Algorithmic trading strategy, based on GARCH (1, 1) volatility and volume weighted average price of asset

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1 IOSR Journal of Business and Managemen (IOSR-JBM) ISSN: 78-87X. Volume, Issue (Sep-Oc. ), PP 3-35 Algorihmic rading sraegy, based on GARCH (, ) volailiy and volume weighed average price of asse Simranji Singh Kohli, Nikunj Makwana, (Compuer Engineering, Sardar Pael Insiue of Technology/ Mumbai Universiy, India) (elecronics and Communicaion Dep., Nirma Insiue of Technology, India) Absrac : Algorihmic rading sraegies have one of he mos significan roles for he new era of financial marke. Various Hedge funds, Muual funds and oher invesmen banks are widely using various algorihmic rading sraegies for risk managemen and fuure volailiy esimaion for financial insrumens. In his paper, we focus upon one algorihmic approach wih he aspec of special case of GARCH model, is abiliy o deliver volailiy forecass and moving average wih muliple weighed price of asse. This model is useful no only for modeling he hisorical process of volailiy bu also in giving us muli-period ahead forecass and helps o give exac enry price. Keywords: Algorihmic rading,garch,volailiy, risk managemen,mahemaical modeling I. Inroducion Algorihmic rading has been one of he mos prominen recen rends in financial indusry. I is widely used by invesmen banks, muual funds, and oher buy side (invesor driven) insiuional raders. Sell side raders, such as marke makers and somehedge funds, provide liquidiy o he marke, generaing and execuing orders auomaically. This has resuled in a dramaic change of he marke microsrucure, paricularly in he way liquidiy is provided. A hird of all European Union and Unied Saes sock rades in 6 were driven by auomaic programs, or algorihms, according o Boson-based financial services indusry research and consuling firm Aie Group. As of 9, HFT firms accoun for 73% of all US equiy rading volume.this indicaes he significance of robus and reliable need of rading algorihms and mahemaical models for his new era of financial marke o model volailiy.the GARCH (p, q) model, inroduced by Bollerslev (986), ofen provides a parsimonious represenaion of he volailiy dynamics in financial ime series []. One radiional difficuly in consrucing GARCH based models is ha he volailiy process is inherenly unobservable. We surmoun his problem by using a proxy of monhly volailiy calculaed using daily daa. Moreover GARCH models rea heeroscedasiciyas a variance o be modeled. As a resul, no only are he deficiencies of leas squares correced, bu a predicion is compued for he variance for each error erm [].We have more faih in he reliabiliy of hese volailiy esimaes. II. Need for forecasing volailiy for a model The main purposes of forecasing volailiy are measuring he poenial fuure profi and losses of a porfolio of financial asses. Moreover i helps a lo in asse pricing phenomenon. One of he mos common use of volailiy for any commodiy, opions, socks are o find ou nex day s volailiy based on hisorical volailiy. This helps o know approximae value of reruns over invesmen. Several recen sudies have found ha he volailiy of daily U.S. dollar exchange raes ends o be highly persisen and well approximaed by an inegraed or long memory-ype GARCH process[3].in asse allocaion, he Markowiz approach of minimizing risk for a given level of expeced reurns has become a sandard approach, and of course an esimae of he variance-covariance marix is required o measure risk. Perhaps he mos challenging applicaion of volailiy forecasing, Seasonaliy in financial-marke volailiy is pervasive. he hisorical variance of he Sandard and Poor's composie sock-price index in Ocober is almos en imes he variance for March; see also Schwer (99) and Glosen, Jagannahan, and Runkle (993) [].So in oday s highly volaile marke,i is imporan o specify various parameers which help o derive volailiy more accuraely. High kurosis exiss wihin financial ime series of high frequencies (observed on daily or weekly basis). This confirms he fac ha disribuion of reurns generaed by GARCH(p,q) model is always lepokuric, even when normaliy assumpion is inroduced. Righ combinaion of volailiy parameer will help o give more reliable and accurae value of volailiy. I is imporan o noe ha kurosis is boh a measure of peak and fa ails of he disribuion. So we have ried o make i as accurae as possible.in he vas empirical finance lieraure models are well known wihin he GARCH framework where alernaive assumpions on he condiional disribuion have been suggesed and exensively analyzed [5]. 3 Page

2 Algorihmic rading sraegy, based on GARCH (, ) volailiy and volume weighed average price of III. Kurosis Of Garch(,) Process GARCH models are very popular for represening he dynamic evaluaion ofvolailiy of financial reurns.[see, e.g., Bollerslev, Engle, and Nelson (99), Engle (99), Bera and Higgins (995), Diebold and Lo pez (995), and McAleer and Oxley (3), among many ohers [6]. GARCH(,) process: has been assumed r u ; u i. i. d. N,, () The second momen of innovaion process E Var, () equals: While he fourh momen is given as: 3 E (3) 3 From covariance saionary condiion of GARCH(,) process, and sricly posiively condiional variance:, () Follows ha he second momen of process exiss. To assure he exisence of he fourh momen, apar from condiions in (), i is necessary in relaion (3) o saisfy his resricion: 3. (5) Since kurosis is defined as: E k, (6) E hen expression (6) becomes: 3 k. (7) 3 Afer some rearrangemen in (7) we can wrie: k (8) From relaion (8) follows ha disribuion of reurns generaed from GARCH(,) process always resuls in excess kurosis, i.e. Fisher's kurosis ( k 3) even normaliy assumpion is inroduced, if and only if condiions in () are saisfied. These condiions also could be saisfied when parameer. Only in ha case innovaions disribuion would be normally shaped ( k 3). Therefore, he kurosis is very sensiive on value of parameer.in general kurosis increases much inensively wih larger parameer in comparison o parameer. IV. Degrees Of Freedom Esimaion Generally, here are hree parameers ha define a probabiliy densiy funcion: (a) locaion parameer, (b) scale parameer and (c) shape parameer. The mos common measure of locaion parameer is he mean. The scale parameer measure variabiliy of probabiliy densiy funcion (pdf), and he mos commonly used is variance or sandard deviaion. The shape parameer (kurosis and/or skewness) deermines how he variaions are disribued abou he locaion parameer. If he daa are heavy ailed, he VaR calculaed using normal assumpion differs significanly from Sudens -disribuion. Therefore, we find ha kurosis and degrees of freedom from Suden's disribuion are closely relaed. 3 Page

3 Algorihmic rading sraegy, based on GARCH (, ) volailiy and volume weighed average price of The densiy funcion of no cenral Suden -disribuion is given as: df df x f x (9) df df df Where is locaion parameer, scale parameer and df shape parameer, i.e. degrees of freedom, and is gamma funcion. Sandard Suden's -disribuion assumes ha,, wih ineger degrees of freedom. However, degrees of freedom can be esimaed as non-ineger, relaing o kurosis: 6 k 3 df. () df From relaion () i's obvious ha sandard -disribuion has heavier ails han normal disribuion when df 3. Hence, if empirical disribuion is more lepokuric esimaed degrees of freedom would be smaller. The second and fourh cenral momen of funcion (9) are defined as: df E E x x df 3 df df df, () wih Fisher's kurosis: 6 k 3. () df Therefore, we may apply mehod of momens and ge consisen esimaors: 6 dfˆ kˆ, (3) ˆ 3 kˆ ˆ 3 kˆ Where he sample variance is biased esimaor of.to ge unbiased esimaor of sandard deviaion we use correcion facor: 3 kˆ 3 kˆ, () which is equivalen o: dfˆ. (5) dfˆ In pracice, he kurosis is ofen larger han six, leading o esimaion of non-ineger degrees of freedom beween four and five. However, kurosis will depend on volailiy persisence. Volailiy persisence is defined in GARCH(,) model. as he sum of parameers If we rearrange condiion variance equaion of GARCH(,) model as follows:, (6) shows he ime which is needed for shocks in volailiy o die ou. If his Then he sum of parameers sum is close o long ime is needed for shocks o die ou. However, if he sum is equal o uniy he covariance saionary condiion is no saisfied and GARCH(,) model follows inegraed GARCH process of order one, i.e. IGARCH(,). If we subsiue v han saionary condiion occurs from ARMA(,) represenaion of GARCH(,) model: v v. (7) 3 Page

4 Algorihmic rading sraegy, based on GARCH (, ) volailiy and volume weighed average price of V. Proposed Algorihm 5. Volailiy Calculaion Volailiy parameers of GARCH(,) like variance covariance marix, Kurosis, probabiliy densiy funcion are calculaed on basis of hisorical daa. We have shown empirical resuls for las years for S&P 5 in Table. 5. Boundary value calculaion Once volailiy parameers are calculaed, hey are fed o he equaion from which boundary value is obained. Boundary values are weighed average price of he financial Insrumen over muliple ime periodsas Volume Weighed Average Price (VWAP) of an asse is a well-esablished benchmark [7]. This boundary value acs as he reference for he decision suppor sysem. 5.3 Decision Suppor Sysem Economic heory frequenly suggess ha economic agens respond no only o he mean, bu also o higher momens of economic random variable [8]. On obaining he boundary value he order is execued by he decision making sysem. If he opening value is greaer han boundary value he conclusion is reached ha he insrumen is overvalued. Hence shor posiion is aken wih selling poin being he difference beween opening value and prediced volailiy. On he oher hand if he opening value is less han he boundary value he conclusion is reached ha he insrumen is undervalued. Hence, long posiion is aken wih he selling poin being he sum of volailiy and opening. I can be noed ha for boh cases he profi is volailiy. 5. Risk Managemen Process for he algorihm The risk managemen process mus also ake place in real ime, alongside wih he algorihm parameer oupus and order managemen. The value of he open posiions and cash available mus be carefully correlaed wih he erms seled when he algorihm sars during any sor of securiy rading sop loss is necessary. In he risk managemen module he curren prices are moniored consanly and checked agains a sop loss value. See figure for Risk Managemen Process. One imporan issue is he one of uning he parameers of algorihm. Even if in pracice here are some sandard values for hem, he performance of he algorihm can be much affeced by a non-opimum parameer value chosen. The main facors ha lead o parameers changing are he volailiy esimaion parameers. Fuzzy logicor neural nework algorihms can also be used o overcome he issue of choosing righ parameers among muliple combinaion ses. The risk managemen process is responsible no only for he loss limiaion bu also for cashing he profi. So i has immense significance for real-ime environmen. Figure. Algorihm Flowchar 33 Page

5 Algorihmic rading sraegy, based on GARCH (, ) volailiy and volume weighed average price of VI. Empirical Resuls: The findings wih he algorihms are presened here. The performance of he S&P 5 has been analyzed. Mone-Carlo simulaion has been performed over random subse of socks. We use daily daa o make forecass for he nex day and have overnigh ime inerval from he close of rading open of he nex day, riskfree raes are used. The ransacion cos incurred is % when we change our posiion. I will be based on omorrow s closing price because we focus on ou-of-sample predicion and we assume ha we place our order o buy or sell immediaely before he close of rading omorrow. Of course we may use omorrow s opening price or high frequency daa in pracice. We believe ha here he sraegy will be more profiable because of more flexibiliy and less delay. The leverage feedback effec has magnified he flucuaion in he marke caused by he exreme evens. For example, afer Lehman Brohers declared is bankrupcy on Sepember h, 8, a series of bank and insurance company failures riggered he global financial crisis in which he marke flucuaes dramaically. I is he exreme even, i.e., he declaraion of Lehman Brohers bankrupcy, ogeher wih he volailiy clusering plus he leverage feedback effec caused by Lehman Brohers bankrupcy news resul in he caasrophic financial crisis in 8 [9].So leverage and ransacion cos canno be negleced. Our empirical resuls ake boh of hese parameers in accoun. Figure 3 shows Cumulaive classic reurn obained via logarihmic reurn for S&P 5 from 99 o.table shows all evaluaion parameers of he es in deail. Figure 3: Cumulaive classic reurn obained via logarihmic reurn for S&P 5 from 99 o 3 Page

6 Algorihmic rading sraegy, based on GARCH (, ) volailiy and volume weighed average price of Figure : Time series of difference beween cumulaive logarihmic reurns of he sraegy and he proxy porfolio wih S&P 5 from 99 o Table : Performance of he sraegy in erms of daily logarihmic reurns for S&P 5 Parameers Value Logarihmic Reurn 95% Classic Reurn 58% Mean Daily Logarihmic reurn.8% Sandard Deviaion.75 Skewness.36 Kurosis.85 VII. Conclusion Andersen and Bollerslev (998) and Chrisodoulakis and Sachell (998, 3) have argued ha he poor forecasing from GARCH models are oo smooh o capure he enire variaion of volailiy. So we have moved one sep forward. We have inroduced he concep of weighed value of asse wih muliple period based boundary condiion along wih volailiy. We have focused on one aspec of GARCH models and heir abiliy o deliver one period ahead forecass of volailiy. We have analyzed hese forecass o acual volailiy calculaed using daily sock reurns. Weighed price and boundary value gives us exac enry poin. In he furher work he inegraion wih a Geneic Algorihm or neural nework wih regression analysis can be a highly desirable soluion in order o une up he parameers of he algorihm in a quick and reliable way, bu also can be used in he discovery of new rading rules in a quick and reliable way. References [] Bollerslev Tim Generalized Auoregressive condiional heeroscedasiciy, Journal of Economerics 3(3), 986, [] Rober F. Engle, The use of ARCH/GARCH Models in Applied Economerics, Journal of Economic Perspecives 5(),, [3] Engle Rober Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica 5(), 98, [] Glosen L Jagannahan R and Runkle D, On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks Journal of Finance 8(5), 993, [5] Senof L American Opion Pricing Using GARCH Models and he Normal Inverse Gaussian Disribuion,Journal of Financial Economerics 6(), 8,5-58. [6] Carnero M, A Penam, D Riuz E, Persisence and Kurosis in GARCH and Sohasic Volailiy Models, Journal of Financial Economerics Oxford Universiy Press, (),, [7] Madhavan, A. VWAP Sraegies, Insiuional Invesor Guides: Trading Insiuional Invesor Inc.,, 3-9. [8] Hall P Yao Q Inference in ARCH and GARCH Models wih Heavy-Tailed Errors, Economerica 7(), 3, [9] Lie-Jane Kao, Po-Cheng Wu and Cheng-Few Lee, A Time-changed NGARCH model on he leverage and volailiy clusering effecs by exreme evens: Evidence from he S&P 5 index over he 8 financial crisis, Proc 3rd Annual Financial Engineering, and Risk Managemen Conference Naional Chiao Tung Universiy. 35 Page