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## THE BEST VOLATILITY FORECAST THAT WE CAN EXPECT TO HAVE

What are the limits that we can expect for volatility forecasts?  This is a simple issue that needs some comments because when we are measuring volatility forecast accuracy, we might be tempted to imagine some arbitrarily small errors but there is a simple limit:  the dispersion of the future realized volatility is out of our control.  So I am considering a historical forecast performance on all daily options for EMC in the period 2008-2013 across all maturities and strikes and consider the aggregate forecasting performance against the actual realized volatility till maturity for each option.  The error for implied volatility is $| \sigma_{iv} - \sigma_{act}| = 0.0168 \pm 0.025$ and that for our prediction is $|\sigma_{mod} - \sigma_{act}| = 0.025 \pm 0.28$.  The performance is worse than we would hope for but what should we hope for?  The dispersion of $\sigma_{act}$ is 0.014 so this is the best we can realistically expect for error in this case.

Now if we do a t-test of $\sigma_{iv}$ versus $\sigma_{act}$ we find a t-stat of 105 (p-val 0), while the corresponding t-stat for $\sigma_{mod}$ versus $\sigma_{act}$ is 11 (p-val close to zero) while a t-test of IV versus Mod has no distinguishability (t-stat  0.5, pval 0.61).  By this measure in aggregate there is no clear superiority of IV versus model.

When we instead consider the distributions of $| \sigma_{act} - \sigma_{iv}|$ and $|\sigma_{act} - \sigma_{mod}|$ then there is a t-stat of -11 and a significant difference, so iv is better than the model at prediction; this is because the dispersion of the $\sigma_{mod}$ at 0.28 is much higher than that of the implied volatility although the means are quite close 0.0288 versus 0.0257.  The point of this note is that we should not expect that a better forecasting produce a figure smaller than 0.014 with any significance, the dispersion of the actual realized volatility.  This is an extremely elementary issue but worth remembering when deep in the midst of attempting to find better volatility forecasts: the elementary limits of volatility forecasting is beyond our control.  Of course in this case we pooled together all the options so the dispersion is wider but the same principle provides bounds for forecast for each maturity.

A closer inspection dividing up all the forecasts by maturity and considering the mean iv and model error shows us that the mean error of implied volatility is generally lower than that of the predictions but for some maturities, specifically 52-177 days.

(MaturityDays, SigmaIV-SigmaAct, SigmaPred-SigmaAct, (+++ Pred better), FutureVariation)

(2, 0.024172310448901786, 0.15972749938652375, ‘—‘, 2.1684043449710089e-19)
(3, 0.070056596835876353, 0.11763854215921844, ‘—‘, 0.0018238420721619073)
(4, 0.064145483341686196, 0.10504128456093891, ‘—‘, 0.00190061746213314)
(5, 0.052304876897702843, 0.089090800800922076, ‘—‘, 0.0019495714783539231)
(7, 0.023199120715727505, 0.078486261316037906, ‘—‘, 6.5052130349130266e-19)
(8, 0.042268562766518046, 0.068226486918324364, ‘—‘, 0.0025188651778728969)
(9, 0.041769989550517132, 0.061109410821761141, ‘—‘, 0.002335036139523035)
(10, 0.038663767204308253, 0.056352826918731259, ‘—‘, 0.0023614924501384475)
(11, 0.038773693656503426, 0.052722395438021881, ‘—‘, 0.0023076930003403319)
(12, 0.038490014338616665, 0.051481307930143667, ‘—‘, 0.0025287860982503383)
(15, 0.032312466178189586, 0.051991653590845045, ‘—‘, 0.002820641197926385)
(16, 0.03146954730241857, 0.0411357753387777, ‘—‘, 0.0028472654401452643)
(17, 0.029082839170770723, 0.040216640927696799, ‘—‘, 0.0028530378440195666)
(18, 0.027012829619652311, 0.037908281188094105, ‘—‘, 0.0027248860635238365)
(19, 0.027148980258722352, 0.038353934551419125, ‘—‘, 0.0027434249292434551)
(22, 0.023965843950084081, 0.033646230863660337, ‘—‘, 0.0030495504646409874)
(23, 0.022500165965920547, 0.030579414080095031, ‘—‘, 0.0032735592065773079)
(24, 0.024356247327641044, 0.032247575544433402, ‘—‘, 0.0034268921085080116)
(25, 0.024178082978761197, 0.031600146991541581, ‘—‘, 0.003518457005196545)
(26, 0.02282462689288782, 0.029615820386211047, ‘—‘, 0.0035306972938309428)
(29, 0.021758566585867323, 0.027733011886957218, ‘—‘, 0.0038404425088714943)
(30, 0.022254985785000989, 0.027821327289769132, ‘—‘, 0.0038695359428596436)
(31, 0.021777911147101092, 0.025742890336100548, ‘—‘, 0.0039408022079992396)
(32, 0.021094873397903297, 0.025497150362846003, ‘—‘, 0.003877674263962703)
(33, 0.019846512335427571, 0.022868790336281582, ‘—‘, 0.0038103064894774896)
(36, 0.019120792200925878, 0.023458043874288646, ‘—‘, 0.0039810055065764493)
(37, 0.01910748457839713, 0.022878584135864605, ‘—‘, 0.0039764668476124053)
(38, 0.019636716659231293, 0.022591935378831484, ‘—‘, 0.0038854695214154771)
(39, 0.019236414771224335, 0.022107818628671061, ‘—‘, 0.0038829645291345851)
(40, 0.019268217193092917, 0.021439022126745123, ‘—‘, 0.003991282417703408)
(43, 0.017333437778307894, 0.020175684773905047, ‘—‘, 0.0041339965816982486)
(44, 0.01656104781647743, 0.018224776079915253, ‘—‘, 0.0037996458000323701)
(45, 0.017633351612872602, 0.019244827901725055, ‘—‘, 0.004181921182133984)
(46, 0.017650340044447155, 0.018619829313119299, ‘—‘, 0.0042080850692840337)
(47, 0.017601731471907349, 0.01992017309862355, ‘—‘, 0.0043129809929092627)
(50, 0.015880331481152182, 0.019033492616964225, ‘—‘, 0.0043787551295468453)
(51, 0.015252784944541796, 0.015601016052715274, ‘—‘, 0.0045828653201411057)
(52, 0.015654554788499933, 0.01542064993013403, ‘+++’, 0.0047478654044919534)
(53, 0.016682451534300792, 0.015012793313993231, ‘+++’, 0.0047576254254881982)
(54, 0.015311341421396481, 0.014910317441982981, ‘+++’, 0.0049252479818798394)
(57, 0.015604113962084646, 0.014453577415390519, ‘+++’, 0.0049559177747770739)
(58, 0.014223907376730269, 0.014503833308739661, ‘—‘, 0.0050075876021050325)
(59, 0.014106491441364688, 0.013756272643951928, ‘+++’, 0.0051742681311972138)
(60, 0.014959041309084233, 0.01315688922463731, ‘+++’, 0.0052062759091143305)
(61, 0.013285841962135959, 0.011357311278186859, ‘+++’, 0.0050139491873453252)
(64, 0.015958142508475874, 0.01325950058050359, ‘+++’, 0.0046987652127190652)
(65, 0.016004317316661125, 0.012802378768107704, ‘+++’, 0.0049639760034975349)
(66, 0.016383000822810943, 0.011534343267058692, ‘+++’, 0.0049115119462546011)
(67, 0.014890034078678204, 0.011365140314703395, ‘+++’, 0.0048926934267266957)
(68, 0.015314792770427606, 0.011058847633656568, ‘+++’, 0.0049721427570954156)
(71, 0.013773595357462037, 0.011673497729328398, ‘+++’, 0.0050820505657128544)
(72, 0.014405051416367019, 0.010287587156247571, ‘+++’, 0.0046571800831287724)
(73, 0.013597315470104379, 0.011581924929250536, ‘+++’, 0.0047058213756686878)
(74, 0.012498842461723865, 0.011239012010066565, ‘+++’, 0.0047490146006070079)
(75, 0.012721200808504765, 0.010999309905648846, ‘+++’, 0.0047504218841780566)
(78, 0.013577352459064957, 0.011785977089110384, ‘+++’, 0.0045984903295322385)
(79, 0.01307753806473559, 0.0099917383980444845, ‘+++’, 0.0045808572774368519)
(80, 0.012483798976589168, 0.0097489918495005153, ‘+++’, 0.0062493902219653485)
(81, 0.013206788170171534, 0.0095256318461472787, ‘+++’, 0.0062009529105988039)
(82, 0.012498477811402911, 0.0099485732914771381, ‘+++’, 0.0062023660047013825)
(85, 0.012109076850390333, 0.0089233358634807202, ‘+++’, 0.0065007136577899622)
(86, 0.011298216097772731, 0.009429175610557004, ‘+++’, 0.0064899225902399667)
(87, 0.012317759080942297, 0.0093516591312403727, ‘+++’, 0.0065164172588171769)
(88, 0.011540621970343015, 0.0096717202676735083, ‘+++’, 0.0066390560978534857)
(89, 0.010777689751186313, 0.0075100833262664295, ‘+++’, 0.0067653580375781784)
(92, 0.011518803230438697, 0.0087447300455133197, ‘+++’, 0.0068108459239094943)
(93, 0.011288431705802977, 0.0086708656379046768, ‘+++’, 0.0067595013978819533)
(94, 0.011423764137639323, 0.0068645073811051706, ‘+++’, 0.0066994846805006249)
(95, 0.010833477746596468, 0.0065423502669258761, ‘+++’, 0.0066607269037501993)
(96, 0.010896219118345681, 0.0065512896334279976, ‘+++’, 0.0067268697583007745)
(99, 0.010992537263413826, 0.0073325599440100062, ‘+++’, 0.0069426057166501118)
(100, 0.010434433908368354, 0.0060716558427773067, ‘+++’, 0.0067355997298879271)
(101, 0.011002338764369511, 0.005801698509047876, ‘+++’, 0.0067141392758106903)
(102, 0.010415714860669327, 0.0072120672626790963, ‘+++’, 0.0067034644613088142)
(103, 0.010610880584825707, 0.0082385497328633338, ‘+++’, 0.0069021403891890752)
(106, 0.011407739582179981, 0.0053694674630942766, ‘+++’, 0.007376611379997719)
(107, 0.010708625405092499, 0.0063134112852608832, ‘+++’, 0.0051413136120935137)
(108, 0.010754975997044594, 0.0051321623258705068, ‘+++’, 0.0074427638219121701)
(109, 0.0099067232626448409, 0.0051825445766684533, ‘+++’, 0.0072171764821908442)
(110, 0.011104271806522919, 0.0046636366836995489, ‘+++’, 0.007457003309578456)
(113, 0.010177206586060332, 0.0056228001995389789, ‘+++’, 0.0074768716737942838)
(114, 0.010141368075635929, 0.0044875397584101008, ‘+++’, 0.007801183772368105)
(115, 0.010219895051615449, 0.0056996086520678052, ‘+++’, 0.0075772393301733638)
(116, 0.010442927712410385, 0.0060587764243964963, ‘+++’, 0.0076193259157247657)
(117, 0.010122399772485953, 0.0059676219422872745, ‘+++’, 0.0076711566357667467)
(120, 0.0089577303830815618, 0.0052143183757884463, ‘+++’, 0.0078854850172536788)
(121, 0.0088734724448912092, 0.0054069056103227915, ‘+++’, 0.0078117320345587734)
(122, 0.0092367634826975402, 0.0048232421728528355, ‘+++’, 0.0078230000038980504)
(123, 0.0090745560982097365, 0.0069755435273008027, ‘+++’, 0.0077918175017330097)
(124, 0.0086796284580979814, 0.0056389551222172712, ‘+++’, 0.0062614877892271294)
(127, 0.0088594900116636052, 0.005038800119737368, ‘+++’, 0.0063435622489589677)
(128, 0.0086583898141907693, 0.0060723832282464389, ‘+++’, 0.0078579534568916972)
(129, 0.0091295246821703445, 0.0050749644395704898, ‘+++’, 0.007843808625425646)
(130, 0.0095100235287899973, 0.0062493800270962772, ‘+++’, 0.0078253807200785331)
(131, 0.0085070980458181473, 0.0061323111636472525, ‘+++’, 0.0079059367620874007)
(134, 0.0090496183062996841, 0.0055485552765108422, ‘+++’, 0.0078968213110221924)
(135, 0.009221964527180165, 0.0057343457541309774, ‘+++’, 0.007323685667519105)
(136, 0.0086237704257309013, 0.0056220618085633433, ‘+++’, 0.0079055001503350129)
(137, 0.008850750144182639, 0.0055492788177673672, ‘+++’, 0.0078916918529028032)
(138, 0.0081059851120347335, 0.006026881982338742, ‘+++’, 0.0080785970974689061)
(141, 0.0084328733351147576, 0.0054505128817328745, ‘+++’, 0.0065782462929298829)
(142, 0.0086348939281571651, 0.0062253804692258162, ‘+++’, 0.0068562856546898858)
(143, 0.0079070291318077395, 0.0056535926838907954, ‘+++’, 0.0081196475781159117)
(144, 0.0086563554610453904, 0.0052439264518623257, ‘+++’, 0.0080955315522601122)
(145, 0.0089315402849473452, 0.0052636861991823833, ‘+++’, 0.0073866195697601623)
(148, 0.0081484824608710326, 0.0057315379284853521, ‘+++’, 0.0081963391714325647)
(149, 0.008669629690946366, 0.0056950298757696121, ‘+++’, 0.0082861895029342041)
(150, 0.0077584524311156986, 0.0062699712666623008, ‘+++’, 0.0083085403084314762)
(151, 0.0078873805775945566, 0.0061948885495324247, ‘+++’, 0.0083071856120393341)
(152, 0.0089338347750740829, 0.0063412585769902574, ‘+++’, 0.0081865499998219349)
(155, 0.0078484074061250068, 0.0063700309046388079, ‘+++’, 0.007818393329221544)
(156, 0.0079992601269848022, 0.0062273934641041227, ‘+++’, 0.0083963956378522182)
(157, 0.0079417669450422331, 0.0068212110408617339, ‘+++’, 0.0083751170908349659)
(158, 0.0077586561923895011, 0.0061579758487784846, ‘+++’, 0.0083396690674496267)
(159, 0.0070716121319859731, 0.0063307595462288946, ‘+++’, 0.0084016392071821107)
(162, 0.0079994851099806901, 0.0070528716623753695, ‘+++’, 0.0084564096194508828)
(163, 0.0074092409432632645, 0.006164938647200217, ‘+++’, 0.0081182421555909788)
(164, 0.0076676542247929875, 0.0066602070661324559, ‘+++’, 0.0081019175388608814)
(165, 0.0072877539779074825, 0.0060842390849736618, ‘+++’, 0.0081324834676038925)
(166, 0.0072830722800429149, 0.0066226386680768275, ‘+++’, 0.0081264807304510366)
(169, 0.0070339822930199081, 0.0073473492517871827, ‘—‘, 0.0075945667935078051)
(170, 0.0080416106968340326, 0.79363044691851259, ‘—‘, 0.0075698760594386872)
(171, 0.007630988317294498, 0.0072812345639979657, ‘+++’, 0.010921729151433497)
(172, 0.0083744755716249186, 0.0079308042648167723, ‘+++’, 0.010172040910748957)
(173, 0.0078864870944130912, 0.0069140057566178838, ‘+++’, 0.0099210841683079823)
(176, 0.0080946383136790128, 0.0073363133628692396, ‘+++’, 0.010383587613381706)
(177, 0.0079336243086647919, 0.0082995729589364717, ‘—‘, 0.01037548638621489)
(178, 0.0082255743182282279, 0.0071634339571982489, ‘+++’, 0.010495445658705123)
(179, 0.0075099925907255232, 0.0079835824108167672, ‘—‘, 0.010624468240364384)
(180, 0.0080955380744853669, 0.0084699820978799637, ‘—‘, 0.011251940222577438)
(183, 0.0076593729500787503, 0.0091985751549777516, ‘—‘, 0.011136209922544228)
(184, 0.0073621957647872485, 0.0087519390936397594, ‘—‘, 0.010824118573306601)
(185, 0.0082034857091791528, 0.0090107200136364078, ‘—‘, 0.010771542463973762)
(186, 0.0072304314845381351, 0.0080687244355780105, ‘—‘, 0.010686658802446725)
(187, 0.0072332118101138534, 0.0091768159420294888, ‘—‘, 0.010983872960267933)
(190, 0.0070009956149992445, 0.0097835535769986298, ‘—‘, 0.010977383170251527)
(191, 0.0076599691477123952, 0.0089398664928575858, ‘—‘, 0.010763045652453796)
(192, 0.0072882620955142712, 0.0087075595241502511, ‘—‘, 0.0107557475893174)
(193, 0.0076490719872876773, 0.0094509073256782289, ‘—‘, 0.010725724329131985)
(194, 0.0075595368866729377, 0.0090122845261302736, ‘—‘, 0.010765545606014114)
(197, 0.0073042037181907599, 0.0095953118477137816, ‘—‘, 0.011904470049529525)
(198, 0.0071751631599000637, 0.0085160007041404102, ‘—‘, 0.0067727498822466687)
(199, 0.0075631676541938857, 0.0094368900263129551, ‘—‘, 0.011004775230355446)
(200, 0.0077616235903134583, 0.0093137139152903413, ‘—‘, 0.010640690010936904)
(201, 0.0077416352290482771, 0.009924204097735331, ‘—‘, 0.01136888366998293)
(204, 0.0081837388494380003, 0.0098501868067726317, ‘—‘, 0.010930546642019917)
(205, 0.0075997722478879553, 0.010310991041387198, ‘—‘, 0.011962792041536278)
(206, 0.0077923683211384032, 0.010500716627493868, ‘—‘, 0.011861126941430891)
(207, 0.0075170876059418032, 0.011131815507376722, ‘—‘, 0.011939119451115855)
(208, 0.0077955078310069526, 0.01083524031650109, ‘—‘, 0.012266618916102845)
(211, 0.0077718438664198357, 0.0098609206363096339, ‘—‘, 0.012161821910468076)
(212, 0.007476838997400456, 0.011325105456845024, ‘—‘, 0.012100903852171177)
(213, 0.0077906495150384123, 0.011373054329497648, ‘—‘, 0.012127616810839148)
(214, 0.0078013938118160321, 0.011027551928440944, ‘—‘, 0.012173016503477437)
(215, 0.0071499680198448623, 0.0087037115359201757, ‘—‘, 0.0086896638427371511)
(218, 0.0070859184997445089, 0.009774764693736419, ‘—‘, 0.0088099596770366828)
(219, 0.0077271288033079191, 0.011242275099805663, ‘—‘, 0.012263783162180859)
(220, 0.0077603169602125403, 0.011094483899708758, ‘—‘, 0.01228387961421998)
(221, 0.0075832404796830587, 0.011097053431177912, ‘—‘, 0.012263594412048332)
(222, 0.0081293072079558423, 0.012139051252621375, ‘—‘, 0.012409847313184733)
(225, 0.0079175706296126299, 0.011401700417378428, ‘—‘, 0.012374738879496801)
(226, 0.0076606605589732693, 0.011192287422213801, ‘—‘, 0.012386421026080251)
(227, 0.0074012930004120261, 0.01214606310100157, ‘—‘, 0.012431058607616296)
(228, 0.0073939364899463771, 0.011896019948107829, ‘—‘, 0.012420854450936474)
(229, 0.0077592358287989329, 0.011648695588194168, ‘—‘, 0.01272572616466699)
(232, 0.0071773135958589579, 0.010061491956577257, ‘—‘, 0.008994783331569214)
(233, 0.0070274958095997665, 0.010774635799086319, ‘—‘, 0.0092248875790461194)
(234, 0.0073276540911178124, 0.012797650762436311, ‘—‘, 0.012755842414107607)
(235, 0.0077450251455636072, 0.012257165487981619, ‘—‘, 0.012756283641658249)
(236, 0.006237053739864655, 0.011208071810488326, ‘—‘, 0.010706287611132407)
(239, 0.008454669754690532, 0.015225026555932705, ‘—‘, 0.013998792633735041)
(240, 0.0089608254526203461, 0.015229395454776732, ‘—‘, 0.014314236267385164)
(241, 0.0091367506043319984, 0.01543933194304407, ‘—‘, 0.014100494614839809)
(242, 0.0089169076903905138, 0.015269778030494507, ‘—‘, 0.014306027309618462)
(243, 0.0089089057003622889, 0.014870338048435783, ‘—‘, 0.014247094927764674)
(246, 0.0087159688315426084, 0.015780600634942861, ‘—‘, 0.015203546498451053)
(247, 0.0095150865533772909, 0.015304001135470152, ‘—‘, 0.015196110790154061)
(248, 0.0092685931724283289, 0.015891450359527708, ‘—‘, 0.015259127033780137)
(249, 0.0095936719687691688, 0.015227862793883835, ‘—‘, 0.015537897007733885)
(250, 0.0088777423487115432, 0.016279806513392434, ‘—‘, 0.015652243247408715)
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