An Improved Composite Forecast For Realized Volatility

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1 Joural of Statistical ad Ecoometric Methods, vol.3, o.1, 2014, ISSN: (prit), (olie) Sciepress Ltd, 2014 A Improved Composite Forecast For Realized Volatility Isaac J. Faber 1 ad Kelsey Eargle 2 Abstract The purpose of this study is to determie whether a superior forecast for security volatility ca be derived by fidig a balace betwee historical data, implied volatility ad a empirical implied distributio. Data are evaluated from optio cotracts ad historical prices sampled o the first tradig day of every moth over a five year period from 2007 to These data are aalyzed to determie the value of a weighted combiatio of the three sources of iformatio ad to ucover if this approach provides a forecast with a higher correlatio to realized volatility. A liear optimizatio solutio is formulated to determie the best possible composite volatility forecast. The results of the test show that there is statistically sigificat evidece i which the composite volatility forecast is preferred at a 95% cofidece level over idividual forecasts. With a better predictor for security volatility, this optimizatio process could be applied to the creatio of portfolios that better meet ivestor risk preferece. JEL Codes: C12, C14, C58, G17 Keywords: Implied Volatility, Empirical Implied Distributio, Realized Volatility 1 Departmet of Systems Egieerig, Uited States Military Academy, Isaac.faber@us.army.mil 2 Departmet of Systems Egieerig, Uited States Military Academy, Kelsey.eargle@us.army.mil Article Ifo: Received : December 13, Revised : Jauary 21, Published olie : February 7, 2014.

2 76 A Improved Composite Forecast for Realized Volatility 1 Itroductio Moder techical aalysis forecasts of volatility are achieved both by leveragig historical data (statistical methods) ad forward-lookig data (implied volatility from derivative prices). I this study these two sources combied with a ovel approached called a empirical implied distributio (EID) are used i order to create a composite forecast. The hypothesis is that a certai combiatio of volatility predictios will create a improved forecast. Liear optimizatio will be used to determie this composite value ad ucover which of the three forecast cotributes most (or least) to better predictios. The forecast will be back tested for accuracy usig a stadard hypothesis test of the correlatio betwee predictios ad the realized volatility (i.e. what the model predicts ad what actually happes). This area of fiacial egieerig is relatively uexplored to date; leavig the potetial for valuable iformatio towards curret methods of risk maagemet ad related future work. The assets used i this study cosists of four Exchage Traded Fuds (ETFs) icludig the SPDR S&P 500 (SPY), the ishares S&P SmallCap 600 Idex Fud (IJR), the Uited States Oil Idex Fud (USO) ad the SPDR Gold Trust (GLD). These four were chose as they represet a large portio of the market ad will assist i verifyig the results of this study. 2 Backgroud 2.1 Forecastig Volatility Usig Forward Lookig Market Data The primary method techical ivestors use to predict how the markets perform i the future is by aalyzig the past performace of a fiacial asset. By observig the chages i prices ad volume of stocks traded, techical aalysts ofte assume that the log adjusted returs of a security are ormally distributed ad that the expected retur is iterpreted as the mea, symbolized by [1]. The risk of a asset is expressed the stadard deviatio of the returs about the mea, symbolized by (referred to as volatility). These two parameters, alog with the assumptio that the returs follow a radom walk described by Browia Motio (or Weier Process), gave techical aalysts a framework to evaluate the performace of a particular asset or group of assets. The classical approach of asset evaluatio is to estimate these two parameters usig stadard statistical methods o historical data. However, other methods for geeratig forecasts of these parameters have become popular. A example is that volatility ca also be forecast by aalyzig the tradig prices for optios cotracts. I 1973, Fischer Black ad Myro Scholes developed a iovative model for pricig a Europea-style call optio. The popular calculatio for determiig volatility is derived from this model kow as the Black-Scholes (B-S) equatio:

3 Isaac J. Faber ad Kelsey Eargle 77 CSt (,) = Nd ( ) S Nd ( ) Ke + 2 S l( ) + ( r± )( T t) d±= K 2 T t rt ( t) The equatio has five parameters, which iclude the strike price (K), the uderlyig price at time t (S), the risk free rate (r), the cotract time to maturity (T-t), ad the volatility of the uderlyig asset () [2]. Equatio 1 ca oly evaluate the price for a call optio (C). I order to price a put optio the use of put-call parity ca used with arbitrage assumptios as show i equatio (2). (- rtt ( - )) PSt (,) = Ke S+ CSt (,) (2) Oe of the reasos this equatio is used as a basis for estimatio is that all of the parameters except the volatility of the uderlyig asset ca be directly observed. So give a market price you ca back out the volatility value ad iterpret what the price is implyig about it (volatility), hece implied volatility (IV). These equatios have flaws i reality as they fail to accout for sigificat chages i the market such as the crashes experieced i 1984 ad As such actual market prices deviate from those predicted by the model. Oe primary reaso for these iaccuracies stems from ormality assumptio that favors returs closer to the mea istead of the tails of the distributio. The B-S equatio assumes that the variace remais costat across all strike prices i a give chai. Therefore, it would be expected for the resultig implied volatilities to form a straight lie whe plotted, i reality, they form a curved lie kow as the volatility smile [4]. However some gais have bee made by attemptig to adjust for this skew with more complex models ad assumptios [5]. Despite these facts, the model ad its derivative work provide useful iformatio for predictive purposes i may scearios. Because implied volatility is so readily iterpreted from a market price; simply solvig for this loe parameter yields a predictio. That is to say that a give price implies a future volatility as expressed through B-S. However ucoverig IV is ot trivial, there is o closed from solutio for solvig this parameter so estimatio methods must be employed. For example, oe of the most commo approaches is to apply the Newto-Raphso method (a recursio approach) for each strike price i a optios chai [3]. (1) 2.2 Empirical Implied Distributios A empirical implied distributio (EID) is a iterpretatio of the price premiums of a optios chai for a give time to expiratio. It is calculated by lookig at the price dyamics of out of the moey premiums withi a give chai. It is importat to poit out that i this cotext the EID is ot a formal probability

4 78 A Improved Composite Forecast for Realized Volatility distributio but a rage of premiums iterpreted by a potetial realizatio of retur (if the price reaches a give strike). Implied probability distributio literature is importat, for example, Shiratsuka [6] determied that a implied distributio does cotai iformatio useful i creatig forecasts for future price movemets, however, the accuracy of these predictios rely heavily o choice of sample period ad are ot as powerful as the historical data. A excellet approach to developig implied distributios is give by Rubistei ad Jackwerth (1996). The focus i these works is to ucover probability (hece a formal probability distributio) where as a EID is oly cocered with price dyamics. Empirical implied distributio volatility (E), ca be extracted from the optios premiums by usig equatios (3), (4) ad (5). Ki (l )( Ci ) i= 1 St () µ = (3) C i= 1 K i 2 (l µ ) ( Ci ) 2 i= 1 St () = C i= 1 i i= 1 = Ki (l )( Ci ) St () C As is obvious from the above expressios the EID is costructed by stadard statistical first ad secod momet estimatios. A example of how these distributios are costructed is detailed i the followig sectio. i= 1 i i (4) (5) 3 Methodology The example for this sectio will use the SPDR S&P 500 (SPY) which is a ETF desiged to track the performace of the S&P 500. The first tradig day of each moth was used as a referece poit for the followig 30 caledar days. All of the returs were calculated as the logarithmically adjusted rate of the close price o the respective dates. Fially, all of the estimates ad realized volatility were aualized so that they could be compared for aalysis. First, a historical forecast for the volatility of SPY was made usig a 24-moth movig average. A 24-moth warm up period was icluded i the data set to allow for a statistical baselie. The square root of this variace was used as for the predicted stadard deviatio of the returs usig historical data. For forward lookig volatility estimates, oly out-of-the-moey optios (both calls ad puts) were used because they are more liquid ad also result i a uimodal

5 Isaac J. Faber ad Kelsey Eargle 79 distributio of prices. Ay optios that displayed arbitrage opportuities were elimiated. Midpoit prices where used ad optios chais where trucated after two sequetial idetical prices less tha $.05 where observed. The implied volatility estimatio was gaied through the use of the Newto-Raphso method o the B-S model averagig the first four out of the moey values for calls ad puts for the first tradig day of each moth. Sice the first tradig day of each moth does ot always occur exactly 30 days later, the E ad IV were iterpolated as a 30-day average. The two optios chais to be used (ear ad ext) are the two chais expirig betwee 0-30 days ad days, respectively. The followig calculatio was used: 30 TTM ear Var = Varear + ( Varext Varear )( ) TTM TTM where TTM represets the time to maturity i days. A demostratio of this calculatio will use data from December 1st 2010 (Figure 1). O this particular day, the variace of the ear edig optio chai was ad was set to expire i 29 days while the variace for the ext optios chai was ad set to expire i 61 days. After aualizig the calculated variace, the result was Figure 1 shows the price distributio used for oe of the data poits used i the estimatio with puts beig represeted with returs lower tha zero ad calls with returs higher. As expected, the distributio has a heavy, egative skew towards puts as recet ivestors geerally value isurace o their holdigs over the potetial gais from calls. ext ear (6) Figure 1: SPY Implied Distributio (DEC 1st 2010) Fially, the realized volatility was calculated by aalyzig daily data rather tha mothly data i order to calculate the actual stadard deviatio of the ext 30 (caledar) days so that it could be used to compare with the other two estimates.

6 80 A Improved Composite Forecast for Realized Volatility For example, the realized volatility for March 2009 was calculated as the stadard deviatio of the returs from the 21 tradig days i March. Fially, the implied volatility, empirical implied distributio volatility, historical data, ad realized volatility were plotted o the Figure 2 for observatio ad compariso. As show by the Figure 2 above, we ca see the how both the implied volatility ad the empirical implied distributio are much more accurate at predictig the realized volatility tha the historical data usig the 24-moth movig average. Despite this observatio, it is importat to ote that historical data has bee foud to be a good predictor of future performace [7]. The 24-moth movig average has a large smoothig effect due to its relatively log time period ad oweighted characteristics. The time frame of this study also ecompasses the massive fiacial crisis the U.S. experieced i late Figure 2: SPY Volatility Chart Comparig the Three Separate Estimates vs. Realized Volatility. The coefficiet of correlatio, r, was calculated to measure the liear associatio betwee the realized volatility ad each estimate as see i Table 1. The ext step was to coduct a hypothesis test for correlatio to determie if the results were statistically sigificat. The hypotheses ad test statistic were formulated as follows. H0 : r = 0 H : r 1 1 r 2 Tstat = 2 1 r (7)

7 Isaac J. Faber ad Kelsey Eargle 81 Table 1: Hypothesis Test for Correlatio Historical Data IV E r t T stat Results FTR Reject Reject The value for t was obtaied with a α =.05 ad 60 degrees of freedom. The degrees of freedom varied from security to security as the umber of data poits chaged, but the α value remaied costat. While all three correlatios were positive, the implied volatility ad empirical implied distributio correlatios were much stroger. The test statistic for the implied volatility ad implied distributio were much greater tha the t value meaig that we should reject the ull hypothesis with statistical sigificace. The ext step was to costruct a liear optimizatio model to predict for the ideal weighted forecast betwee the three estimates for a composite estimate of the three approaches. A composite volatility estimate,, was used as a ew parametric defied by equatio (8) so log as the weights (w i ) met the costrait i equatio (9). = w IV IV + w + w HV H V (8) s.t.: w IV + w + w HV = 1 (9) I equatios (8) ad (9), the weights of each respective estimate are varied whe lookig for the optimal combiatio. Similarly, the same weights ca be used to determie the mea of the composite volatility estimate. µ = w IV µ IV + w µ + w HV µ HV (10) The ext step was to calculate the stadard deviatios for the realized volatility ad the composite volatility estimate. 2 ( x µ ) real = (11) 1 = w + w + w H (12) IV IV HV V The objective fuctio was set to maximize the correlatio betwee the realized volatility ad the ewly created. Max : ρ real, real, real, where = E( real, ) + µ µ = real real (13)

8 82 A Improved Composite Forecast for Realized Volatility Figure 3: The Composite Volatility Estimate vs. Realized Volatility. Table 2: Hypothesis Test for the Composite Volatility Estimate for SPY Value r T T stat Results FTR Figure 3 depicts the composite volatility estimate compared to the realized volatility for SPY. Simple observatio shows that the composite volatility estimate teds to overestimate the realized volatility. This is true at every aalyzed time frame except for three occasios, two beig times whe the stock market was goig through the crash i 2008 ad subsequet effects experieced i late Oe of the most promisig coclusios that ca be draw from this graph is that the composite volatility estimate should make for a good forecastig parametric of future volatility. The peaks ad valleys of the composite volatility closely mirror the realized volatility across the observed data. Further aalysis should be doe to determie the predictive capabilities of this approach across various asset domais. Additioally, the correlatio betwee the two was over 81% ad had a test statistic of as show i Table 2, both strog values. 4 Results ad Coclusio The same process explaied i the previous sectio was repeated for three additioal securities to iclude IJR, GLD, ad USO. The compiled summary ca

Isaac J. Faber ad Kelsey Eargle 83 be see i Figure 4 ad Table 3. Figure 4: Stacked Bar Chart of the Resultat Weights Figure 4 provides isight as to what average weight each ETF forecast cosisted of.

9 Isaac J. Faber ad Kelsey Eargle 83 be see i Figure 4 ad Table 3. Figure 4: Stacked Bar Chart of the Resultat Weights Figure 4 provides isight as to what average weight each ETF forecast cosisted of. Values could be egative because the weight simply is the coefficiet by which the origial respective estimate is multiplied by as equatio (8) showed. SPY preferred the implied volatility ad empirical implied distributio while it discouted the historical data. IJR, o the other had, did ot discout ay of the estimates usig roughly 50% historical data, 30% implied volatility ad 20% implied distributio volatility. The GLD ETF highly favored the implied volatility estimatio. Fially, USO utilized roughly 83% of the implied volatility ad 36% historical data i exchage for subtractig about 18% of the empirical implied distributio estimate. Table 3: Summary Correlatios ad Improvemet. SPY IJR GLD USO Previous Best Correlatio Estimate IV IV IV Correlatio Improvemet 9.70% 3.23% 4.14% 5.38% It was foud that the correlatio ca be improved sigificatly by usig this composite forecast method. All four ETFs experieced a icrease i correlatio after the methodology was applied. O average, this method improved the correlatio 5.61%. At a 95% cofidece level, a paired t test shows that these improvemets are statistically sigificat. This cofirms our origial hypothesis that certai combiatios of volatility estimatios ca be foud through the use of liear optimizatio to create a improved estimate for the volatility of a security s

10 84 A Improved Composite Forecast for Realized Volatility returs. The composite volatility estimate represets a statistically superior estimate. More accurate volatility forecasts will allow ivestors, particularly risk ad portfolio maagers, to make better ivestmets decisios. Refereces [1] C. Kirkpatrick ad J. Dahlquist, Techical Aalysis: The Complete Resource for Fiacial Market Techicias, Fiacial Times Press, 3, (2006). [2] F. Black ad M. Scholes,The Pricig of Optios ad Corporate Liabilities, Joural of Political Ecoomy, 81, (1973), [3] I. Faber, The Iformatioal Cotet of Optios Prices, Uiversity of Washigto, [4] G. Jiag ad Y. Tia, The Model-Free Implied Volatility ad Its Iformatio Cotet, The Society for Fiacial Studies, 18, (2012), [5] T. Aderso ad O. Bodareko, Costructio ad Iterpretatio of Model-Free Implied Volatility, Uiversity of Copehage, [6] S. Shiratsuka, Iformatio Cotet of Implied Distributios: Empirical Studies of Japaese Stock Price Idex Optios, Moetary ad Ecoomic Studies, (2001), [7] Joseph Piotroski, Value Ivestig: The Use of Historical Fiacial Statemet Iformatio to Separate Wiers from Losers, Joural of Accoutig Research, 38, (2000), [8] S. Beiga, Fiacial Modelig, MIT Press, Cambridge, Massachusetts, , [9] CBOE (2003), VIX white paper, dowloaded 9 Ja [10] M. Capiski ad T. Zastawiak, Mathematics for Fiace, Spriger, Lodo, [11] M. Rubistei ad J.C. Jackwerth, Recoverig Probability Distributios from Optio Prices, Joural of Fiace, 51, (1996),

Statistics for Economics & Business

Statistics for Economics & Business Statistics for Ecoomics & Busiess Cofidece Iterval Estimatio Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio How to determie