{"id":522,"date":"2017-06-19T21:29:31","date_gmt":"2017-06-19T19:29:31","guid":{"rendered":"http:\/\/dobretrejdy.com\/?p=522"},"modified":"2023-04-09T09:22:42","modified_gmt":"2023-04-09T07:22:42","slug":"gamma","status":"publish","type":"post","link":"https:\/\/dobretrejdy.com\/?p=522","title":{"rendered":"Gamma"},"content":{"rendered":"

\u00a0 \u00a0 K\u00a0dokonalej\u0161\u00edmu pochopen\u00ed z\u00e1vislosti ceny opce na pohybu podkladov\u00e9ho aktiva vyj\u00e1d\u0159en\u00e9ho \u0159eck\u00fdm p\u00edsmenem Delta, jak jsem jej popisoval v\u00a0\u010dl\u00e1nku Delta<\/strong><\/a><\/span> je zapot\u0159eb\u00ed hlub\u0161\u00edho ponoru do problematiky \u0159eck\u00fdch p\u00edsmen, kdy\u017e t\u00edm nejbli\u017e\u0161\u00edm p\u00edsmenem k\u00a0popisovan\u00e9 Delta je \u0159eck\u00e9 p\u00edsmeno Gamma<\/u><\/strong>. Tak jako Rumburak byl \u010darod\u011bj druh\u00e9 kategorie, tak tak\u00e9 na Gamma m\u016f\u017eeme pohl\u00ed\u017eet jako na \u0159eck\u00e9 p\u00edsmeno \u201edruh\u00e9 kategorie\u201c. Pro\u010d takov\u00e1 degradace? Popisoval jsem d\u0159\u00edve, \u017ee cenu opc\u00ed vypo\u010d\u00edt\u00e1vaj\u00ed matematick\u00e9 modely, kde nejzn\u00e1m\u011bj\u0161\u00ed z\u00a0nich \u2013 Black-Scholes\u016fv model<\/strong>, vyu\u017e\u00edv\u00e1 p\u0159i stanoven\u00ed ceny op\u010dn\u00edch kontrakt\u016f \u0161est z\u00e1kladn\u00edch \u00fadaj\u016f \u2013 Cenu podkladov\u00e9ho aktiva<\/strong>, Hodnotu strike,<\/strong> D\u00e9lku \u017eivota op\u010dn\u00edho kontraktu<\/strong>, Implied Volatilitu<\/strong>, sou\u010dasnou \u00darokovou m\u00edru<\/strong> a hodnotu p\u0159\u00edpadn\u011b vypl\u00e1cen\u00e9 Dividendy<\/strong> za dobu \u017eivota opce. Proto\u017ee z\u00a0t\u011bchto \u0161esti \u00fadaj\u016f m\u016f\u017eeme Hodnotu strike, \u00darokovou m\u00edru a zji\u0161t\u011bn\u00ed, zda se vypl\u00e1c\u00ed Dividenda, br\u00e1t jako zji\u0161t\u011bn\u00e9 konstanty, potom n\u00e1m cenu op\u010dn\u00edho kontraktu ur\u010duj\u00ed zbyl\u00e9 t\u0159i nezn\u00e1m\u00e9, kter\u00e9 m\u016f\u017eeme pozorovat a p\u0159em\u00fd\u0161let o m\u00ed\u0159e jejich vlivu na cenu opce (Cena podkladov\u00e9ho aktiva, D\u00e9lka \u017eivota op\u010dn\u00edho kontraktu a Implied Volatilita). Pro posuzov\u00e1n\u00ed t\u011bchto t\u0159\u00ed nezn\u00e1m\u00fdch se pak v\u017eilo ozna\u010den\u00ed vyjad\u0159ovan\u00e9 \u0159eck\u00fdmi p\u00edsmeny. M\u00edru vlivu zm\u011bny Ceny podkladov\u00e9ho aktiva vyjad\u0159uje hodnota Delta, zm\u011bnu D\u00e9lky \u017eivota opce vyjad\u0159uje Th\u00e9ta a zm\u011bnu Implied Volatility zachycuje Vega. Tato t\u0159i \u0159eck\u00e1 p\u00edsmena pak zachycuj\u00ed p\u0159\u00edmou m\u00edru vlivu t\u011bchto nezn\u00e1m\u00fdch na cenu op\u010dn\u00edho kontraktu, jsou to tedy \u0159eck\u00e1 p\u00edsmena \u201eprvn\u00ed kategorie\u201c. V\u0161echna ostatn\u00ed \u0159eck\u00e1 p\u00edsmena, se kter\u00fdmi se sezn\u00e1m\u00edte v\u00a0n\u011bjak\u00e9 literatu\u0159e jsou odvozeninami, slou\u010deninami nebo vyjad\u0159uj\u00ed jin\u00e9 komplexn\u011bj\u0161\u00ed vztahy mezi t\u011bmito z\u00e1kladn\u00edmi p\u00edsmeny nebo jejich vztahy k\u00a0jin\u00fdm matematick\u00fdm veli\u010din\u00e1m spojen\u00fdch n\u011bjakou vazbou s\u00a0cenou op\u010dn\u00edho kontraktu.<\/span><\/p>\n

\u00a0\u00a0 Gamma<\/strong> je nejzn\u00e1m\u011bj\u0161\u00ed \u201eodvozen\u00e9\u201c \u0159eck\u00e9 p\u00edsmeno. Jeho hodnota vyjad\u0159uje, jak rychle se m\u011bn\u00ed hodnota Delta<\/strong> vzhledem ke zm\u011bn\u011b ceny podkladov\u00e9ho aktiva<\/strong>. Pro jednoduchou p\u0159edstavu bych si k\u00a0hodnot\u011b Delta u ka\u017ed\u00e9 opce na ur\u010dit\u00e9m strike mohl p\u0159i\u0161roubovat tachometr, vlastn\u011b \u201eGammametr\u201c a ten by mi ukazoval, jek rychle se bude hodnota Delta m\u011bnit v\u00a0souvislosti se zm\u011bnou ceny podkladov\u00e9ho aktiva. O tom, jak se takov\u00fd Gammametr vyr\u00e1b\u00ed, jak funguje a jak\u00e9 pak mohou b\u00fdt jeho hodnoty a co n\u00e1m nam\u011b\u0159en\u00e9 hodnoty \u0159\u00edkaj\u00ed, bude dne\u0161n\u00ed \u010dl\u00e1nek. \u00a0 \u00a0<\/span><\/p>\n

\n

\u00a0\u00a0 Matematika Gamma<\/strong><\/span><\/h2>\n

\u00a0\u00a0 Pokud p\u0159ijmu p\u0159edpoklad, \u017ee cenu opc\u00ed lze vypo\u010d\u00edtat pomoc\u00ed Black-Scholesova oce\u0148ovac\u00edho modelu, kter\u00fd pracuje s\u00a0hodnotami \u0161esti faktor\u016f vstupuj\u00edc\u00edch do ceny op\u010dn\u00edho kontraktu (Cena podkladov\u00e9ho aktiva, Hodnota strike, D\u00e9lka \u017eivota op\u010dn\u00edho kontraktu, Implied Volatilita, \u00darokov\u00e1 m\u00edra a hodnota p\u0159\u00edpadn\u011b vypl\u00e1cen\u00e9 Dividendy za dobu \u017eivota opce), tak bych m\u011bl tak\u00e9 pomoc\u00ed tohoto modelu b\u00fdt schopen vypo\u010d\u00edtat hodnotu v\u0161ech z\u00e1kladn\u00edch \u0159eck\u00fdch p\u00edsmen a pop\u0159\u00edpad\u011b jejich odvozenin. V\u00a0p\u0159ede\u0161l\u00fdch \u010dl\u00e1nc\u00edch jsem ukazoval, jak lze takov\u00fd matematick\u00fd model \u201erozkl\u00ed\u010dovat\u201c a vyu\u017e\u00edt jej k\u00a0vlastn\u00edmu zji\u0161\u0165ov\u00e1n\u00ed hodnoty Call i Put opc\u00ed a tak\u00e9 jak si nap\u0159\u00edklad ve vlastn\u00edm Excelu vytvo\u0159it sv\u016fj vlastn\u00ed op\u010dn\u00ed \u0159et\u011bzec, v\u00a0n\u011bm pak bylo mo\u017en\u00e9 podle jednoho z\u00a0\u010dl\u00e1nk\u016f dopo\u010d\u00edtat tak\u00e9 hodnotu Delta takov\u00fdch opc\u00ed.<\/span><\/p>\n

\u00a0\u00a0 Extrahov\u00e1n\u00edm z\u00a0Black-Scholesova modelu, kter\u00e9 m\u016f\u017eete v\u00a0literatu\u0159e naj\u00edt, je hodnota Gamma vypo\u010dtena na z\u00e1klad\u011b tohoto n\u00ed\u017ee uveden\u00e9ho vzorce.<\/span><\/p>\n

\"\"<\/span>\u00a0\u00a0 Proto\u017ee vzorec nevypad\u00e1 nijak zvl\u00e1\u0161\u0165 pochopiteln\u011b, lze si jej pro lep\u0161\u00ed pochopen\u00ed \u201ep\u0159elo\u017eit do \u010de\u0161tiny\u201c a pot\u00e9 bude jeho interpretace mo\u017en\u00e1 jednodu\u0161\u0161\u00ed. Z\u00e1m\u011brn\u011b jsem ji\u017e v\u00a0tomto vzorci ozna\u010dil \u010derven\u00fdm obd\u00e9ln\u00edkem pravou \u010d\u00e1st vzorce. Tato prav\u00e1 \u010d\u00e1st toti\u017e bez n\u011bjak\u00e9ho slo\u017eit\u00e9ho po\u010de\u0161\u0165ov\u00e1n\u00ed vyjad\u0159uje Distribu\u010dn\u00ed funkci norm\u00e1ln\u00edho rozd\u011blen\u00ed<\/strong> pro hodnotu d1<\/strong>. Tuto hodnotu jsem ji\u017e zji\u0161\u0165oval a ukazoval jej\u00ed matematick\u00fd tvar v\u00a0\u010dl\u00e1nku Cena opc\u00ed<\/strong><\/a><\/span>, tam jsem tak\u00e9 vyu\u017e\u00edval jej\u00ed hodnotu p\u0159i ur\u010dov\u00e1n\u00ed ceny opce. Jej\u00ed v\u00fdpo\u010det nen\u00ed slo\u017eit\u00fd a Excel si s\u00a0n\u00ed um\u00ed poradit.<\/span><\/p>\n

\u00a0\u00a0 Co tedy uvid\u00edm, dosad\u00edm-li si za nezn\u00e1m\u00e9 v\u00fdrazy \u201e\u010desk\u00e1 slova\u201c. Na n\u00ed\u017ee uveden\u00e9m obr\u00e1zku je pak vid\u011bt \u010desk\u00e1 skladba vzorce v\u00fdpo\u010dtu hodnoty Gamma pro opci na n\u011bjak\u00e9m strike.<\/span><\/p>\n

\"\"<\/p>\n

\u00a0\u00a0 Z\u00a0\u010desk\u00e9ho vzorce zejm\u00e9na vypl\u00fdv\u00e1 velmi zjednodu\u0161uj\u00edc\u00ed skute\u010dnost. V\u00a0\u010ditateli zlomku je exponenci\u00e1ln\u00ed funkce, jej\u00edm\u017e exponentem je sou\u010din hodnoty Dividendy a D\u00e9lka \u017eivota opce. Proto\u017ee jsem v\u00a0\u010dl\u00e1nc\u00edch v\u00fdpo\u010dtu ceny opc\u00ed a v\u00a0\u010dl\u00e1nc\u00edch o Dividend\u00e1ch uv\u00e1d\u011bl, \u017ee hodnota Dividendy vstupuje do Black-Scholesova vzorce ve sv\u00e9m anualizovan\u00e9m tvaru, m\u016f\u017eeme se na hodnotu exponentu exponenci\u00e1ln\u00ed funkce d\u00edvat, jako by byl t\u00e9m\u011b\u0159 nulov\u00fd a cokoliv t\u00e9m\u011b\u0159 na nultou je t\u00e9m\u011b\u0159 jedna. Cel\u00fd v\u00fdpo\u010det Gamma je pak jednoduch\u00fdm pod\u00edlem<\/strong> Distribu\u010dn\u00ed funkce norm\u00e1ln\u00edho rozd\u011blen\u00ed<\/strong> a sou\u010dinu t\u0159\u00ed \u201eprom\u011bnn\u00fdch\u201c cenotvorn\u00fdch hodnot<\/strong>. Takovou matematickou interpretaci si lze jednodu\u0161e p\u0159ev\u00e9st do Excelu a pot\u00e9 hodnotu Gamma zkou\u0161et modelovat za r\u016fzn\u00fdch tr\u017en\u00edch situac\u00ed. Obr\u00e1zek z\u00a0takov\u00e9ho Excelu je vid\u011bt n\u00ed\u017ee, kdy po zm\u011bn\u00e1ch parametr\u016f v\u00a0modr\u00fdch bu\u0148k\u00e1ch m\u016f\u017eete pozorovat nejen zm\u011bnu ceny opc\u00ed a jejich Delta, ale tak\u00e9 hodnoty Gamma. Funkce Gamma je pak do takov\u00e9ho Excelu naprogramov\u00e1na ve VBA a lze ji kdekoliv v\u00a0tomto Excelu vyvolat v\u00a0jak\u00e9koli bu\u0148ce jej\u00edm vyvol\u00e1n\u00edm \u201e=GammaOpce\u201c. Jednoduch\u00fd Excel s\u00a0touto funkc\u00ed je ke\u00a0st\u00e1\u017een\u00ed zde<\/a><\/strong><\/span>. M\u016f\u017eete si rovn\u011b\u017e prohl\u00e9dnou jej\u00ed script ve VBA a tento nap\u0159\u00edklad kop\u00edrov\u00e1n\u00edm voln\u011b p\u0159en\u00e1\u0161et do jin\u00fdch vlastn\u00edch se\u0161it\u016f.<\/span><\/p>\n

\"\"<\/span><\/p>\n

\n

Logika Gamma<\/span><\/strong><\/h2>\n

\u00a0\u00a0 Matematicky jsem tedy schopen si velikost Gamma spo\u010d\u00edtat, je to ale k\u00a0ni\u010demu, kdy\u017e si takov\u00e9 hodnoty nem\u016f\u017eu interpretovat a posuzovat, co mi vlastn\u011b \u0159\u00edkaj\u00ed.<\/span><\/p>\n

\u00a0\u00a0 Gamma<\/strong>\u00a0tedy vyjad\u0159uje, jak se zm\u011bn\u00ed hodnota Delta, kdy\u017e se cena podkladov\u00e9ho aktiva zv\u00fd\u0161\u00ed o jeden dolar<\/strong> (nebo bod u futures kontrakt\u016f). Toto je jednoduch\u00e1 interpretace Gamma jako \u010d\u00edsla.<\/u><\/span><\/p>\n

\u00a0\u00a0 Pokud budu m\u00edt akcii JPM, kter\u00e1 se dnes obchoduje za 86,18 USD, tak Long Call opce na strike 86 bude m\u00edt Delta +52 (hodnota Delta Long Call je kladn\u00e1) a hodnotu Gamma 13. Pokud se cena JPM zv\u00fd\u0161\u00ed o 1 USD na cenu 87,18 USD, potom bude nov\u00e1 hodnota Delta Long Call opce +52 + 13 = 65. Pokud cena JPM klesne 1 USD na hodnotu 85,18 potom bude nov\u00e1 hodnota Long Call Delta +52 – 13 = 39.<\/span><\/p>\n

\u00a0\u00a0 Stejn\u00fd v\u00fdpo\u010det pak plat\u00ed pro Put opce. Za stejn\u00fdch podm\u00ednek pro akci JPM, kter\u00e1 se dnes obchoduje za 86,18 USD, tak Long Put opce na strike 86 bude m\u00edt Delta -48 (hodnota Delta Long Put je z\u00e1porn\u00e1) a hodnotu Gamma 13. Pokud se cena JPM zv\u00fd\u0161\u00ed o 1 USD na cenu 87,18 USD, potom bude nov\u00e1 hodnota Delta Long Put opce -48 + 13 = 35. Pokud cena JPM klesne 1 USD na hodnotu 85,18 potom bude nov\u00e1 hodnota Long Put Delta -48 – 13 = -61<\/span><\/p>\n

\u00a0 \u00a0Gamma<\/strong> lze tak\u00e9 interpretovat jako budouc\u00ed zm\u011bnu Delta v\u00a0procentn\u00edm vyj\u00e1d\u0159en\u00ed ke zm\u011bn\u011b ceny podkladov\u00e9 akcie<\/strong>. Toto by mohla b\u00fdt druh\u00e1 jednoduch\u00e1 interpretace Gamma jako \u010d\u00edsla vyj\u00e1d\u0159en\u00e9ho jako procento<\/u>.<\/span><\/p>\n

\u00a0\u00a0 Pokud budu m\u00edt akcii JPM, kter\u00e1 se dnes obchoduje za 86,18 USD, tak Long Call opce na strike 86 bude m\u00edt Delta 52 a hodnotu Gamma 13. Gamma si pak mohu vylo\u017eit jako procento zm\u011bny ceny Delta p\u0159i pohybu akcie JPM. Pokud se cena JPM zv\u00fd\u0161\u00ed o 2 USD na cenu 88,18 USD, potom bude nov\u00e1 hodnota Delta vy\u0161\u0161\u00ed o 13% pohybu podkladov\u00e9 akcie JPM. 13% ze dvoudolarov\u00e9ho pohybu na akcii JPM = 0,26 USD. Nov\u00e1 hodnota Delta pak bude 52 +26 = 78.<\/span><\/p>\n

\u00a0 \u00a0Toto je z\u00e1kladn\u00ed n\u00e1hled na hodnotu Gamma. Proto\u017ee hodnota Gamma vyjad\u0159uje hodnotu zm\u011bny Delta, je pak jasn\u00e9, jak\u00e9 hodnoty m\u016f\u017eu \u010dekat na \u201eGammametrech\u201c namontovan\u00fdch na Delta na jednotliv\u00fdch strike. Z\u00a0p\u0159edch\u00e1zej\u00edc\u00edho \u010dl\u00e1nku o Delta vypl\u00fdv\u00e1, \u017ee Delta opc\u00ed hluboce \u201ev pen\u011bz\u00edch\u201c se bl\u00ed\u017e\u00ed k\u00a0hodnot\u011b 100<\/strong> a tedy, \u017ee pohyb podkladu o jeden dolar bude znamenat zv\u00fd\u0161en\u00ed ceny opce tak\u00e9 o stejnou hodnotu, cena opce tak p\u0159esn\u011b kop\u00edruje hodnotu podkladu. Pokud tedy p\u0159i cen\u011b JPM 86.18 USD budu m\u00edt ITM Long Call opci na strike 70 (z\u00e1vratn\u00fdch 16 bod\u016f \u201ev pen\u011bz\u00edch\u201c), tak tato opce bude m\u00edt Delta 100 a hodnotu Gamma 0. Pokud se cena JPM zv\u00fd\u0161\u00ed o +5 USD na cenu 91.18 USD, z\u016fstane Delta m\u00e9 opce st\u00e1le 100 a hodnota Gamma bude st\u00e1le nula. Delta v\u00edce ne\u017e 100 b\u00fdt nem\u016f\u017ee, proto\u017ee se opce chov\u00e1 identicky jako podkladov\u00e1 akcie (podklad m\u00e1 v\u017edy hodnotu Delta 100). Proto\u017ee k\u00a0hodnot\u011b Delta ve v\u00fd\u0161i 100 ji\u017e nen\u00ed mo\u017en\u00e9 \u201enic dal\u0161\u00edho p\u0159idat\u201c, tak hodnota Gamma bude nulov\u00e1. Hluboce<\/strong> ITM opce maj\u00ed Gamma zanedbatelnou<\/strong>, hodnota Gamma tak kles\u00e1 sou\u010dasn\u011b s\u00a0pono\u0159ov\u00e1n\u00edm \u201edo pen\u011bz\u201c \u2013 \u010d\u00edm v\u00edce je opce v\u00a0pen\u011bz\u00edch, t\u00edm m\u00e1 men\u0161\u00ed Gamma<\/strong>.<\/span><\/p>\n

\u00a0\u00a0 Plat\u00ed to samoz\u0159ejm\u011b i naopak o velmi vzd\u00e1len\u00fdch opc\u00edch, kter\u00e9 jsou \u201emimo pen\u00edze\u201c. Pokud tedy p\u0159i cen\u011b JPM 86.18 budu m\u00edt OTM Long call opci na strike 110 (zna\u010dn\u011b vzd\u00e1len\u00fdch \u010dtrn\u00e1ct bod\u016f \u201eod pen\u011bz\u201c), tak tato opce bude m\u00edt Delta nula a hodnotu Gamma nula. Pokud se cena JPM zv\u00fd\u0161\u00ed o 5 USD na cenu 91.18 USD, z\u016fstane Delta m\u00e9 opce st\u00e1le nula a hodnota Gamma bude st\u00e1le nula. Pohyb je velmi mal\u00fd vzhledem k\u00a0m\u00e9mu velmi vzd\u00e1len\u00e9mu strike a cena opce v\u016fbec nereaguje na pohyb podkladu, je to pro cenu m\u00e9 OTM opce pohyb tak zanedbateln\u00fd, \u017ee nem\u00e1 na cenu pra\u017e\u00e1dn\u00fd vliv, jako by takov\u00fd pohyb v\u016fbec nenastal. Velmi vzd\u00e1len\u00e9 OTM opce maj\u00ed Gamma zanedbatelnou<\/strong>, hodnota Gamma pak roste sou\u010dasn\u011b s\u00a0p\u0159ibli\u017eov\u00e1n\u00edm strike opce k\u00a0sou\u010dasn\u00e9 cen\u011b podkladov\u00e9 akcie, \u010d\u00edm v\u00edce se bl\u00ed\u017e\u00ed \u201ek pen\u011bz\u016fm\u201c \u2013 t\u00edm m\u00e1 v\u011bt\u0161\u00ed Gamma<\/strong>.<\/span><\/p>\n

\u00a0 \u00a0Z\u00a0v\u00fd\u0161e uveden\u00e9ho pozorov\u00e1n\u00ed hodnoty Gamma na \u201eextr\u00e9mn\u00edch\u201c strike tedy vypl\u00fdv\u00e1, \u017ee nejvy\u0161\u0161\u00ed hodnota Gamma<\/strong> je u opc\u00ed, kter\u00e9 jsou \u201ena pen\u011bz\u00edch\u201c<\/strong>, tedy na strikes, kter\u00e9 jsou nejbl\u00ed\u017ee sou\u010dasn\u00e9 obchodovan\u00e9 ceny podkladov\u00e9ho aktiva. Gamma pak kles\u00e1 v\u00a0obou sm\u011brech, jak se vzdaluje od t\u011bchto ATM strikes, a\u0165 u\u017e se strike \u201epono\u0159uje do pen\u011bz\u201c nebo se od t\u011bchto \u201epen\u011bz vzdaluje\u201c.<\/span><\/p>\n

\u00a0 \u00a0S\u00a0touto prvn\u00ed poznanou \u201ekvalitou\u201c Gamma je spojena tak\u00e9 jej\u00ed druh\u00e1 vlastnost, kter\u00e1 souvis\u00ed se samotnou podstatou Delta. Hodnota Delta u opc\u00ed na stejn\u00fdch strike<\/strong> ale v\u00a0r\u016fzn\u00fdch expirac\u00edch sm\u011brem do budoucnosti kles\u00e1<\/strong>, kles\u00e1<\/strong> tak samoz\u0159ejm\u011b tak\u00e9 hodnota Gamma<\/strong> t\u011bchto opc\u00ed na t\u011bchto stejn\u00fdch strike. Na namontovan\u00fdch \u201eGammagrafech\u201c bych pak sm\u011brem do budoucnosti u opc\u00ed na stejn\u00fdch strikes vid\u011bl jin\u00e9 \u2013 ni\u017e\u0161\u00ed hodnoty. Na n\u00ed\u017ee uveden\u00e9m obr\u00e1zku jsou vid\u011bt hodnoty Gamma pro jednotliv\u00e9 strike opc\u00ed akcie JPM, kter\u00e9 plasticky zobrazuj\u00ed v\u00fd\u0161e uveden\u00e9 vlastnosti Gamma.<\/span><\/p>\n

\"\"<\/p>\n

\u00a0 \u00a0Na obr\u00e1zku je zn\u00e1zorn\u011bn\u00e1 \u010dervenou \u0161ipkou aktu\u00e1ln\u00ed hodnota akcie JPM v\u00a0dob\u011b tvorby grafu. \u010cerven\u00e1 k\u0159ivka<\/strong> p\u0159edstavuje hodnotu Gamma pro jednotliv\u00e9 strike opce s\u00a0nejbli\u017e\u0161\u00ed expirac\u00ed<\/strong>, modrou k\u0159ivkou<\/strong> je zn\u00e1zorn\u011bn\u00e1 hodnota Gamma pro jednotliv\u00e9 strike opce se vzd\u00e1len\u011bj\u0161\u00ed expirac\u00ed<\/strong>. Je z\u0159eteln\u011b vid\u011bt nejvy\u0161\u0161\u00ed hodnoty Gamma na strike co nejbl\u00ed\u017ee sou\u010dasn\u00e9 cen\u011b a tak\u00e9 je patrn\u00e9, \u017ee modr\u00e1 k\u0159ivka hodnot Gamma opc\u00ed se vzd\u00e1len\u011bj\u0161\u00ed expirac\u00ed je ni\u017e\u0161\u00ed ne\u017e k\u0159ivka hodnot Gamma s\u00a0bli\u017e\u0161\u00ed expirac\u00ed.<\/span><\/p>\n

\u00a0 \u00a0Stoj\u00ed za pov\u0161imnut\u00ed, \u017ee Gamma je stejn\u00e1 pro Call i Put opce<\/strong> na stejn\u00e9m strike. Je to zp\u016fsobeno t\u00edm, \u017ee absolutn\u00ed hodnota sou\u010dtu Delta Call i Put opc\u00ed na stejn\u00e9m strike je roven 100. Pokud se podklad pohne n\u011bjak\u00fdm sm\u011brem, tak v\u017edy jedna z\u00a0opc\u00ed na sv\u00e9 hodnot\u011b p\u0159id\u00e1 a druh\u00e1 o stejnou hodnotu klesne, Delta jedn\u00e9 z\u00a0opc\u00ed se zv\u00fd\u0161\u00ed, zat\u00edmco Delta druh\u00e9 opce adekv\u00e1tn\u011b klesne, aby pak celkov\u00fd sou\u010det Delta byl st\u00e1le 100. Tato Call-Put parita<\/strong> n\u00e1m zaji\u0161\u0165uje, \u017ee Gamma, kter\u00e1 \u010d\u00edseln\u011b reprezentuje takov\u00e9 zvy\u0161ov\u00e1n\u00ed Delta na jedn\u00e9 opci a kles\u00e1n\u00ed Delta na druh\u00e9 opci mus\u00ed m\u00edt stejnou hodnotu.<\/span><\/p>\n

Gamma Long opc\u00ed je kladn\u00e1 a naopak Gamma Short opc\u00ed je z\u00e1porn\u00e1.<\/span><\/strong><\/p>\n

\u00a0\u00a0 Pozitivn\u00ed (kladn\u00e1) Gamma Long Call znamen\u00e1, \u017ee pohyb podkladov\u00e9ho aktiva sm\u011brem nahoru zvy\u0161uje kladnou Delta t\u011bchto Long Call opc\u00ed sm\u011brem k\u00a0hodnot\u011b +100 a pokud cena podkladu kles\u00e1, tak se hodnota Delta Long Call sni\u017euje sm\u011brem k\u00a0hodnot\u011b nula. Pozitivn\u00ed Gamma Long Put opc\u00ed zp\u016fsobuje, \u017ee z\u00e1porn\u00e1 Delta t\u011bchto Long Put opc\u00ed se sni\u017euje sm\u011brem k\u00a0nule, kdy\u017e cena podkladu roste a naopak pokud cena podkladu pad\u00e1, tak z\u00e1porn\u00e1 Delta Long Put opc\u00ed je je\u0161t\u011b v\u00edce negativn\u011bj\u0161\u00ed a\u017e k\u00a0maxim\u00e1ln\u00ed hodnot\u011b -100.\u00a0\u00a0<\/span><\/p>\n

\u00a0 Negativn\u00ed (z\u00e1porn\u00e1) Gamma Short Call znamen\u00e1, \u017ee pohyb podkladov\u00e9ho aktiva sm\u011brem nahoru zvy\u0161uje negativitu Delta t\u011bchto Short Call opc\u00ed sm\u011brem k\u00a0hodnot\u011b -100 a pokud cena podkladu kles\u00e1, tak se hodnota Delta Short Call sni\u017euje sm\u011brem k\u00a0hodnot\u011b nula. Negativn\u00ed Gamma Short Put opc\u00ed zp\u016fsobuje, \u017ee kladn\u00e1 Delta t\u011bchto Short Put opc\u00ed se sni\u017euje sm\u011brem k\u00a0nule, kdy\u017e cena podkladu roste a naopak pokud cena podkladu pad\u00e1, tak kladn\u00e1 Delta Short Put opc\u00ed je je\u0161t\u011b v\u00edce pozitivn\u011bj\u0161\u00ed a\u017e k\u00a0maxim\u00e1ln\u00ed hodnot\u011b +100.<\/span><\/p>\n

\u00a0 \u00a0Shrnut\u00edm t\u011bchto v\u00fdznam\u016f Gamma m\u016f\u017eu pak pou\u017e\u00edt na jednotliv\u00e9 op\u010dn\u00ed strategie nebo cel\u00e1 op\u010dn\u00ed portfolia. Sou\u010dtem Gamma v\u0161ech op\u010dn\u00edch kontrakt\u016f<\/strong> pak mohu dost\u00e1vat odpov\u011b\u010f na ot\u00e1zku<\/strong>, jak\u00e9mu riziku vypl\u00fdvaj\u00edc\u00edmu z\u00a0pohybu podkladu je vystaveno nap\u0159\u00edklad cel\u00e9 m\u00e9 op\u010dn\u00ed portfolio,<\/strong> jak moc se zm\u011bn\u00ed hodnota Delta portfolia, kdy\u017e se zm\u011bn\u00ed hodnota podkladu. Obecn\u011b pak bude platit, \u017ee pokud bude sou\u010det Gamma kladn\u00fd, tak pohyb \u201eje m\u016fj p\u0159\u00edtel\u201c a naopak, pokud bude Gamma cel\u00e9ho portfolia z\u00e1porn\u00e1, bude pohyb podkladov\u00e9ho aktiva m\u00edt nep\u0159\u00edzniv\u00fd vliv na op\u010dn\u00ed portfolio, toto v\u0161echno samoz\u0159ejm\u011b za p\u0159edpokladu, \u017ee by v\u0161echny ostatn\u00ed tr\u017en\u00ed okolnosti p\u016fsob\u00edc\u00ed na na\u0161e portfolio z\u016fstaly konstantn\u00ed (volatilita a \u010das). Ne\u017e uk\u00e1\u017eu, jak si takovou skute\u010dnost vyhodnocovat pro cel\u00e9 portfolio, shrnu p\u0159edch\u00e1zej\u00edc\u00ed tvrzen\u00ed o Gamma na jednotliv\u00fdch uk\u00e1zk\u00e1ch. <\/span><\/p>\n

\u00a0 \u00a0Na n\u00ed\u017ee uveden\u00e9m obr\u00e1zku je vid\u011bt uk\u00e1zka dvou pozic Long Straddle<\/strong>, po\u0159\u00edzen\u00fdch na ATM strike, ka\u017ed\u00e1 pozice je v\u00a0jin\u00e9 expiraci<\/strong>, prvn\u00ed m\u00e1 expiraci tento t\u00fdden (23.7.), druh\u00e1 pozice p\u0159\u00ed\u0161t\u00ed t\u00fdden (30.7.).<\/span><\/p>\n

\"\"<\/p>\n

\u00a0 \u00a0Na obr\u00e1zku je vid\u011bt Gamma po\u0159\u00edzen\u00e9 pozice v\u00a0\u010derven\u00fdch obd\u00e9ln\u00edc\u00edch za situace, kdy podkladov\u00e1 akcie AAPL se obchoduje za 143.80 USD. Je vid\u011bt, \u017ee Gamma je kladn\u00e1<\/strong>, proto\u017ee jsou po\u0159\u00edzeny Long Call a Long Put<\/strong> opce tvo\u0159\u00edc\u00ed ATM Long Straddle. Je tak\u00e9 patrn\u00e9, \u017ee Gamma je vy\u0161\u0161\u00ed na bli\u017e\u0161\u00ed expiraci (+26) a ni\u017e\u0161\u00ed na vzd\u00e1len\u011bj\u0161\u00ed expiraci (+16). Celkov\u00e1 Gamma<\/strong> cel\u00e9 op\u010dn\u00ed pozice dvou Long Straddle v\u00a0r\u016fzn\u00fdch expirac\u00edch je pak +42<\/strong>. Plat\u00ed tak v\u00fd\u0161e v\u00a0\u010dl\u00e1nku uveden\u00e9, \u017ee opce na stejn\u00fdch strike, ale ve vzd\u00e1len\u011bj\u0161\u00edch expirac\u00edch maj\u00ed Gamma ni\u017e\u0161\u00ed ne\u017e na bli\u017e\u0161\u00edch expirac\u00edch. Na prav\u00e9 stran\u011b obr\u00e1zku je vid\u011bt simulovan\u00e1 situace, kdy podkladov\u00e1 akcie po uptrendu pos\u00edlila o +5 USD<\/strong> na cenu 148.80 USD. V\u00a0hn\u011bd\u00fdch obd\u00e9ln\u00edc\u00edch je pak vid\u011bt hodnota Gamma po takov\u00e9m uptrendu. Gamma dramaticky poklesla<\/strong> u obou opc\u00ed Long Straddle. Je to proto, \u017ee ob\u011b Long Call<\/strong> obou Long Straddle se dostaly v\u00edce \u201edo pen\u011bz\u201c<\/strong>, kde\u017eto ob\u011b Long Put<\/strong> obou Long Straddle se dostaly v\u00edce \u201emimo pen\u00edze\u201c<\/strong>. Op\u011bt tak plat\u00ed v\u00fd\u0161e v\u00a0\u010dl\u00e1nku uveden\u00e9, \u017ee nejvy\u0161\u0161\u00ed Gamma je u opc\u00ed, kter\u00e9 jsou nejbl\u00ed\u017ee obchodovan\u00e9 cen\u011b na ATM strike a jak se opce se sv\u00fdmi strike vzdaluj\u00ed od t\u00e9to ceny, tak hodnota Gamma kles\u00e1, a\u0165 u\u017e se jedn\u00e1 o pohyb \u201ev\u00edce do pen\u011bz\u201c nebo o pohyb \u201ed\u00e1le od pen\u011bz\u201c. Je tak\u00e9 patrn\u00e9, jak je r\u016fzn\u00e1 Delta po takov\u00e9m pohybu u opc\u00ed s\u00a0bli\u017e\u0161\u00ed expirac\u00ed a vzd\u00e1len\u011bj\u0161\u00ed expirac\u00ed. Cel\u00e1 pozice je v\u00a0profitu +322 USD. Nutno tak\u00e9 podotknout, \u017ee simulace nepo\u010d\u00edt\u00e1 s\u00a0jin\u00fdmi vlivy, jako je zm\u011bna Implied Volatility (Vega) a p\u016fsoben\u00ed \u010dasu (Th\u00e9ta).<\/span><\/p>\n

\u00a0 \u00a0Opa\u010dnou situaci se z\u00e1pornou Gamma cel\u00e9ho op\u010dn\u00edho portfolia a simulac\u00ed pohybu m\u016f\u017eu pozorovat na dal\u0161\u00edm obr\u00e1zku, jedn\u00e1 se o opa\u010dnou pozici k\u00a0p\u0159edchoz\u00ed uk\u00e1zce, kdy byly po\u0159\u00edzeny dv\u011b pozice Short Straddle<\/strong>, po\u0159\u00edzen\u00fdch na ATM strike, ka\u017ed\u00e1 pozice je op\u011bt v\u00a0jin\u00e9 expiraci<\/strong>, prvn\u00ed m\u00e1 expiraci tento t\u00fdden (23.7.), druh\u00e1 pozice p\u0159\u00ed\u0161t\u00ed t\u00fdden (30.7.).<\/span><\/p>\n

\"\"<\/p>\n

\u00a0\u00a0 Na obr\u00e1zku op\u011bt vid\u00edm Gamma po\u0159\u00edzen\u00e9 pozice v\u00a0\u010derven\u00fdch obd\u00e9ln\u00edc\u00edch za situace, kdy podkladov\u00e1 akcie AAPL se obchoduje za 143.80 USD. Je vid\u011bt, \u017ee Gamma je z\u00e1porn\u00e1<\/strong>, proto\u017ee jsou po\u0159\u00edzeny Short Call a Short Put<\/strong> opce tvo\u0159\u00edc\u00ed ATM Short Straddle. Je tak\u00e9 op\u011bt patrn\u00e9, \u017ee Gamma je vy\u0161\u0161\u00ed na bli\u017e\u0161\u00ed expiraci (-23) a ni\u017e\u0161\u00ed na vzd\u00e1len\u011bj\u0161\u00ed expiraci (-15). Celkov\u00e1 Gamma<\/strong> cel\u00e9 op\u010dn\u00ed pozice dvou Long Straddle v\u00a0r\u016fzn\u00fdch expirac\u00edch je pak -38<\/strong>. Plat\u00ed tak v\u00fd\u0161e v\u00a0\u010dl\u00e1nku uveden\u00e9, \u017ee opce na stejn\u00fdch strike, ale ve vzd\u00e1len\u011bj\u0161\u00edch expirac\u00edch maj\u00ed Gamma ni\u017e\u0161\u00ed ne\u017e na bli\u017e\u0161\u00edch expirac\u00edch. Na prav\u00e9 stran\u011b obr\u00e1zku je vid\u011bt simulovan\u00e1 situace, kdy podkladov\u00e1 akcie po uptrendu pos\u00edlila o +5 USD<\/strong> na cenu 148.80 USD. V\u00a0hn\u011bd\u00fdch obd\u00e9ln\u00edc\u00edch je pak vid\u011bt hodnota Gamma po takov\u00e9m uptrendu. Gamma op\u011bt dramaticky poklesla<\/strong> u obou opc\u00ed Short Straddle. Je to proto, \u017ee ob\u011b Short Call<\/strong> obou Short Straddle se dostaly v\u00edce \u201edo pen\u011bz\u201c<\/strong>, kde\u017eto ob\u011b Short Put<\/strong> obou Short Straddle se dostaly v\u00edce \u201emimo pen\u00edze\u201c<\/strong>. Op\u011bt tak plat\u00ed v\u00fd\u0161e v\u00a0\u010dl\u00e1nku uveden\u00e9, \u017ee nejvy\u0161\u0161\u00ed Gamma je u opc\u00ed, kter\u00e9 jsou nejbl\u00ed\u017ee obchodovan\u00e9 cen\u011b na ATM strike a jak se opce se sv\u00fdmi strike vzdaluj\u00ed od t\u00e9to ceny, tak hodnota Gamma kles\u00e1, a\u0165 u\u017e se jedn\u00e1 o pohyb \u201ev\u00edce do pen\u011bz\u201c nebo o pohyb \u201ed\u00e1le od pen\u011bz\u201c. Cel\u00e1 pozice je ve ztr\u00e1t\u011b -232 USD. Nutno tak\u00e9 op\u011bt podotknout, \u017ee simulace nepo\u010d\u00edt\u00e1 s\u00a0jin\u00fdmi vlivy, jako je zm\u011bna Implied Volatility (Vega) a p\u016fsoben\u00ed \u010dasu (Th\u00e9ta).<\/span><\/p>\n

\n

Gamma a Delta-Neutral<\/span><\/strong><\/h1>\n

\u00a0\u00a0 Jak takov\u00e9 poznatky vyu\u017e\u00edt pro m\u016fj vlastn\u00ed trading nebo alespo\u0148 co z\u00a0nich vlastn\u011b ve shrnut\u00ed vypl\u00fdv\u00e1? V\u00a0\u010dl\u00e1nku Straddle a Delta-Neutral<\/strong><\/a>\u00a0<\/span>jsem popisoval, jak za\u0159\u00eddit, abych m\u011bl portfolio Delta neutr\u00e1ln\u00ed<\/strong> a vyrobit pro dan\u00fd okam\u017eik t\u00e9to neutrality nez\u00e1vislost pozice na pohybu podkladu. P\u0159i zapojen\u00ed poznatk\u016f o Gamma se posunu o kousek d\u00e1l.<\/span><\/p>\n

\u00a0\u00a0 Pokud vytvo\u0159\u00edm portfolio<\/strong>, kter\u00e9 je Delta-Neutral<\/strong>, nemohu objektivn\u011b v\u016fbec v\u011bd\u011bt, kter\u00fdm sm\u011brem se vyd\u00e1 podkladov\u00e9 aktivum, zda poroste nebo bude klesat. Co v\u0161ak mohu zjistit je skute\u010dnost, jak\u00fd vliv tento n\u00e1sledn\u00fd pohyb bude m\u00edt na budouc\u00ed hodnotu Delta. Tuto skute\u010dnost mi napov\u00edd\u00e1 Gamma cel\u00e9 komplexn\u00ed op\u010dn\u00ed pozice<\/strong>.<\/span><\/p>\n

Gamma pozitivn\u00ed<\/span><\/strong><\/h2>\n

\u00a0\u00a0 Na n\u00ed\u017ee uveden\u00e9m obr\u00e1zku je vid\u011bt stejn\u00e1 pozice dvou Long Straddle<\/strong> na ATM opc\u00edch ve dvou nejbli\u017e\u0161\u00edch expirac\u00edch se simulac\u00ed, \u017ee cena akcie AAPL vzrostla o +5 USD.<\/span>\u00a0\u00a0\"\" <\/span><\/p>\n

\u00a0 \u00a0Z\u00a0obr\u00e1zku je patrn\u00e9, \u017ee jsem k\u00a0t\u011bmto dv\u011bma Long Straddle po\u0159\u00eddil je\u0161t\u011b +37 Long akci\u00ed AAPL,<\/strong> proto\u017ee op\u010dn\u00ed pozice vykazovala -37 Delta. Takovou neutralizac\u00ed jsem dos\u00e1hl nulov\u00e9 hodnoty Delta. Gamma<\/strong> m\u00e9 po\u0159\u00edzen\u00e9 Delta-Neutral pozice je kladn\u00e1 (+24)<\/strong>. To je velmi d\u016fle\u017eit\u00e9 zji\u0161t\u011bn\u00ed. Pokud cena podkladu \u2013 akcie AAPL poroste \u2013 poroste tak\u00e9 Delta cel\u00e9 pozice. Proto\u017ee je Delta kladn\u00e1 \u2013 bude kladn\u00e1 je\u0161t\u011b v\u00edce, jej\u00ed hodnota poroste, \u010d\u00edm v\u011bt\u0161\u00ed pohyb t\u00edm bude v\u00fdkonnost vy\u0161\u0161\u00ed. Proto\u017ee je Delta kladn\u00e1 a cena podkladu roste, chov\u00e1 se cel\u00e1 pozice jako bych byl „cel\u00e9 portfolio Long“, a to je skv\u011bl\u00e1 konstelace.\u00a0M\u00e1m takov\u00e9 op\u010dn\u00ed portfolio, kter\u00e9 p\u0159i uptrendu<\/strong> m\u00e1 Delta kladnou, tak\u017ee vyd\u011bl\u00e1v\u00e1<\/strong>. Na prav\u00e9 stran\u011b obr\u00e1zku je vid\u011bt simulace m\u00e9 Delta-Neutral pozice s\u00a0kladnou Gamma p\u0159i uptrendu o +5 USD<\/strong> na akci\u00edch AAPL. Delta se vy\u0161plhala z nuly na hodnotu +149 a celkov\u00fd profit je +424 USD. Proto\u017ee to je jednoduch\u00e1 a \u010diteln\u00e1 pozice, tak v\u00edm, \u017ee vyd\u011bl\u00e1vaj\u00ed ob\u011b Long Call a tak\u00e9 Long akcie, Long Put opce na hodnot\u011b ztr\u00e1cej\u00ed.<\/span><\/p>\n

\u00a0 Na dal\u0161\u00edm obr\u00e1zku je vid\u011bt identickou Delta-Neutral pozici dvou Long Straddle<\/strong> v\u00a0r\u016fzn\u00fdch expirac\u00edch s po\u0159\u00edzen\u00fdmi 37 x Long akciemi, ov\u0161em se simulac\u00ed, \u017ee cena akcie AAPL klesla o -5 USD<\/strong>.<\/span><\/p>\n

\"\"<\/p>\n

\u00a0\u00a0 Gamma<\/strong> m\u00e9 po\u0159\u00edzen\u00e9 Delta-Neutral pozice<\/strong> je samoz\u0159ejm\u011b tak\u00e9 op\u011bt kladn\u00e1 (+24)<\/strong>. Pokud cena podkladu \u2013 akcie AAPL bude klesat \u2013 bude klesat tak\u00e9 Delta cel\u00e9 pozice a bude se propadat do negativn\u00edho p\u00e1sma. Proto\u017ee se stane Delta z\u00e1pornou \u2013 bude se cel\u00e1 pozice chovat, jako bych byl \u201ecel\u00e1 pozice Short\u201c, \u010d\u00edm v\u011bt\u0161\u00ed propad – t\u00edm bude v\u00fdsledek a v\u00fdkonnost op\u010dn\u00edho portfolia lep\u0161\u00ed, a to tak\u00e9 nen\u00ed v\u016fbec \u0161patn\u00e9. \u00a0M\u00e1m takov\u00e9 op\u010dn\u00ed portfolio, kter\u00e9 p\u0159i downtrendu vyd\u011bl\u00e1v\u00e1<\/strong>. Na prav\u00e9 stran\u011b obr\u00e1zku je vid\u011bt simulace m\u00e9 Delta-Neutral pozice s\u00a0kladnou Gamma p\u0159i downtendu o -5 USD<\/strong> na akci\u00edch AAPL. Delta klesla z nuly na hodnotu -103 a celkov\u00fd profit je +240 USD. Proto\u017ee to je op\u011bt jednoduch\u00e1 a \u010diteln\u00e1 pozice, tak v\u00edm, \u017ee vyd\u011bl\u00e1vaj\u00ed ob\u011b Long Put, Long akcie a ob\u011b Long Call na hodnot\u011b ztr\u00e1cej\u00ed.<\/span><\/p>\n

Gamma Negativn\u00ed<\/span><\/strong><\/h2>\n

\u00a0 Jin\u00fd obr\u00e1zek pak mohu vid\u011bt, pokud sestroj\u00edm portfolio, kter\u00e9 bude sice Delta-Neutral<\/strong> tak\u00e9, ale bude m\u00edt ve sv\u00e9m sou\u010dtu Gamma negativn\u00ed (z\u00e1pornou)<\/strong>. Takov\u00e9 portfolio je vid\u011bt na obr\u00e1zku, jedn\u00e1 se o pozice dvou Short Straddle na ATM opc\u00edch ve dvou nejbli\u017e\u0161\u00edch expirac\u00edch. Budu pak simulovat pohyb podkladov\u00e9 akcie sm\u011brem nahoru o +5 USD.<\/span><\/p>\n

\"\"<\/p>\n

\u00a0 \u00a0Abych op\u010dn\u00ed pozici s\u00a0Delta +37 neutralizoval, musel jsem po\u0159\u00eddit -37 x Short akcie AAPL. Z\u00a0lev\u00e9 \u010d\u00e1sti obr\u00e1zku je pak patrn\u00e9, \u017ee Delta je nulov\u00e1 ale Gamma je z\u00e1porn\u00e1 na hodnot\u011b -24<\/strong>. Pokud cena podkladu poroste, Delta cel\u00e9 pozice bude z\u00e1porn\u00e1<\/strong>, \u010d\u00edm v\u00edce poroste, t\u00edm v\u00edce bude negativn\u00ed. Cel\u00e9 portfolio se d\u00edky z\u00e1porn\u00e9 Delta bude chovat, jako bych byl \u201ecel\u00e9 portfolio Short\u201c, a to p\u0159i r\u016fstu ceny podkladu je p\u0159esn\u011b to, co nem\u016f\u017eu pot\u0159ebovat. Na prav\u00e9 stran\u011b obr\u00e1zku je simulace r\u016fstu ceny akcie AAPL o +5 USD<\/strong>, kdy je vid\u011bt, \u017ee p\u016fvodn\u011b nulov\u00e1 Delta je nyn\u00ed na hodnot\u011b -145 a celkov\u00e1 ztr\u00e1ta je -388 USD. Proto\u017ee to je jednoduch\u00e1 a \u010diteln\u00e1 pozice, tak v\u00edm, \u017ee sice vyd\u011bl\u00e1vaj\u00ed ob\u011b Short Put, ale prod\u011bl\u00e1vaj\u00ed ob\u011b Short Call a tak\u00e9 m\u00e9 Short akcie.<\/span><\/p>\n

\u00a0\u00a0 Nejinak tomu bude p\u0159i p\u00e1du podkladov\u00e9 akcie a Delta-Neutral<\/strong> pozic\u00ed se z\u00e1pornou Gamma<\/strong>. Na n\u00ed\u017ee uveden\u00e9m obr\u00e1zku je vid\u011bt, jak by to dopadlo v\u00a0takov\u00e9m p\u0159\u00edpad\u011b, kdy budu simulovat pohyb akci\u00ed sm\u011brem dol\u016f o -5 USD.<\/span><\/p>\n

\"\"<\/p>\n

\u00a0 \u00a0K\u00a0neutralizaci pozice jsem musel pou\u017e\u00edt -7 x Short akcie AAPL. Delta-Neutral<\/strong> vytvo\u0159en\u00e1 pozice na lev\u00e9 stran\u011b obr\u00e1zku m\u00e1 Gamma z\u00e1pornou na hodnot\u011b -25.<\/strong> Na prav\u00e9 stran\u011b obr\u00e1zku je op\u011bt simulace pohybu ceny akcie – p\u00e1du ceny akcie AAPL o –5 USD,\u00a0<\/strong>kdy je vid\u011bt, \u017ee Delta se z nuly vy\u0161plhala na \u00a0+112 a celkov\u00e1 ztr\u00e1ta je -224 USD. Z\u00e1porn\u00e1 Gamma zp\u016fsobila, \u017ee Delta cel\u00e9 pozice se st\u00e1v\u00e1 pozitivn\u00ed a roste.<\/strong> \u010c\u00edm v\u00edce bude cena akcie AAPL klesat, t\u00edm v\u00edce bude Delta \u201ekladn\u011bj\u0161\u00ed\u201c. Cel\u00e1 op\u010dn\u00ed pozice se p\u0159i padaj\u00edc\u00edm cen\u011b AAPL bude chovat jako bych m\u011bl po\u0159\u00edzeno \u201ecel\u00e9 portfolio Long\u201c, a to je opravdu to, co si nemohu p\u0159\u00e1t. Proto\u017ee to je op\u011bt velmi jednoduch\u00e1 a \u010diteln\u00e1 pozice, tak v\u00edm, \u017ee sice vyd\u011bl\u00e1vaj\u00ed ob\u011b Short Call a tak\u00e9 m\u00fdch sedm Short akci\u00ed, ale prod\u011bl\u00e1vaj\u00ed ob\u011b Short Put.<\/span><\/p>\n

\u00a0\u00a0 Toto jsou velmi jednoduch\u00e9 p\u0159\u00edklady, kdy jsem schopen odhadnout, jak\u00e9 jednotliv\u00e9 pozice m\u00e1m (nap\u0159\u00edklad 2 x Long Straddle). Pokud ale m\u00e1m slo\u017eit\u011bj\u0161\u00ed multikontraktn\u00ed pozice, potom mohu vyhodnocen\u00edm celkov\u00e9 Gamma z\u00edskat kvalitn\u011bj\u0161\u00ed odpov\u011b\u010f na jednoduchou ot\u00e1zku, jak\u00e1 je expozice takov\u00e9ho komplexn\u011bj\u0161\u00edho portfolia v\u016f\u010di pohyb\u016fm na trhu podkladov\u00e9ho aktiva. Nemus\u00edm pak b\u00fdt n\u011bjak\u00fd \u201ev\u011bdec\u201c, abych nap\u0159\u00edklad p\u0159i pohledu na v\u011bt\u0161\u00ed portfolio na obr\u00e1zku n\u00ed\u017ee mohl ihned posoudit, \u017ee jak\u00fdkoliv v\u00fdrazn\u011bj\u0161\u00ed pohyb takov\u00e9ho Delta-Neutral portfolia nebude nic, co bych s\u00a0nad\u0161en\u00edm uv\u00edtal, proto\u017ee toto portfolio m\u00e1 velmi z\u00e1pornou Gamma.<\/span><\/p>\n

\"\"<\/p>\n

\n

\u00a0 \u00a0Z\u00a0v\u00fd\u0161e uveden\u00fdch p\u0159\u00edklad\u016f by pak m\u011bl vyplynout celkem jednozna\u010dn\u00fd z\u00e1v\u011br, \u017ee vytv\u00e1\u0159et pozice nebo portfolia se z\u00e1pornou Gamma jsou k\u00a0ni\u010demu a \u017ee by pak tedy pouze sta\u010dilo b\u00fdt Delta-Neutral s\u00a0pozitivn\u00ed Gamma a nemohl bych vlastn\u011b v\u016fbec prod\u011blat. Nen\u00ed to pravda. V\u00fd\u0161e uveden\u00e9 p\u0159\u00edklady jsou vytvo\u0159en\u00e9 \u201ev izolaci\u201c, kde op\u010dn\u00ed pozice nebyly vystaveny jin\u00fdm tr\u017en\u00edm vliv\u016fm<\/strong>, zejm\u00e9na vlivu Implied Volatility<\/strong> a b\u011bhu \u010dasu<\/strong>. Pozice, kter\u00e9 jsou sice Gamma pozitivn\u00ed, jako nap\u0159\u00edklad Long Straddle pr\u00e1v\u011b trp\u00ed t\u011bmito dal\u0161\u00edmi vlivy, kter\u00e9 je nutno p\u0159i jejich dr\u017een\u00ed eliminovat. Nedostatek pohybu na podkladu, rozpad \u010dasov\u00e9 slo\u017eky ceny op\u010dn\u00edho kontraktu v\u00a0z\u00e1vislosti na zkracuj\u00edc\u00ed se dob\u011b \u017eivota opce nebo pokles volatility jsou okolnosti, kter\u00e9 takov\u00e9 pozice t\u00e1hnou do ztr\u00e1t a je t\u0159eba se nau\u010dit je eliminovat. Naopak n\u011bkter\u00e9 jednodu\u0161\u0161\u00ed, typicky Gamma negativn\u00ed pozice, jako nap\u0159\u00edklad Iron Condor, Credit Vertical Spread mohou z\u00a0takov\u00fdch efekt\u016f t\u011b\u017eit. Nepohyb, pokles Implied Volatility a rozpad \u010dasov\u00e9 slo\u017eky pr\u00e9mia mohou p\u0159isp\u011bt k\u00a0jejich mo\u017en\u00e9 profitabilit\u011b :c)<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\u00a0 \u00a0 K\u00a0dokonalej\u0161\u00edmu pochopen\u00ed z\u00e1vislosti ceny opce na pohybu podkladov\u00e9ho aktiva vyj\u00e1d\u0159en\u00e9ho \u0159eck\u00fdm p\u00edsmenem Delta, jak jsem jej popisoval v\u00a0\u010dl\u00e1nku Delta je zapot\u0159eb\u00ed hlub\u0161\u00edho ponoru do problematiky \u0159eck\u00fdch p\u00edsmen, kdy\u017e t\u00edm nejbli\u017e\u0161\u00edm p\u00edsmenem k\u00a0popisovan\u00e9 Delta je \u0159eck\u00e9 p\u00edsmeno Gamma. Tak jako Rumburak byl \u010darod\u011bj druh\u00e9 kategorie, tak tak\u00e9 na Gamma Continue Reading<\/i><\/a><\/p>\n","protected":false},"author":1,"featured_media":539,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[2],"tags":[17,26,32],"class_list":["post-522","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-opcni-zaklady","tag-black-scholes","tag-delta","tag-gamma"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/dobretrejdy.com\/wp-content\/uploads\/2017\/06\/180750308.jpg","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/s8zd8A-gamma","jetpack-related-posts":[{"id":3202,"url":"https:\/\/dobretrejdy.com\/?p=3202","url_meta":{"origin":522,"position":0},"title":"Delta Neutral – VI.","author":"dobretrejdy :c)","date":"4.5.2019","format":false,"excerpt":"\u00a0 \u00a0Je pond\u011bl\u00ed 25.3.2019 a j\u00e1 sleduji t\u0159et\u00ed d\u00edl \u010dtvrt\u00e9 s\u00e9rie kultovn\u00edho seri\u00e1lu Billions. Taylor rozepisuje na tabuli matematickou \u00favahu, kter\u00e1 demonstruje spletitost, slo\u017eitost a sofistikovanost vyd\u011bl\u00e1v\u00e1n\u00ed pen\u011bz na finan\u010dn\u00edch trz\u00edch. Snaha je pravd\u011bpodobn\u011b uk\u00e1zat, \u017ee pokud to nem\u00e1te v\u00a0hlav\u011b matematicky v\u00a0po\u0159\u00e1dku, nem\u00e1te \u0161anci se takov\u00fdm investorsk\u00fdm veli\u010din\u00e1m rovnat. \u00a0\u00a0\u2026","rel":"","context":"V \u201eOp\u010dn\u00ed kombinace\u201c","block_context":{"text":"Op\u010dn\u00ed kombinace","link":"https:\/\/dobretrejdy.com\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/05\/mean.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/05\/mean.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/05\/mean.jpg?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/05\/mean.jpg?resize=700%2C400&ssl=1 2x"},"classes":[]},{"id":574,"url":"https:\/\/dobretrejdy.com\/?p=574","url_meta":{"origin":522,"position":1},"title":"Vega","author":"dobretrejdy :c)","date":"2.7.2017","format":false,"excerpt":"\u00a0\u201e\u2026 p\u0159esto\u017ee se v\u017eilo, \u017ee popis faktor\u016f p\u016fsob\u00edc\u00edch na tvorbu op\u010dn\u00ed ceny jsme zvykl\u00ed ozna\u010dovat \u0159eck\u00fdmi p\u00edsmeny Delta, Gamma, Th\u00e9ta, Rho, Vega, tak mus\u00edm podotknout, \u017ee Vega nen\u00ed p\u00edsmenem \u0159eck\u00e9 abecedy\u2026\u201c, v\u00a0tomto smyslu v\u011bt\u0161inou b\u00fdvaj\u00ed uv\u00e1d\u011bny \u010dl\u00e1nky o \"Vega\" zab\u00fdvaj\u00edc\u00ed se vlivem zm\u011bny Implied Volatility na cenu op\u010dn\u00edho kontraktu, z\u0159ejm\u011b\u2026","rel":"","context":"V \u201eOp\u010dn\u00ed z\u00e1klady\u201c","block_context":{"text":"Op\u010dn\u00ed z\u00e1klady","link":"https:\/\/dobretrejdy.com\/?cat=2"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/07\/30.06.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/07\/30.06.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/07\/30.06.jpg?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/07\/30.06.jpg?resize=700%2C400&ssl=1 2x"},"classes":[]},{"id":551,"url":"https:\/\/dobretrejdy.com\/?p=551","url_meta":{"origin":522,"position":2},"title":"Th\u00e9ta","author":"dobretrejdy :c)","date":"25.6.2017","format":false,"excerpt":"\u00a0 Nev\u00edm, jestli Th\u00e9ta Kaz\u00ed Libu\u0161i, proto\u017ee nejsem vojvoda Krok, ale proto\u017ee jsem op\u010dn\u00ed obchodn\u00edk, tak v\u00edm, jak\u00a0Th\u00e9ta m\u016f\u017ee zkazit i velmi dobrou op\u010dn\u00ed pozici. V\u00a0p\u0159edch\u00e1zej\u00edc\u00edch \u010dl\u00e1nc\u00edch jsem ji\u017e popisoval, jak se m\u011bn\u00ed cena op\u010dn\u00edho kontraktu v\u00a0souvislost\u00ed s\u00a0pohybem podkladov\u00e9ho aktiva a jak tuto z\u00e1vislost zachycujeme pomoc\u00ed \u0159eck\u00e9ho p\u00edsmene Delta. V\u00a0n\u00e1vaznosti\u2026","rel":"","context":"V \u201eOp\u010dn\u00ed z\u00e1klady\u201c","block_context":{"text":"Op\u010dn\u00ed z\u00e1klady","link":"https:\/\/dobretrejdy.com\/?cat=2"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/06\/theta.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/06\/theta.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/06\/theta.jpg?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/06\/theta.jpg?resize=700%2C400&ssl=1 2x"},"classes":[]},{"id":2911,"url":"https:\/\/dobretrejdy.com\/?p=2911","url_meta":{"origin":522,"position":3},"title":"Delta Neutral – II.","author":"dobretrejdy :c)","date":"3.3.2019","format":false,"excerpt":"\u00a0 \u00a0Pro\u010d se v\u016fbec zab\u00fdvat ot\u00e1zkou Delta Neutrality, co mi p\u0159in\u00e1\u0161\u00ed a k\u00a0\u010demu vlastn\u011b v\u016fbec je, by mohl b\u00fdt z\u00e1kladn\u00ed dotaz po p\u0159e\u010dten\u00ed minul\u00e9ho \u010dl\u00e1nku. Jako op\u010dn\u00ed obchodn\u00edk, vybaven z\u00e1kladn\u00edmi znalostmi o tvorb\u011b ceny op\u010dn\u00edho kontraktu v\u00edm, \u017ee cena opce je vypo\u010d\u00edt\u00e1v\u00e1na matematick\u00fdmi modely, do kter\u00fdch vstupuj\u00ed jednotliv\u00e9 cenotvorn\u00e9 komponenty,\u2026","rel":"","context":"V \u201eOp\u010dn\u00ed kombinace\u201c","block_context":{"text":"Op\u010dn\u00ed kombinace","link":"https:\/\/dobretrejdy.com\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/03\/Delta-pozitive-1.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/03\/Delta-pozitive-1.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/03\/Delta-pozitive-1.jpg?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/03\/Delta-pozitive-1.jpg?resize=700%2C400&ssl=1 2x"},"classes":[]},{"id":308,"url":"https:\/\/dobretrejdy.com\/?p=308","url_meta":{"origin":522,"position":4},"title":"Delta","author":"dobretrejdy :c)","date":"4.5.2017","format":false,"excerpt":"\u00a0 \u00a0Delta je rozd\u00edl. Delta Praha \u2013 Brno je 200 (kilometr\u016f), o tom ale \u010dl\u00e1nek nebude. Bude to o Delta op\u010dn\u00edch kontrakt\u016f. Porozum\u011bn\u00ed Delta je jednou ze z\u00e1kladn\u00edch dovednost\u00ed, kterou by m\u011bl disponovat ka\u017ed\u00fd op\u010dn\u00ed trader. Toto jednoduch\u00e9 \u0159eck\u00e9 p\u00edsmeno ozna\u010duje m\u00edru z\u00e1vislosti ceny opce na pohybu podkladov\u00e9ho aktiva. Vyjdu\u2026","rel":"","context":"V \u201eOp\u010dn\u00ed z\u00e1klady\u201c","block_context":{"text":"Op\u010dn\u00ed z\u00e1klady","link":"https:\/\/dobretrejdy.com\/?cat=2"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/05\/Delta.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/05\/Delta.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/05\/Delta.jpg?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2017\/05\/Delta.jpg?resize=700%2C400&ssl=1 2x"},"classes":[]},{"id":3383,"url":"https:\/\/dobretrejdy.com\/?p=3383","url_meta":{"origin":522,"position":5},"title":"Delta Neutral \u2013 IX.","author":"dobretrejdy :c)","date":"30.6.2019","format":false,"excerpt":"\u00a0 \u00a0Zachycen\u00ed v\u00fdznamu \u0159eck\u00fdch p\u00edsmen, kter\u00e9 popisuj\u00ed vlastnosti m\u00e9ho op\u010dn\u00edho kontraktu, vypad\u00e1 jako velmi zaj\u00edmav\u00e1 discipl\u00edna a m\u016f\u017ee b\u00fdt pravd\u011bpodobn\u011b dobrou v\u00fdchoz\u00ed situac\u00ed pro m\u00e9 op\u010dn\u00ed obchody. Prozat\u00edm jsem se zab\u00fdval pouze vybran\u00fdm segmentem situac\u00ed, kdy jsem zejm\u00e9na op\u010dn\u00ed kontrakty nakupoval a pozoroval, jak\u00e9 jsou hodnoty vybran\u00fdch \u0159eck\u00fdch p\u00edsmen Delta,\u2026","rel":"","context":"V \u201eOp\u010dn\u00ed kombinace\u201c","block_context":{"text":"Op\u010dn\u00ed kombinace","link":"https:\/\/dobretrejdy.com\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/06\/main-1.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/06\/main-1.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/06\/main-1.jpg?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/dobretrejdy.com\/wp-content\/uploads\/2019\/06\/main-1.jpg?resize=700%2C400&ssl=1 2x"},"classes":[]}],"_links":{"self":[{"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=\/wp\/v2\/posts\/522","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=522"}],"version-history":[{"count":16,"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=\/wp\/v2\/posts\/522\/revisions"}],"predecessor-version":[{"id":1883,"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=\/wp\/v2\/posts\/522\/revisions\/1883"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=\/wp\/v2\/media\/539"}],"wp:attachment":[{"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=522"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=522"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dobretrejdy.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=522"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}