Market risk VaR historical simulation model with autocorrelation effect: A note

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1 Inernaional Journal of Banking and Finance Volume 6 Issue 2 Aricle Marke risk VaR hisorical simulaion model wih auocorrelaion effec: A noe Wananee Surapaioolkorn SASIN Chulalunkorn Universiy Follow his and addiional works a: hp://epublicaions.bond.edu.au/ijbf Recommended Ciaion Surapaioolkorn, Wananee (29) "Marke risk VaR hisorical simulaion model wih auocorrelaion effec: A noe," Inernaional Journal of Banking and Finance: Vol. 6: Iss. 2, Aricle 9. Available a: hp://epublicaions.bond.edu.au/ijbf/vol6/iss2/9 This Journal Aricle is brough o you by he Faculy of Business a epublicaions@bond. I has been acceped for inclusion in Inernaional Journal of Banking and Finance by an auhorized adminisraor of epublicaions@bond. For more informaion, please conac Bond Universiy's Reposiory Coordinaor.

2 Surapaioolkorn: Marke risk VaR hisorical simulaion model wih auocorrelaion The Inernaional Journal of Banking and Finance, 28/9 Vol. 6. Number 2: 29: IJBF MARKET RISK VaR HISTORICAL SIMULATION MODEL WITH AUTOCORRELATION EFFECT: A NOTE Wananee Surapaioolkorn SASIN Chulalunkorn Universiy, Thailand Absrac The modern marke risk model using Value a Risk (VaR) mehod in he banking area under he BASEL II Accord can ake differen forms of simulaion. In his paper, hisorical simulaion will be applied o he VaR model comparing he wo differen approaches of Geomeric Brownian Moion (GBM) process and Boosrapping mehods. The analysis will use correlaion plos and examine he effecs of he auocorrelaion funcion for sock reurns. Keywords: BASEL II Accord, Marke Risk Model, VaR Model, Sochasic Process, Hisorical Simulaion, Boosrapping JEL Classificaion: C5, C5, C88. Inroducion In all aspecs of life, we ofen wan o have he abiliy o predic he fuure wih some cerainy. The mos reliable echniques used for forecasing he fuure are probabiliy heory and saisical applicaions. In he banking and finance world, we ofen combine quaniaive engineering echniques such as quanum physics, classical economerics heory and operaional research echniques o help wih predicions of he movemen of liquidiy price in financial markes. In he area of financial marke risk, he main echnique used o predic he movemen of prices is based on he so-called Value a Risk (VaR) model. This model has been used widely in he banking indusry afer he compleion of he BASEL I accord in 996 and ever since i was publicly inroduced by J. P. Morgan in 997 (see Pearson (22) for more deails). The mahemaical mehods behind his model are based on he sochasic process using geomeric Brownian moions which will be inroduced in he nex secion. More deailed explanaions of he BASEL accord and he marke risk model can be found in Surapaioolkorn (27). Published by epublicaions@bond, 29

3 Inernaional Journal of Banking and Finance, Vol. 6, Iss. 2 [29], Ar Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: The main feaure of he VaR model is he performance by calculaing risk saisics using a bank s porfolio value wih a correlaion mehod like he parameric VaR or simulaion mehods using he hisorical or he Mone Carlo (MC) echniques. Two essenial crieria for choosing he simulaion mehod are he accuracy of he esimaed oupu and he iming required o creae such oupu (see Glasserman (24) for more deails). Simulaion mehod requires fas compuer machine and sofware echnology as well as he bes risk model specialis (ofen referred o as financial engineer ) who can handle he daa applicaions, he sochasic process echniques, he simulaion mehods and who can undersand he marke risk drivers well. In his analysis, he hisorical simulaion mehod will be used. I will be explained in Secion 3. One of he mos imporan conceps in modelling marke risk is he use of adequae fuure financial marke price reurns. Financial marke daa required for he esimaion of VaR ofen come in erms of marke prices and spread raes such as ineres raes, foreign exchange (FX) raes, equiy index raes, and implied volailiies. These prices and raes shall be referred o as marke reurn raes. Marke price reurns provided by he bank ofen come in daily high frequency dae raes. In his quaniaive analysis, he equiy index raes will be used. They are inroduced in Secion 4. In his paper, he esimaion of fuure expeced value of financial asses using financial marke daa will be analysed using he Auocorrelaion (ACF) produced from he hisorical simulaion of he VaR Model. The sochasic processes involved in he VaR model will be inroduced in he nex secion. The aim is o see he effecs of he simulaion pah generaed from financial marke daa. 2. Sochasic Process for Marke Risk VaR Model There are a lo of VaR applicaions used for he calculaion of risk facors in measuring financial marke risk. Among hem is McNeil e al. (25). The purpose of using he VaR Model is o find he expeced loss of a porfolio over some ime horizon wih a given level of probabiliy a he maximum level using sochasic processes. In his secion, we begin wih he key heory behind he VaR model known as he sochasic process. In he world of finance, he uncerainy of he fuure can be quanified using he so-called sochasic process. Many genius mahemaicians and scienis like Gauss, Wiener, Levy, and Io have conribued he mos exraordinary mahemaical echniques such as Gaussian disribuion, Wiener process, Brownian moions, Levy processes, and Io s Lemma respecively o he modern world of financial banking. See Mikosch (26) for furher deails. All of hese echniques are based on he mehod of sochasic process which is proven o be he mos powerful ool for forecasing he fuure of imporan financial applicaions in he banking secor. Sochasic process is used in simulaion mehods o obain he maximum values of he required marke risk saisics. The simples sochasic process can be defined using mahemaical noaions as hp://epublicaions.bond.edu.au/ijbf/vol6/iss2/9 2

4 Surapaioolkorn: Marke risk VaR hisorical simulaion model wih auocorrelaion Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: Y Y ( ), T, A () where Y is he random variable which can be any marke reurn rae; refers o ime inerval frequency ype of daa like monhly, daily, hourly, second-bysecond (or ick-by-ick); and A is defined as some space in he process. A. Geomeric Moion Processes One of he mos useful sochasic processes ha are well used in financial engineering is he geomeric moion. The word geomeric ofen refers o exponenial form of mahemaics. There are hree useful geomeric moions used in he marke risk banking sysem. We will define hem in his secion. In he marke risk sysem, Y can ake differen forms of sochasic process depending on he ype of marke daa reurns. Suppose ha Y represens he sock price process a ime. In financial mahemaics, he sock price process Y is ofen of he Geomeric exponenial form given by Y Y e X or X log( Y ),, (2) (3) where, X is defined as he rae of log-reurn of he sock price. The log-reurn process X can be modelled by a sochasic diffusion equaion (SDE), for example an Ornsein-Uhlenbeck (OU) process, or a Levy process, for example Variance Gamma process. The hree geomeric moions are defined as followed: (i) Geomeric Brownian Moion (GBM) Process Geomeric Brownian Moion (GBM) process is named afer Rober Brown, he biologis whose research daes o he 82s. I was Norber Wiener (923) who inroduced Brownian moion o he mahemaical world. If X is a Brownian wih or wihou drif, hen equaion (3) is called Geomeric Brownian Moion (GBM). The SDE for Y can be wrien as dy Y d Y dz, (4) where, Z is a sandard Brownian Moion. In his case, Y has a Log-Normal disribuion, since X has a normal disribuion. Therefore, equaion (4) is someimes called Log-Normal model. In pracice, we ofen use his geomeric process for equiy index raes or sock price reurns. Published by epublicaions@bond, 29 3

5 Inernaional Journal of Banking and Finance, Vol. 6, Iss. 2 [29], Ar Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: (ii) Geomeric Ornsein-Uhlenbeck (OU) Process The Ornsein-Uhlenbeck (OU) Process is named afer Leonard Salomon Ornsein and George Eugene Uhlenbeck, he Duch physicis whose research daes o he 92s. I is also known as he mean revering sochasic process. In any marke risk model, he log-reurn X is an OU Process wih dynamics SDE as follows d log( Y ) = α ( µ log( Y ) d + σdz, (5) or dx = α µ X ) d + σdz, ( (6) where, Z() is a sandard Brownian Moion, α is he speed of reversion, µ is he (log) mean reversion level, and σ is he volailiy. Then equaion (3) may be called Geomeric OU Process. In pracice we ofen use his geomeric process for ineres raes. (iii) Geomeric Levy Process The Levy Process is named afer he French mahemaician Paul Levy. This is he sochasic coninuous ime process based in probabiliy heory. Wiener Process is one of he mos well-known examples of he Levy Process which has o have saionary independen incremens as well as saring he value a zero. Anoher good example of he Levy Process is he Poisson Process. If X is a Levy Process, for example X is a Variance Gamma Process, hen a Brownian Moion is a special Levy process, herefore (4) is also a Geomeric Levy Process. 3. Hisorical Simulaion and Boosrapping Mehod Hisorical simulaion is one of he mos well-known simulaion mehods used for marke risk modelling. In hisorical simulaion, raher han generaing random numbers from a machine, he acual pas hisory of marke daa are used so as o reflec realiies like fa ails, persisence, and oher common sylized facs of volailiy. In hisorical simulaion, we assume ha pas hisory is repeaed and predic he fuure using he pas daa. The daa used can be referred o as nonoverlapping or overlapping daa. In his secion, we firsly analyse hese wo ypes of daa using he hisorical simulaion. Boosrapping mehod used o help wih he correlaion beween pahs as well as he limiaion of daa series will be inroduced a he end of his secion. hp://epublicaions.bond.edu.au/ijbf/vol6/iss2/9 4

6 Surapaioolkorn: Marke risk VaR hisorical simulaion model wih auocorrelaion Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: (i) Non-overlapping daa For non-overlapping daa, consider he following: The s simulaion run is o be consruced from [,,n] The 2 nd simulaion run is o be consruced from [n+,, 2n] and so on unil.. The N h / n (if N is a muliple of n) simulaion run is o be consruced from [N- n+,, N] In pracice, more pahs are generaed in order o increase accuracy of he VaR. This is where he overlapping hisorical periods can be used. (ii) Overlapping daa For overlapping daa, consider a n day VaR calculaion (e.g. days). Obviously, he simulaion mus consruc many pahs composed of n observaions. Assume ha for each marke risk facor here exis N hisorical observaions. Currenly he pahs will be sampled as follows. Observaions [,,n] can be used o consruc he firs pah. Observaions [2,, (n+)] can be used o consruc he second pah, and so on unil The N-n+ pah is consruced from observaions [(N-n+),, N]; so ha N-n+ (> N/n) simulaion pahs are generaed from his overlapped daa. Le ake a discree example. Suppose ha he sock prices for he las 3 days were, 5.3,.63 and oday s sock price is 9.72, so ha he daily sock reurns for he pas 3 days were 5%, 6% and 7%. If his daa is used o run he hisorical simulaion and predic he sock price for omorrow and 3 days from oday, we have: A ime node for day : 9.72 * exp(.5) = 25.86, and A ime node 2 for day 3 : * exp(.6) * exp(.7) = * exp(.3) = So he rae change daa used for ime node is 5% and ha for ime node 2 is 3%. For example, assume ha here are 9 days of pas daa and we wan o esimae he values for he nex 3 days. From he daa of days, 2 and 3, we can generae one pah for fuure 3 days, from days 4, 5, 6 for he 2nd pah, and from days 7, 8, 9 for he 3rd pah. Now, we have 3 hisorical simulaion pahs, and a each fuure ime node of day, 2 and 3, we have 3 samples each so ha we can compue he saisics for he values. In hisorical simulaion, he process ofen allows for overlapping. This means ha i is possible o use [days, 2, 3] for he s pah, [days 2, 3, 4] for he 2nd pah, [days 3, 4, 5] for he 3rd pah, ec. This way, more simulaion pahs are Published by epublicaions@bond, 29 5

7 Inernaional Journal of Banking and Finance, Vol. 6, Iss. 2 [29], Ar. 9 6 Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: generaed. The only disadvanage is ha he pahs are correlaed. The purpose of his analysis is o see if any disorion resuls due o his correlaion. To avoid he correlaion problem, (for example, 3-day daa can be randomly picked from 9 days of daa) he boosrapping mehods may be inroduced. Boosrapping Mehod In hisorical simulaion mehod, he main problem ofen found is he lack of hisorical daa. To avoid his problem, we inroduce a saisical echnique known as Boosrapping. Boosrapping mehod assumes ha daily reurns are independen and can be used in our hisorical simulaion. For example, we can randomly sample he rae of reurns from he hisorical daa and repea many imes o exrac more informaion from he given daa. The basic idea of he boosrapping mehod is o ake some real daa and hen creae a large number of replicae daa ses by sampling wih replacemen (wih equal probabiliy) from he original sample. Thus each new, replicae daa se is a slighly perurbed version of he original one. Sampling wih replacemen means ha every sample is reurned o he daa se afer sampling. Therefore a paricular daa poin from he original daa se could appear muliple imes in a given boosrap sample. 4. Quaniaive Analysis for Marke Risk Model In his secion, we will ouline he correlaion and auocorrelaion funcions, proposed analysis and he analysis carried ou for his paper. The US equiy reurns using he asses of observaions is used as financial marke daa. MATLAB sofware is chosen o help wih consrucing he marke risk VaR model in order o generae he simulaed pahs using he hree differen mehods of sampling from he hisorical daa. (i) Correlaion and Auocorrelaion Funcions (ACF) The linear correlaion of wo variables X, and Y can be defined as Cov( X, Y ) = xy, where Cov ( X, Y ) = E (7) var X var Y X. Y E X. E Y 2 The correlaion value lies beween o +, where high posiive or negaive correlaion is possible depending on he marke movemens. Le us consider he following hree examples. In he overlapping daa, if he marke is keep moving up, so ha he rae of reurn for a week is.,.2,.4,.5,.7, hen he one-day lagged daa could be.2,.4,.5,.7,.8. Obviously hey are highly posiively correlaed. If he marke moves up and down, he correlaion beween he iniial daa and he lagged daa could be negaively correlaed. For example, hp://epublicaions.bond.edu.au/ijbf/vol6/iss2/9 6

8 Surapaioolkorn: Marke risk VaR hisorical simulaion model wih auocorrelaion Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: if he daily reurns are.,.,.2,.,.2, and he one-day lagged daa are.,.2,.,.2,., hen he correlaion is highly negaive. For Non-overlapping real marke daa, i is possible o obain eiher posiive or negaive correlaion values. If he raes were moving up las week and moving up again his week, hen here is a posiive correlaion beween las week and his week. If he raes were moving up las week and moving down his week, hen here is a negaive correlaion beween las week and his week. The Auo-Correlaion Funcion (ACF) or serial correlaion occurs when we know ha observaions ha are recorded sequenially over ime are no independen of one anoher. For example, if a series X = (X, X - ), hen X, X - share informaion. This happened in overlapped hisorical periods. The ACF can be defined as Cov( X, X ) = var( X ) var( X ) 2, (8) (ii) Proposed Analysis I is proposed ha he auocorrelaion effec can be quanified by consrucing a simple sand-alone simulaion of his process in MATLAB. In paricular, hisorical marke equiy daa will be used o consruc he sample pahs (in he manner oulined above). The erminal value of each sample pah will be used o value a simple conrac (for example a vanilla opion) and he VaR of his conrac calculaed. The simulaion will run for calculaing he VaR for a -day horizon. We will hen analyse he resuling sample pahs generaed for ACF or serial auocorrelaion. In addiion o his, each of he simulaed pahs will be analysed for evidence of sysemaic bias in heir correlaions. For example, he correlaions beween pahs and heir auocorrelaion funcions can be deermined. This mehod will hen be conrased wih he hree furher procedures for sampling from he hisorical daa provided. (i) Selecing non-overlapping periods. Tha is, observaions (...n) will be used o consruc he firs pah, observaions (n+)...,2n will be used o consruc he second pah, and so on unil he N / n h pah is consruced from observaions. N-n+,, N. (ii) A sampling procedure based on he Boosrap mehod will be uilised (where samples of n observaions will be chosen repeaedly a random from he se of N observaions). (iii) Each of hese procedures can also be underaken using simulaed equiy daa generaed from a GBM process as deailed in subsecion 2.. Published by epublicaions@bond, 29 7

9 Inernaional Journal of Banking and Finance, Vol. 6, Iss. 2 [29], Ar Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: (iii) Analysis For procedures (i) and (ii) using he non-overlapping and he overlapping periods via he Boosrapping mehod, here exiss a new window N w for each generaed new pah. The number of new windows can be considered as Tob Horizon N W = +, Tau (9) where Tob represens he number of oal observaions, Horizon represens he sample size wihin each window, and Tau represens number of incremens for he nex simulaion pah (i.e. day, 2 days, ec.). Noe ha when horizon equals Tau, we have a non-overlapping pah. For procedure (iii), by generaing simulaed daa using a GBM process, we will perform he same analysis on his simulaed daa (which has he same momens as he acual marke daa) as we do for he hisorical daa for comparison (so ha he lengh of he simulaed ime series is idenical o ha of he hisorical daa). The GBM process can be considered as follows: S 2 σ = S µ exp + σ ε 2 () where S S - = S is he change in sock price, S is he small inerval of ime,, and ε is a random drawing from a sandardised normal disribuion. A parameer µ represens he expeced rae of reurn per uni ime from he sock and a parameer σ represens he volailiy of he sock price. 5. Empirical Resuls For Boosrapping mehods, simulaed values of variables menioned in equaion (9) for he hisorical pahs give a mixure of negaive and posiive values. Using he wo following examples, he resuls are shown below. (i) Taking he Tau, Horizon o be 5, and he Tob o be 25, here are he non-overlapping hisorical periods. The number of windows is 5. The correlaion values is displayed below:. Using he GBM process, hp://epublicaions.bond.edu.au/ijbf/vol6/iss2/9 8

10 Surapaioolkorn: Marke risk VaR hisorical simulaion model wih auocorrelaion Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: Using he Boosrapping mehod, By examining he correlaion plos (no shown) for hese oupus i was observed ha he plos are no adequae. This shows ha we should increase he number for variables Tau and Horizon. The second analysis is deailed below. (ii) Taking he Tau =, Horizon = 5, and Tob = 25 here are overlapping hisorical periods. The number of window is. The correlaion values for he firs 5 columns (i.e 5 windows) and he firs 5 pahs are displayed below.. Using GBM process, Using he Boosrapping mehod, The correlaion plos displayed in Figure represen he simulaion of Pah. These plos sugges ha here exiss a serial correlaion beween pahs around zero. The exreme poin of value comes from he diagonal marix of where each pah is correlaed o iself. This is jus one example of one paricular pah; however, i is rue for all hisorical simulaion pahs. A his sage, le us examine he auocorrelaion (ACF) plos below o see wheher our model is a suiable model for our daa se. By aking he lag a 5, he ACF plos for Pah show lile correlaion for he pahs a all lag. Slighly more auo-correlaion a lag 4 occurred in he second plo below. Again, he ACF plos for all pahs are similar o he Pah. This means ha he hisorical simulaion model may be of value in fiing wih our financial sock daa. Published by epublicaions@bond, 29 9

11 Inernaional Journal of Banking and Finance, Vol. 6, Iss. 2 [29], Ar Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: GBM: s Window GBM: 2nd Window Tob Tob Boosrapping: s Window Boosrapping: 2nd Window Tob Tob Figure : Correlaion Plos using he GBM process and he Boosrapping mehod Sample Auocorrelaion Funcion (ACF) For GBM Sample Auocorrelaion Lag Sample Auocorrelaion Funcion (ACF) For Boosrapp Sample Auocorrelaion Lag Figure 2: ACF Plos using he GBM process and he Boosrapping mehod hp://epublicaions.bond.edu.au/ijbf/vol6/iss2/9

12 Surapaioolkorn: Marke risk VaR hisorical simulaion model wih auocorrelaion Marke risk Var hisorial simulaion model wih auocorrelaion effec: A noe: Conclusion These empirical resuls sugges ha he GBM process and Boosrapping mehods provide similar oupu in erms of correlaion values and he ACF plos. Tha is, he correlaion values are no consisen. The non-overlapping pahs should provide correlaion values o be relaively larger han he overlap pahs. Alhough his is no always he case, i seems ha he values of correlaion are very similar hroughou (i.e. around zero) for all mehods. Auhor saemen: Submiing auhor is Wananee Surapaioolkorn, Sasin Graduae Insiue of Business Adminisraion, Chulalongkorn Universiy, Bangkok, Thailand, 27. Tel: ; Wananee. surapaioolkorn@sasin.edu. References Glasserman, P. (24). Mone Carlo Mehods in Financial Engineering, Applicaion of Mahemaics: Sochasic Modelling and Applied Probabiliy, Springer. McNeil, A. J., Frey, R., and Embrechs P. (25). Quaniaive risk managemen: Conceps, echniques, ools. Princeon Series in Finance, Princeon Universiy Press, Princeon and Oxford. Mikosch, T. (26). Elemenary sochasic calculus wih fi nance in view. Advanced Series on Saisical Science & Applied Probabiliy Vol. 6. Pearson, N. D. (22). Risk budgeing: Porfolio problem solving wih value-a- Risk. John Wiley & Sons. Saunders, A., Boudoukh, J., and Allen, L. (23). Undersanding marke, credi and operaional risk: The value a risk approach. Blackwell Publishing. Surapaioolkorn, W. (27, July-Sepember). Quaniaive review on he presen and fuure of financial risk modelling and he role of BASEL II Accord. SCMS Journal of Indian Managemen, 4 (3). Published by epublicaions@bond, 29