Option trading for optimizing volatility forecasting

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1 Journal of Saisical and Economeric Mehods, vol.6, no.3, 7, ISSN: (prin), (online) Scienpress Ld, 7 Opion rading for opimizing volailiy forecasing Vasilios Sogiakas Absrac This paper invesigaes he forecasing abiliy of several volailiy specificaions ha aim o quanify marke risk. Using an opions rading sraegy on volailiy he comparison is implemened in a dynamic approach, applying he sandardized predicion error crierion. The empirical findings of he paper sugges ha he SPEC crierion ouperforms all volailiy models ha assume normaliy on he daa and exhibis similar forecasing abiliy wih mos of he models ha assume skewed disribuions of asse reurns. JEL classificaion numbers: G, G3, G7 Keywords: SPEC; opion rading; sraddle; marke risk; volailiy forecasing; Black-Scholes. Inroducion Forecasing ime series daa and volailiy has is one of he cornersones for finance and for ha reason has araced he ineres of researchers. While mos of he research focuses on he deerminaion of models based on he minimizaion of he mean squared error or he maximizaion of he likelihood, his paper focuses Universiy of Glasgow, Adam Smih Business School, UK Aricle Info: Received : May 6, 7. Revised : July, 7 Published online : Sepember 5, 7

2 66 Opion rading for opimizing volailiy forecasing on he maximizaion of he profiabiliy of a hypoheical invesor. This is moivaed srongly by he work of Engle (993) who applied a sraddle echnique wih opions ha expire in one day and for which he payoff is direcly linked wih he volailiy of he underlying asse. In ha framework he maximizaion of a hypoheical invesor s payoff would indicae an opimal volailiy forecasing model. The aim of his paper is he invesigaion of he opimum algorihm for choosing models, in erms of predicabiliy power, for quanifying marke risk. For ha reason I examine he payoff of several volailiy specificaions ha aim o forecas he payoff of a sraddle opions rading sraegy. Among he volailiy specificaion, which could be assumed as passive rading sraegies on volailiy, I also apply a dynamic approach ha dicaes a dynamic acive rading approach according o which he volailiy specificaion which is applied a each week is allowed o be differen. The main findings of he paper are in favour of an acive rading sraegy according o which invesors benefi from choosing alernaive volailiy models in each period. This dynamic approach resuls in a profiable payoff for hese invesors who obain similar rewards wih hose invesors who base heir expecaions on skewed disribuions and complicaed volailiy specificaions. The second chaper of he paper explains he simulaion process of he necessary daa, while he hird one he mehodology. The fourh chaper discusses he resuls and finally he fifh concludes he paper. Daa For he purposes of he analysis of he paper wo opions are available; eiher o use real raded daa or simulaed daa as was he case for Engle (993) and

3 Vasilios Sogiakas 67 Degiannakis and Xekalaki (5). Engle (993) focused on he NYSE sock index simulaing he corresponding opion prices and applying many compeiive models such as he MA variance in he squared residuals in order o forecas daily forecass of volailiy. While he former case (real daa) offers a more realisic approach for developing forecasing crieria, he laer is more preferable as i overcomes several issues such as he auocorrelaion of opion prices, he bid-ask spread and he non-synchronous rading. The opion prices of he Ahens Derivaives Exchange exhibi auocorrelaions because raders have he opporuniy o rade opions a heir closing price for a shor period of ime afer he closing ime of he Derivaives Exchange each day. Anoher limiaion for using real daa is ha he execuion of he opions a he Ahens Derivaives Exchange akes place every hird Friday of each monh, resuling in a single observaion over a period of days for opions wih mauriy erm of one day. This consequenly is very likely o rise many economeric quesions when applying GARCH models; see among ohers Dros and Nijman (99). The daa consiss of daily spo prices for he ATHEX/FTSE- during he period form 3// o 3/6/5. The simulaed opion prices are derived from he Black-Scholes (973) model assuming normaliy on our daa: r C S d K e d () r P S d K e d () where d S ln r K and d d

4 68 Opion rading for opimizing volailiy forecasing S : Spo Prices a, Κ: exercise price, σ: variance of he underlying asse, r: he risk free ineres rae (Euribor), and τ: ime o mauriy. From he descripive analysis of Figure of he appendix, i seems ha he Jarqua-Bera saisic cass doub on he normaliy assumpion mainly because of he lepokuric and skewed disribuion of asse reurns. 3 Mehodology The fuure call and pu opion price a +, condiional on he available informaional se up o, wih τ days o expiraion (τ=t-) is denoed wih C and P equaions: and can be quanified according o he B-S formula by he following r C S d K e d (3) r P S d K e d (4) where d S ln r K S ln r K d d is he sandard deviaion during + o T given he informaional se up o.

5 Vasilios Sogiakas 69 Each passive sraegy or agen adops a forecas mehod for variance and rades opions for a unique day invesing on ATHEX/FTSE-. Assuming ha he opion is a-he-money and ha he exercise price is equal o S *exp(r ) hen he price for for an opion ha expires in a day (τ=) would become: d and d (5) r r C S S e e S S S (6) r r P S S e e S (7) Then a expiraion, he long sraddle posiion ha is he volailiy rader would obain a payoff equal o T S K (8) T Consequenly, he payoff of he sraddle long posiion a any would be equal o: r a he money S e (9) The ransacion beween wo agens (i and i*) is execued a he median bid/ask prices according o he following payoff: * ii C C *, for C,, C i *, i i, i C, C *, i, *, i for C C i, i ()

6 7 Opion rading for opimizing volailiy forecasing Hence, whenever he i agen overesimaes he fuure volailiy compared o agen i*, hen his agen (i) would overprice he opion resuling a a long posiion rading posiion expecing a profi equal o: r profi C i C S e * C i i C,, *,, i () This equaion a expiraion would become: r T profit T ST e CT T, i C * T T, i () For he volailiy forecasing process I use he GARCH family models wih several exensions in order o accoun for he asymmeries on boh he volailiy specificaion and he disribuion of he ime series as shown in Table of he appendix: m m m z h and m m m m m m m h g h,..., h p, y y i,..., yq y i (3) where he innovaions ε represen he filered ime series: y y c c y c y (4)... k The one-sep ahead predicion error is given below: or y c, c, y y y (5) Adoping a GARCH(,) model, he one-sep ahead condiional variance forecas equals:

7 Vasilios Sogiakas 7 h Eh,, a a E, E h (6) which a expiraion urns o: ht T Eh T T a a E T E ht (7) For increasing he accuracy of he volailiy forecas I adop also he APARCH(,) model using eiher symmeric disribuions such as he Normal, he -suden and he GED or he skewed disribuion of Gio and Lauren () according o he following equaions: q p i i i j j i j (8) h a y y b h f n n y y s m s h y y m, n h n s n h y / n, n n y y s m s h y y m, n h n s n h (9) where n n m n and s m

8 7 Opion rading for opimizing volailiy forecasing Following Engle (993) we adop he sandardized predicion error crierion (SPEC) for choosing models. SPEC is defined as he sum of he squared deviaions of forecased and observed reurns as shown below: SPEC z i burn i i i h burn i q q m m m burn i q burn i i burn i : : () m where burn: is he sample which is used for making in-sample forecass, q: is he number of one-sep-ahead in-sample forecass ha are used when applying he SPEC. As q decreases, hen he invesor s flexibiliy for using differen models is increased. A (=burn+k*q) according o SPEC he m h model will be chosen ha minimized he sum of he squared sandardized one-sep-ahead predicion errors. 4 Empirical Findings The implemenaion of he sraddle rading sraegy wih mauriy of one day is presened in Figure of he Appendix. There are 3 agencies and he payoffs are esimaed for 74 weeks. According o his figure i is shown ha he firs en agencies exhibi he higher payoffs. The SPEC crierion is applied for he examined series and i is represened in Figure 3 of he appendix. According o SPEC i is obvious ha he res agencies (-3) are hose ha reward invesors wih he minimum predicion error. 5 Conclusion This paper in srongly moivaed by he work of Engle (993) and invesigaes a model choice process ha should be adoped when he predicive power maers

9 Vasilios Sogiakas 73 insead of he model fi per se. I applied he afore-menioned crierion using daa from a relaively small Sock Exchange, he Ahens Sock Exchange. The imporance of his choice is ha during he examined ime period he Greek economy experienced a major developmen because of he Ahens 4 Olympic Games ha were accommodaed a Greece, having araced many invesors during he firs half of he firs decade of his cenury. According o Figure 4, acive rading on volailiy which is he implicaion of adoping he SPEC crierion seems o be an imporan rule for maximizing he predicive power and hus he profiabiliy of invesors. This rule ouperformed passive volailiy rading sraegies ha are based on symmeric models, while i rewarded invesors wih high profiabiliy similar o ha of more advanced economeric approaches ha accoun for asymmeries in he volailiy and he reurns disribuions. References [] Akaike Η. (973). Informaion Theory and he Exension of he Maximum Likelihood Principle. Proceeding of he Second Inernaional Symposium on Informaion Theory, Budapes, [] Black, Fisher and Scholes Myron, 973, The pricing of opions and corporae liabiliies, Journal of Poliical Economy 8, [3] Bollerslev T., Engle R. F. and Nelson D. (994). ARCH Models. Ch. 49 in R.F. Engle and D. L. McFadden eds. Handbook of Economerics, IV Elsevier. [4] Campbell J., Lo A. and MacKinlay A. (997). The Economerics of Financial Markes. Princeon Universiy Press, Princeon. [5] Engle R.F: ARCH wih Esimaes of he Variance of he U.K. Ιnflaion. Economerica, Vol.5, No. 4 (July, 98)

10 74 Opion rading for opimizing volailiy forecasing [6] Engle R. F., Hong C. H., Kane A., Noh J., 993. Arbirage valuaion of variance forecass wih simulaed opions. Adv. Fuures Opions Res. 6, [7] Lamber P. and Lauren S. (). Modelling Skewness Dynamics in Series of Financial Daa. Discussion Paper, Insiu de Saisique, Louvain-la- Neuve. [8] Xekalaki E., Panareos J. and Psarakis S. (3). A Predicive Model Evaluaion and Selecion Approach The Correlaed Gamma Raio Disribuion. Sochasic Musings: Perspecives from he Pioneers of he Lae h Cenury, (J. Panareos, ed.), Lawrence Erlbaum Associaes, 88-. [9] Xekalaki E. and Degiannakis S., 5. Evaluaing volailiy forecass in opion pricing in he conex of a simulaed opions marke. Compuaional Saisics & Daa Analysis 49 (5) 6-69.

11 Vasilios Sogiakas 75 Appendix Lis of Tables Table. The definiion of he applied volailiy models. MODELS Model : Model : Model 3: Model 4: Model 5: Model 6: Model 7: Model 8: Model 9: Model : Model : Model : Model 3: Model 4: Model 5: Model 6: Model 7: Model 8: Model 9: Model : Model : Model : Model 3: GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_Normal GARCH(,)_AR()_T-Suden GARCH(,)_AR()_T-Suden AP-GARCH(,)_AR()_T-Suden AP-GARCH(,)_AR()_T-Suden AP-GARCH(,)_AR()_T-Suden GARCH(,)_AR()_T-Suden GARCH(,)_AR()_T-Suden GARCH(,)_AR()_GED GARCH(,)_AR()_Skewed-T-Suden GARCH(,)_AR()_Skewed-T-Suden AP-GARCH(,)_AR()_Skewed-T-Suden

12 76 Opion rading for opimizing volailiy forecasing Lis of Figures Series: RETURNS Sample /4/ 4/4/5 Observaions 37 Mean -.43 Median Maximum.8686 Minimum Sd. Dev..494 Skewness Kurosis Jarque-Bera 3.5 Probabiliy. Figure. Hisogram of he ATHEX/FTSE- reurns Cummulaive Profiabiliy per week of each Agen during he 74 weeks period Cummulaive Profiabiliy per week weeks (5 days) agens Figure. Cumulaive weekly payoff for he differen volailiy forecasing approaches.

13 Vasilios Sogiakas 77 SPEC of each Agen in a week basis period, during he ime horizon of 74 weeks SPEC weeks (5 days) 5 3 agens Figure 3. SPEC crierion over he whole period for each volailiy model. Cummulaive Annual Profiabiliy of he 3 Agencies and he SPEC Agency 5 Annual Cummulaive Profiabiliy SPEC Agency Figure 4. Cummulaive annual profiabiliy of he differen volailiy specificaions.