing o risk appears o have been a quick fix. The developmens of such opions are now done in house (in large Financial Insiuions) and possibly kep under wraps raher han being under peer review or undergo some in deph scruiny. Such an in deph criique work would be based on years of analysis and research work similar o hose sudying for PhD s. I seems he developmen of he opion and risk pricing models have been a raher quick fix for a major company willing o pay large funds for he developmen of such exoic opions or invesmen derivaives. Such a shor erm process of developmen does no and possible canno lead o a full proof model. The ime framework ha is needed for a new applicaion model o be rigorously developed and esed can be undersood by undersanding for example he Black-Scholes model and even hen heir work is limied in heir applicaions. Clearly, he ime aken for he developmen of ranches ha seems o have sared he GFC in US would no have passed his es of ime aken for analysis. Serious models in science have required years developmen o a model sage ha would subsequenly have undergone exreme esing over years so ha proofs could be developed. In his example, serious consideraion of correlaions or aking in accoun incidens such as exreme cases or all oher possible risk evens as noed by Taleb (1) in his book Black Swan could no have been conduced in his framework. I is also imporan o noe ha here is much research done in house in recen imes. In many insances research developmens in large insiuions are no published especially if hey involve cuing-edge research ha has possibly large moneary gains for a company. As such he research work may only be available for criical analysis afer he money making opporuniy has passed. This is a real problem for finance and business more generally in modern imes. Anoher issue is o do wih he provision of large funds of money usually fed by larger insiuions paricularly o arac high achievers in mahemaics. Such a soluion works mos of he ime bu can fail as i did in he case above. In mos cases i is due o he lack of appreciaion of he criical aspecs of he real financial space of operaion on he modeler s par. A suden in mahemaics may no be oally aware of all on goings in he real financial world. I is during his criical financial space analysis when he need for he exper or bes of he financiers is required o inerrogae he model. The exper financiers herefore mus ake responsibiliy for hey also had a role in he global financial crisis in ha hey should have appreciaed serious flaws in he risk analysis and modelling process. Clearly, he expers in he finance and economics fields did no and one possible cause for his may be aribued o he lack of knowledge and appropriae raining in he mahemaics of finance. If his is he case, i can be raced back o he learning phase of he expers during which heir insrucors failed o show he criical imporance and undersanding he developmen of more complicaed mahemaical e-jbest Vol.7, Iss.1 (13) 51

10 Tularam Volume 7, Issue 1 (13) models. If he bes of he financiers could no mahemaically oversee, analyse or criically es he models or any assumpions underpinning hem bu insead assume hem o be correc hen here is cerainly a serious need o rehink financial raining. Finance has also changed mahemaics research and eaching There is no doub ha financial mahemaics has changed mahemaics research and mahemaics eaching (Roca and Tularam, 1; Roca e al, 1). I would difficul o find mahemaics faculies no conducing any research in finance and economics around he globe (Roca and Tularam, 1; Roca e al, 1; Tularam e al, 1). Almos all new algebra and calculus exs now wrien include financial applicaion examples. Mahemaics ingrained naure in finance and he use of mahemaical conceps has changed finance research and eaching (as i should have) bu a he same ime finance has influenced mahemaics research and eaching considerably. There are a number of courses in universiies ha inegrae he eaching of finance heory and mahemaics. Priche and Feinsein (1999) argued ha ha his inegraed naure produces a deeper undersanding in boh relaed fields. I is no only simply a one way granger causaliy beween finance and mahemaics bu since he pracice of finance has changed mahemaical eaching and research a wo way g- causaliy exis!!! (no proven). However, i is clear ha modern finance simply canno do wihou mahemaics bu reverse is debaable (you may wish o consider he reverse proposiion more carefully). How mahemaics has changed financial research in paricular can be comprehended by a recen saemen on research opics for finance heses: I is possible o keep he hesis compleely heoreical (i.e. no compuers & numbers), bu i is more naural o involve some numerical work. In fac, depending on how deep you go ino he PDE-approach, you can ge "as heavy numerics as you like Even hen many coninue o argue ha mahemaics is no a way in which financial work should be undersood and ha i should no be he focus or he use of models ough o be discouraged and so on. There have been oher srong commens made in his regard in recen imes in many cases blaming he mahemaics and modeling using hem. One he mos famous mahemaicians of all ime Berrand Russell lecure would have behaved differenly i seems; his behaviour is worh quoing from Imre Salusinszky's aricle (1996) Academic squabbles are feud for hough : Berrand Russell, afer coninual inerjecion during a Cambridge lecure by Ludwig Wigensein dragged him across he floor o he exi, and slammed he door on his head several imes. There were no furher dispuaions..., noed Russell in his diary ha nigh. The auhor hopes ha dispues and concerns are deal wih a much beer fashion oday han hose days! bu he recen heaed discussion abou mahemaics and is imporance in finance is a relevan poin. The criical views expounded earlier appear o be mainly by hose who have no undersood mahemaics or learning of i. There is nohing ha can be furher from he ruh given he risk analysis of he ranches should have been more wisely reviewed or assessed by he world s bes financiers even when he calculaions were developed by mahemaicians. The bes financiers and managers are being paid large funds for heir work, and hus should have been well versed in he modeling processes. The financial researchers should have criiqued he model wih all heir assumpions given ha hey are he so called world s bes financiers bu clearly his did no occur. I seems ha his was due o heir inabiliy o undersand he mahemaical complexiy of he exoics bu more imporanly i was probably due o heir lack of appreciaion of he underlying assumpions in he modeling process. The mahemaicians working in he field are rained in applied mahemaics and herefore were able o easily develop exoic money making opions or develop valuaion models for opions - ranches. However, wha appears o be occurring is ha very high paid modeling posiions are adverised by he world s major e-jbest Vol.7, Iss.1 (13) 5

11 Tularam Volume 7, Issue 1 (13) financial insiuions - so called big players who invesing large amouns of funding for he developmen of exoica such as ranches in he firs place using young guns who may be less experienced in he finance space of operaion. Evidence of he exisence of such financial expers in boh quaniaive and quaniaive are readily found. If one searches he inerne here are many financial expers and holding raher presigious posiions afer having a highly disinguished careers, for example: Case 3: One of he many expers found in financial engineering and referred o as he maser of saic hedging: Dr. X is a Managing Direcor a Morgan Sanley wih 15 years of experience in he derivaives indusry. was also a finance professor for 8 years a Cornell Universiy, afer obaining his PhD from UCLA. is he Execuive Direcor of he Mah Finance program a NYU's Couran Insiue, he Treasurer of he Bachelier Finance Sociey, and a rusee for he Museum of Mahemaics in New York. has over 7 publicaions in academic and indusry-oriened journals and serves as an associae edior for 8 journals relaed o mahemaical finance. was seleced as Quan of he Year by Risk Magazine in 3 and shared in he ISA Medal for Science in 8. Las December, he Inernaional Associaion of Financial Engineers (IAFE) and Sungard joinly announced ha hey have seleced Dr. X as is 1 Financial Engineer of he Year. This case is jus o show an example of expers in he field of quaniaive analysis in finance. Clearly, he case is no presened o blame or poin fingers a any paricular person bu o highligh he experise, achievemens and raher high public sandings of he bes financiers around he world. Since here are so many expers in he field here should be some who would have highlighed he difficulies he ranches posed bu his was only done afer he even and indeed by mosly blaming he mahemaicians or so called quans in he field. Simple ideas in finance may lead o complex mahemaics no easily undersood by finance personnel As a poin of ineres raher simplisic ideas in finance may be ranslaed ino mahemaics easily bu soluion o he mahemaics problems may be raher difficul o find wihou making cerain simplificaions or assumpions. Black has saed in one of his inerviews ha i ook hem approximaely 6 monhs o develop he mahemaical equivalen problem for asse pricing and is change over ime; and hen anoher 3 years o successful solve he same mahemaically. An ofen quoed example is sufficien o bring o aenion how a simple ype of problem can lead o a complex mahemaics one ha requires higher order mahemaics o solve. For example, here is a lo of daa now available in he sock marke and i is easily possible o collae all low, high and closing price for a number of years. If he bes esimae of he price p was required from he daa hen order saisics and maximum likelihood mehods would be sufficien o presen a mahemaical model: 3 i p, 1 n 1 1 yi p, p i1 i1 1 Q p l High p Low p i e-jbest Vol.7, Iss.1 (13) 53 e, 1 yi p, where y 1, y, funcion. y Close p 3, and Q is he sandard error This equaion appears o be raher difficul o solve when posed in his form bu a simplified form leads o he bes esimae being he average of he high and low prices ha is no so surprising. Many such examples can be saed o show ha simple

12 Tularam Volume 7, Issue 1 (13) financial ideas can lead o much more complex mahemaical proposiions o deal wih. Once posed in a mahemaical framework many of he ools of mahemaics ha has been developed over cenuries can be applied o solving he problems bu he answers found may or may no be useful in he real world. Wha is gained however is a proof of wha may have been only a hunch or belief of financiers in he firs place; in his way mahemaics allows such beliefs or hunches o be proved based on sound logical saemens and argumens. I may seem ha he exreme values or ouliers are no aken ino accoun in his averaging process bu whenever new informaion becomes available, a new high or low is formed ha in iself now conains he mos recen and imporan informaion conen and he average capures all he relevan informaion. I should be noed ha a mahemaical soluion can be developed by simplifying he model ha allows soluions ha are based a somewha simple bu realisic model. The soluion depends on sound saisical ools ha can be applied wih reasonable approximaions. In mos cases, mahemaicians can provide soluions ha may be simple bu mos imporanly and in mos cases allow significan insighs ino he real processes a work. The nex secion highlighs he naure of skills ha are required boh in finance and mahemaics and i will be argued ha he base skills and processes may be easily gained by mahemaicians over years of learning of proofs, accuracy in problem solving and applicaion of mahemaical echniques o novel conexs. When seen from his perspecive i seems ha modern sudens in finance may lack many of hese paricularly in heir preparaion years. The skills and processes required are based on problem solving mehods rouinely or daily praciced in mahemaics classrooms. Indeed, many of he skills overlap wih he generic ype of he so called higher order and criical hinking skills ha are criically imporan and required for experise in problem solving in any field of sudy. The nex few secions address his issue in some deail and show in how pure and applied mahemaics learning and raining over years provide an edge over he ohers ypes of raining in his area. Tha is, he raining of mahemaicians rouinely involve he use of such criical higher order hinking skills whenever mahemaicians develop and solve eiher simple or complex problems in classrooms, uorials, workshops, and lecures. Higher order hinking skills (HOTS) and mahemaics learning and raining Essenially, mahemaical finance is concerned wih he discipline of finance and economics wih is underlying heory. Financial mahemaicians end o derive, exend and model aspecs of heories and applicaions involved in economics. Clearly, mahemaical finance relies on oher areas such as saisics and compuer science. For example, a financial economis would sudy srucural aspecs o reach a paricular price of a sock while a mahemaician will use his or her knowledge of pure and applied mahemaics o obain he fair value of a derivaive based on an underlying asse. Many of he mahemaical processes involved require he use higher order hinking skills (HOTS) such as creaiviy, logical argumens, deducion, meacogniive analysis and so on, during a single applied problem solving exercise (Tularam, 1997, 1998; Tularam and Kelson, 1998; Tularam, 11, 13). Such an exercise may ake some ime o solve as aply noed in Black s commens earlier; Black saed ha i ook hem 6 monhs o develop he Black Schole s equaions and hen anoher hree years o solve he model. This shows solving of mahemaical models akes ime someimes years. However, i also shows ha mahemaics expers are highly rained in he use of such higher order hinking skills ha includes he affecive rais of persisence and moivaion o pursue heir goals. Alhough soluions may ake some ime mahemaicians are able o search and develop logical approaches using heir problem solving and laeral hinking o creae soluion opporuniies in mos cases. e-jbest Vol.7, Iss.1 (13) 54

13 Tularam Volume 7, Issue 1 (13) Criical hinking skills include comparison, classificaion, sequencing, cause/effec, paerning, webbing, analogies, deducive and inducive reasoning, forecasing, planning, hypohesizing, and criiquing; while creaive hinking involves flexibiliy, originaliy, fluency, elaboraion, brainsorming, modificaion, imagery, associaive hinking, aribue lising, meaphorical hinking, forced relaionships (Tularam, 1997). More deailed definiion of higher order hinking involved in mahemaics can be saed as criical hinking and problem solving skills involving exercising sound reasoning in undersanding, making complex choices, undersanding he inerconnecions among sysems, and framing, analysing and solving problems. Many of he skills ha are ermed higher order criical hinking skills are ofen praciced rouinely during mahemaics learning and raining as saed earlier. Though hey are generic ype problem solving skills i seems ha lile aenion may be paid o such learning in he rouine sense in oher courses when compared o mahemaics daily learning. Andrew Wiles proved Ferma s las heorem and his work ook 7 years o complee. Evidence of he relaionship beween higher order criical hinking and mahemaical is made raher obvious and clear in his documenary (see references). A deep analysis of his work shows ha mahemaical problem solving involves hinking ha uses evidence skilfully and imparially; organizes houghs and ariculaes hem concisely and coherenly; disinguishers beween logically valid and invalid inferences; suspends judgmen in he absence of sufficien evidence o suppor a decision; undersands he difference beween reasoning and raionalizing; aemps o anicipae he probable consequences of alernaive acions; undersands he idea of degrees of belief; sees similariies and analogies ha are no superficially apparen; can learn independenly and has an abiding ineres in doing so; applies problem-solving echniques in domains oher han hose in which learned; can srucure informally represened problems in such a way ha formal echniques, such as mahemaics, can be used o solve hem; can srip a verbal argumen of irrelevancies and phrase i in is essenial erms; habiually quesions one's own views and aemps o undersand boh he assumpions ha are criical o hose views and he implicaions of he views; is sensiive o he difference beween he validiy of a belief and he inensiy wih which i is held; is aware of he fac ha one's undersanding is always limied, ofen much more so han would be apparen o one wih a non-inquiring aiude; and recognizes he fallibiliy of one's own opinions, he probabiliy of bias in hose opinions, and he danger of weighing evidence according o personal preferences (Schafersman, 1991; Tularam, 1997, 1998,,11, 13); ha are ofen referred o as higher order - criical creaive and problem solving hinking skills. Criical hinking appears o be more lef-brain while creaive hinking appears o be more righ brain. The ofen quoed Bloom (1956) sudied cogniion and mea-cogniion using cogniive, psychomoor, and affecive domains. The cogniive domain included knowledge, comprehension, applicaion as well as analysis, synhesis, and evaluaion. Bloom s analysis showed: 1. Knowledge refers mainly o memory of facs;. Comprehension refers o undersanding of facs, demonsraed by organizing or inerpreing hem; 3. Applicaion refers o using undersanding o solve problems; 4. Analysis refers o recognizing paerns suggesed by facs); 5. Synhesis involves producing somehing new; and 6. Evaluaion involves judging qualiy of a soluion or heory. e-jbest Vol.7, Iss.1 (13) 55

14 Tularam Volume 7, Issue 1 (13) The firs hree are lower order hinking skills and his include applicaion, which many hink is higher order. However, he higher-order hinking skills are he abiliy o analyse, synhesise, and evaluae ha also includes a number of reflecive hinking skills. For he benefi of financial educaion, he auhor has included some more specific hinking skills in each of he six caegories above ha can be easily idenified in he examples in his regard laer (challenge yourself o idenify hem). In addiion, he ofen ignored affecive domain is also included in his higher order skill base. Knowledge collec describe idenify lis show ell abulae define examine label name reell sae quoe enumerae mach read record reproduce copy selec Examples: daes, evens, places, vocabulary, key ideas, pars of diagram; Comprehension associae compare disinguish exend inerpre predic differeniae conras describe discuss esimae group summarize order cie conver explain paraphrase resae race Examples: find meaning, ransfer, inerpre facs, infer cause and consequence, examples Applicaion apply classify change illusrae solve demonsrae calculae complee solve modify show experimen relae discover ac adminiser ariculae char collec compue consruc deermine develop esablish prepare produce repor each ransfer use Examples: use informaion in new siuaions, solve problems Analysis analyze arrange connec divide infer separae classify compare conras explain selec order breakdown correlae diagram discriminae focus illusrae infer ouline prioriize subdivide poins ou inerogae Examples: recognize and explain paerns and meaning, see pars and wholes Synhesis combine compose generalize modify inven plan subsiue creae formulae inegrae rearrange design speculae rewrie adap anicipae collaborae compile devise express faciliae reinforce srucure subsiue inervene negoiae reorganize validae Examples: discuss wha if siuaions, creae new ideas, predic and draw conclusions Evaluaion assess compare decide discriminae measure rank es convince conclude explain grade judge summarize suppor appraise criicize defend persuade jusify reframe Examples: make recommendaions, assess value and make choices, criique ideas Affecive domain acceps aemps challenges defends dispues joins judges conribues praises quesions shares suppors voluneers persiss Domain Aribues: self-moivaion, inerpersonal relaions, emoions, aiudes, appreciaions, and values e-jbest Vol.7, Iss.1 (13) 56

15 Tularam Volume 7, Issue 1 (13) The imporance of he criical and higher order hinking skills a he universiy level canno be oversaed as noed in he research resuls of fourh year physics sudens. The quanum physiciss were used as a conrol group while he aomic physiciss were specially augh HOTS he experimenal group. The resuls showed ha experimenal group demonsraed a high level of HOTS while he conrol group were average. Also, he pre and pos esing showed significan differences in performance beween he groups (Molina e al., 8; Tularam and Amri, 1; Tularam, 13) In mos cases, mahemaics lecurers argue ha more demands should be placed on suden early mahemaics learning and eaching ye i seems lile aenion has been paid generally when boh he lower and higher order skills highlighed above are exposed or augh. Noneheless, wha is clearly eviden in ha mahemaics classrooms, lecures, uorials, workshops or problem solving sessions is ha hey ypically include more han heir fair share of pracice and ime spen on he above skills (Tularam and Amri, 11); much of ime spen in sessions are spen on aciviies ha require he acive use of he above hinking skills. I can be said wih high probabiliy ha he above hinking skills are almos rouinely praciced during mahemaics learning in each and every mahemaics classroom, universiy lecure or during uorial and workshops around he world. Therefore, if simply he lengh of ime spen on learning and pracising as well as raining in he use of higher order skills is considered hen over he many years i is no surprising ha mahemaicians are accomplished in hinking or are considered o be raher sharp in hinking; and alened in he use of such skills when dealing wih demanding problems and when solving quesions posed in absrac prime number heory or real life problem solving and indeed in applied financial mahemaics. Higher order hinking skills in heory of financial mahemaics The use of Io s lemma from discree sochasic calculus has been an imporan breakhrough in mahemaical financial sudies generally and his is a poin canno be dispued and as such he auhor inroduces and exposes Io s hinking in he following. The mahemaics used in Io s work can be demanding in ha he derivaion involves many of he skills menioned above as will be noed in he derivaion of he univariae case below. Consider a sochasic process defined by B() = (more generally B() = B a fixed saring poin) where B( + 1) = B() + ( + 1) and {, 1,, }. The innovaions in B are independen sandard normal random variables: ( + 1) ~ NIID(, 1) for all. Tha is, a special version of a random walk wih normally disribued incremens. If he process is more frequen in each fixed ime inerval hen for example; if = 1/n for some arbirary ineger n > 1 hen B( + ) = B() + ( + ) wih B() = and B = B( + ) B() = ( + ); ha is normally disribued as N(, ). Over n periods of lengh [ie. n = 1] he new process has he same expeced change and he same variance as he original has over one fixed ime inerval or period. Making by n large allows [ie. d] and by leing d be he smalles posiive real number such ha d = whenever >1 he value db() = B( + d) B() = ( + d) can be defined as he incremens in he process B(). db() may be hough of as a normally disribued random variable wih mean and variance d; db is ofen referred o as whie noise and B() is ofen referred o as a sandard Wiener process. A number of rules can hen be developed based on he saisical argumens of mean and variance such as: e-jbest Vol.7, Iss.1 (13) 57

16 Tularam Volume 7, Issue 1 (13) 1. E[dB() ] = d; ha is, E[dB() ] = Var[dB()] = d since db() = (( + d) and ( + d) ~ N(, d) and since E[( + d)] = implies ha E[( + d) ] = Var[( + d)]. Var[dB() ] = ; Var[dB() ] = E[dB() 4 ] (E[dB() ]) = E[( + d) 4 ] d = 3d d = I is noed he erm ( + d) is N(, ) he expecaion: E[( + d) 4 ] = 3 4 (using kur(x) = [E(X 4 ) - 4µE(X) + 6µ E(X ) - 3µ 4 ] / σ 4. The above shows ha db() = d is a consan. If he variance of a funcion of a random variable vanishes he expecaion sign becomes redundan: for example, E[f(dB())] = f(db()) when Var[f(dB())] =. A number of rules can be generaed such as d = since d is small and db()d is also, while db() = d. I can be said ha a sandard Brownian moion denoed by B() is a sochasic process defined by: B() = B (= ) (wih probabiliy 1) and has incremens B s B ha is normally disribued wih N(, s ) for all s and wih s >. The incremens are independen normal random variables; db() is a differenial represenaion of B(). However, he sandard Brownian moion is no a good model for sock price movemens in ha i has no drif. A process ha allows for drif in prices hus allowing he expeced o change over any fuure inerval (non-zero) would be more useful. To inroduce a drif le X be an Io process defined by dx d db where he dependence of (drif) and (diffusion) on (X, ) is normally no shown in equaions (noe: X and B are used for X() and B() respecively). If a funcion (f) is defined on f: [, T] hen a funcion Y = f(x, ) is an Io process. The change in Y (dy) may now be defined using Equaion 1 based on Taylor series as shown laer: dy [ f ( X, ) f ( X, ) f ( X, ) db ; (1) x 1 f xx ( X, ) ] d x f mus be wice coninuously differeniable in boh X and bu we only need f x, f xx, and f o exis and o be coninuous. However, how can Equaion 1 be esablished? Using a nd order Taylor series for f(x + d, + d) ha is close o he poin (X, ), he change dy = f(x +d, + d) f(x, ) can be deermined. f ( X d, d) f ( X, ) f x ( X, ) dx f ( X, ) d () 1 f 1 xx f ( X x, )( dx ( X ), ) dx 1 f ( X, )( d) d R( residual) The remaining erms (R) end o as higher order d s and mos of he higher order dx s ends o. In Equaion, dx, d are non zero while d is zero. The erms (dx ) and dx d is deermined by squaring boh sides of dx (Equaion 3) giving (dx ) o be d ; ( dx ) ( d db ) (3) d he produc db d d; db d dx d is since dx d ( d db ) d. Now Equaion 1 may be derived using dy and Taylor s series expansion and dx d db ; e-jbest Vol.7, Iss.1 (13) 58

17 Tularam Volume 7, Issue 1 (13) dy f( X [ f x d ( X, d ) f( X, ) f ( X, ) f( X, ) 1 f xx dx, d ) f( X ( X, ) ] d f x ( X, ), ) db. (4) In Equaion 4, he higher order erms (R) vanish. In ordinary calculus dx is small enough and dx vanishes as i is much smaller sill. However in sochasic calculus, he dx erm is a random variable and dx canno be assumed o vanish bu raher converges o d. The higher order erms such as dx 3 or dxd ec do in fac vanish. Therefore, Equaion 1 (Equaion 4) have been esablished as a way of represening he change in a funcion Y() derived using Taylors series and Io s heoreical work. A brief examinaion of he above works would sugges many skills have been used such as analysing, quesioning, creaiviy, consrucing, collecing erms, jusifying, persuading, synhesis, and logical presenaion and finally judgemen (for more specific skills see earlier under analysis, synhesis and evaluaion). Developmen of financial models using higher order mahemaical hinking In finance, he rading of opions is an essenial aciviy. An opion is a conrac ha gives he righ o he buyer of he opion o buy or sell a primary asse (a sock or a bond) a a price wih mauriy dae fixed a he ime of purchase. This opion is similar o an insurance conrac in ha he holder of he opion is somewha proeced agains changes in he underlying asse price. A number of finance agens considered he quesion of wheher here exiss a heoreical price of any derivaive or opion in a given economy (Duffie, 1). The process of derivaion of such pricing and model can developed based on Io and sochasic calculus and he noion of no arbirage (no free lunch). In an ideal siuaion he model predics he price o be he iniial amoun of money invesed as a porfolio exacly replicaes he payoff of he opion a is mauriy dae. Clearly, a number of assumpions need o be aken ino accoun in his pricing formula. Jus as in chemisry he ideal gas equaion (pv = nrt) has been used o explain behaviour of gases in general where as in fac none of he gases hemselves ever behave ideally bu vary wih differen degrees from he model predicion. Even when his is he case, a number of advances have been made in chemisry in erms of predicing srucures and properies of gases afer he developmen of he ideal gas equaion. The model is able o provide he opporuniy o develop higher order rules. This model was made possible wih he use of mahemaics and indeed mahemaics models are a he forefron of developmen of cuing edge knowledge in many fields. The model consruced relied on he skills of he developers in ha hey were able o use heir raining in higher order hinking and criical and creaive skills o reach a higher level of absracion and undersanding in anoher field. In many cases, he physiciss and chemiss were mahemaicians firs and applied mahemaicians laer in heir respecive fields. The same is noed in finance wih he work of Io and ohers. The iniial work on Brownian moion and sochasic process is due o Bachelier (19) who modelled sock price bu i is for cerain ha Black and Scholes (1973) can be credied for he use of his knowledge and develop i o a higher level when hey developed he European opions pricing formula. Laer, Harrison and Kreps (1979) and Harrison and Pliska (1981) provided a general concepual framework o he opion pricing problem based on he developed sochasic inegraion heory. The res is hisory bu he developmen of he model is a grea advance in he field and was achieved using mahemaical hinking ha is highlighed in he following. e-jbest Vol.7, Iss.1 (13) 59

18 Tularam Volume 7, Issue 1 (13) Black-Scholes model and modelling derivaive pricing The parial differenial equaion ha changed mahemaical finance forever is analysed in some deail in a case sudy manner o show how higher order criical hinking skills rouinely applies o mahemaical hinking during developmen of creaive ideas and soluion processes. Mos if no all of he models and echniques applied o pricing of asses or valuaion of opions are based on he model developed by Fischer Black and Myron Scholes in 1973 and Rober Meron (1973). The following noes are nealy adaped from Leeds and Srahmore Universiy s lecure noes. One of he mos imporan aspecs of higher order hinking is o deal wih he use of leer for variables and consans. Simply he use of leers does no change he affec mahemaicians as i does when ohers view equaions; his is an essenial aspec of absracion and criical o mahemaical hinking seeing hrough he problems. A sochasic process X is more formally a family of random variablesx T and is defined on a given probabiliy space. In mahemaics a space is defined by a series of mahemaical rules ha he objecs in he space obey o he leer. The process is indexed by a parameer ha varies over an index se T providing he ime-line. I was noed ha Brownian moion can be used o model asse reurns having a zero mean ye i fails as he growh rae of he reurns is nonzero. There was need hen o exend he sandard Brownian moion o have some non-zero drif erm (mean) and a 1D generalised sochasic differenial equaion dx ad bdz, X X was iniiaed where a (drif rae) and (SDE): b (sandard deviaion) are consans and Z is a Wiener process wih being a sandard Wiener process wih mean zero and variance one. Various noaions are noed in exs so in he following dz is defined in place of db. The mean of dx is given by ad and is variance is given by b d (see earlier derivaions). Therefore, for an inerval, he inegraed form he SDE can be given as X s X a ds b dz where X X (5) The soluion in Equaion 5 is given as: X X a bz where Z X X. Here X has a linear rend (or drif) given by X a Z and and he variance erm: b d. The parameers a and b are usually referred o as drif velociy ( ) and volailiy ( ). The auhor has developed some rajecories using Malab of a generalised Brownian moion (Figure 1) and i is noed ha X akes boh posiive and negaive values. This was a major problem wih he Bachelier model as sock prices canno ake below zero values. e-jbest Vol.7, Iss.1 (13) 6

19 Tularam Volume 7, Issue 1 (13) Figure1: Typical rajecories of a generalised 1-D Brownian moion wih drif. 5 and sandard deviaion = 1 and 95% envelope (limiing curves) i.e. rend line, x + a envelope abou he 4 Generalised Brownian Moion.5 Sandard Brownian Moion -Dr G A Tularam X() S Time X Changing parameers a and b o become funcions of he underlying variable and ime allowed researchers o generalize Brownian moion. For example, if a ax, and b bx,, a 1D generalised Iô process can be wrien as dx ax, d bx, dz, where ax, is he insananeous drif rae ha measures he expeced rae of change in X and b X, is he insananeous variance ha measures he amoun of random diffusion. I is assumed ha boh drif and variance are funcions of ime. The expeced drif is a X, d and he expeced variance is given o be b X,d. The change over he inerval, can hen be expressed in an inegraed form as a linear funcion of drif and volailiy erms are considered (Equaion 6): e-jbest Vol.7, Iss.1 (13) 61

20 Tularam Volume 7, Issue 1 (13) X X ax s,sds bx s,sdzs : X (6) In he above, an imporan decision was made based on he observaion ha he opion price is dependen on he sock price; he price movemen over ime mus be relaed o he sock s movemen over ime. Moreover, a geomeric Brownian moion process was also considered for sock price modelling and Iô s lemma allowed he change in he opion price o be deermined as he following SDE shows (Black and Scholes, 1973). In a simple case of a geomeric Brownian moion i can be assumed ha a( X, ) X and b( X, ) X hus forming dx X d X db or dx X d db ; a process wih a consan expeced reurn over ime and a consan variance of reurn. Alhough simplified his is a more naural model of sock prices (why?). dx X ad bdz ; X X (7) S log / X X ads bdz s Here X is he price of a raded asse a ime while X dx is he rae of reurn X and on he asse. Equaion 7 canno be solved direcly since i is no linear in conains pars ha involve sandard Wiener process Z. The inegral however, shows ha he logarihmic sock reurn execues he relaively simple generalised Wiener process now called geomeric Brownian moion. Firs inroduced by Samuelson X X exp S never (1965) i has an advanage over he Bachelier model in ha assumes negaive values as shown in Figure. The creaion of a model ha allowed only non-negaive values was a major improvemen in he modelling of prices (Figure ). e-jbest Vol.7, Iss.1 (13) 6