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Modely Stochastické Volatility MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV MATEMATIKY A STATISTIKY MODELY STOCHASTICKÉ VOLATILITY Diplomová práce Martin Diviš Vedoucí práce: Mgr. Ondřej Pokora, Ph.D. Brno 2015 Bibliografický záznam Autor: Bc. Martin Diviš Přírodovědecká fakulta, Masarykova univerzita Ústav matematiky a statistiky Název práce: Modely stochastické volatility Studijní program: Matematika Studijní obor: Finanční matematika Vedoucí práce: Mgr. Ondřej Pokora, Ph.D. Akademický rok: 2014/2015 Počet stran: IX + 69 Klíčová slova: Evropská opce, volatilita, Blackův-Scholesův model, Wienerův proces, CIR proces, modely stochastické volatility, kalibrace Hestonova modelu, Lévyho procesy Bibliographic Entry Author Bc. Martin Diviš Faculty of Science, Masaryk University Department of Mathematics and Statistics Title of Thesis: Stochastic volatility models Degree programme: Mathematics Field of Study: Financial Mathematics Supervisor: Mgr. Ondřej Pokora, Ph.D. Academic Year: 2014/2015 Number of Pages: IX + 69 Keywords: European option, volatility, Black-Scholes model, Wiener process, CIR process, stochastic volatility models, calibration of the Heston model, Lévy process Abstrakt V diplomové práci se zabýváme oceňováním evropských opcí a jejich závislostí na volatilitě. Blackův-Scholesův model má mnoho slabin. Největší slabinou je předpoklad konstantní volatility. V diplomové práci jsou popsány typy volatilit, které se nevyvíjí konstantně. Zabýváme se zejména stochastickou volatilitou. Ta je sice matematicky nejkomplikovanější, ale také nejpřesnější. Popisujeme modely stochastické volatility, například Hestonův model, 3/2 model či SABR model. Tyto modely si dokážou poradit s častými problémy při oceňování opcí jako je například volatily smile či silné chvosty reálných pravděpodobnostních rozdělení. Nejslavnější Hestonův model kalibrujeme v Excelu. Také se zaměřujeme na Lévyho procesy, které si poradí se skoky podkladového aktiva. Abstract Throughout this diploma thesis we deal with the valuation of European options and their dependence on volatility. The Black-Scholes model has many weaknesses. The biggest weakness is the forecast of the constant volatility. In this diploma thesis, there are described several types of volatility, which do not develop constantly. We deal particularly with the stochastic volatility. On the one hand, it is the most complex from the mathematic point of view, but on the other hand the most accurate. We describe stochastic volatility model, for instance Heston model, 3/2 model or SABR model. These models are able to deal with frequent problems during the valuation of options such as volatility smile or other fat tails of real probability distributions. The most famous Heston model is calibrated in Excel. Also, we focus on Levy's processes, which deal with the jumps of an underlying asset. Poděkování Na tomto místě bych rád poděkoval doktoru Ondřeji Pokorovi za skvělé vedení diplomové práce. Vyzdvihl bych jeho ochotu, cenné rady a motivující přístup. Za výbornou výuku předmětů finanční matematiky bych chtěl poděkovat docentu Martinu Kolářovi a doktoru Pokorovi. Nabyté znalosti z jejich předmětů jsem zúročil nejen při psaní diplomové práce. Rád bych poděkoval také svým rodičům za podporu během studia. Prohlášení Prohlašuji, že jsem svoji diplomovou práci vypracoval samostatně s využitím informačních zdrojů, které jsou v práci citovány. Brno 14. prosinec 2014 ……………………………… Martin Diviš Obsah Úvod ......................................................................................................................... IX 1. Ekonomické a matematické pojmy ......................................................... 1 1.1. Finanční deriváty ................................................................................... 1 1.2. Matematické věty a definice.............................................................. 4 2. Blackův-Scholesův model ........................................................................ 12 2.1. Předpoklady Blackova-Scholesova modelu ............................. 12 2.2. Odvození Blackova-Scholesova modelu .................................... 13 2.3. Garmanův-Kohlhagenův model .................................................... 21 3. Oceňování finančních derivátů .............................................................. 23 3.1. Arbitrážní teorie oceňování ........................................................... 23 3.2. Zajištění................................................................................................... 24 3.3. Risk neutrální míra ............................................................................ 25 3.4. Metoda Monte Carlo .......................................................................... 27 3.4.1. Generátory náhodných čísel ................................................ 31 4. Volatilita .......................................................................................................... 33 4.1. Historická volatilita ........................................................................... 33 4.2. Implikovaná volatilita ....................................................................... 34 4.3. Deterministická volatilita ................................................................ 39 4.4. Stochastická volatilita ....................................................................... 39 4.4.1. Mean-reverting proces .......................................................... 40 4.4.2. CIR proces ................................................................................... 41 4.4.3. Ho-Lee model ............................................................................ 42 5. Metody stochastické volatility ............................................................... 43 5.1. Hestonův model .................................................................................. 44 5.1.1. Hestoův model se skoky ........................................................ 47 5.2. GARCH model ....................................................................................... 48 5.3. 3/2 model .............................................................................................. 49 5.4. Steinův model ....................................................................................... 49 5.5. CEV model .............................................................................................. 50 5.6. SABR volatility model ....................................................................... 51 5.7. Chenův model ....................................................................................... 53 5.8. Shrnutí ..................................................................................................... 54 6. Výpočet a kalibrace Hestnova modelu ............................................... 55 VII 7. Lévyho procesy ............................................................................................ 60 7.1. Mertonův jump diffusion model ................................................... 64 Závěr ....................................................................................................................... 66 Seznam použité literatury ............................................................................. 67 VIII Úvod Na počátku sedmdesátých let dvacátého století dochází k přelomu v oceňování finančních derivátů. Pánové Black, Scholes a Merton publikují Blackův-Scholesův model, za který později obdrží Myron Scholes a Robert Merton Nobelovou cenu za ekonomii. Blackův-Scholesův model je skutečně impozantní model, který většinou dokáže odhadnout správnou cenu evropské opce. Avšak v obdobích finančních krizí, kdy je na trhu vysoká volatilita, model příliš nefunguje. V diplomové práci se zaměříme na slabiny Blackova-Scholesova modelu. Hlavní slabinou je předpoklad, že volatilita se vyvíjí konstantně. Další slabinou je předpoklad lognormálních cen podkladového aktiva, který ne zcela správně popisuje reálné pravděpodobnostní rozdělení v extremních hodnotách. Blackův-Scholesův model také nedokáže popsat skoky ve vývoji podkladového aktiva. Také se zaměříme na důležitý předpoklad risk neutrality a s ním spojenou nutnost zajištění. Na začátků práce si popíšeme finanční deriváty a zadefinujeme si důležité matematické pojmy, které se využívají při oceňování finančních derivátů. V druhé kapitole si odvodíme slavný Blackův-Scholesův model. Dále si popíšeme rizikově neutrální oceňování a zajištění. Na konci třetí kapitoly se budeme věnovat numerické metodě Monte Carlo, která má veliké praktické využití nejen při oceňování finančních derivátů. Čtvrtá kapitola bude patřit volatilitě. Popíšeme si historickou, implikovanou, deterministickou a hlavně stochastickou volatilitu. Ukážeme si na grafech, proč není vhodné, aby se využívala konstantní volatilita. Tyto grafy zachycují volatility smile, skew, term structure a surface. Také se zaměříme na modelování stochastické volatility pomocí mean-reverting procesu, které ještě vylepšíme CIR procesem. Pátá kapitola je pojmenována stejně jako diplomová práce tzn. Modely stochastické volatility. Popíšeme si celkově sedm modelů. Naši pozornost zaměříme na nejslavnější Hestonův model a ukážeme si, jak si poradí se slabinami Blackova-Scholesova modelu. Také popíšeme výhody ostatních modelů stochastické volatility a zmíníme jejich praktické využití. Zaměříme se samozřejmě i na nevýhody těchto komplikovaných modelů. S matematickou složitostí a přesností bohužel souvisí i obtížná kalibrace. S ní si pokusíme poradit v předposlední kapitole při kalibraci Hestonova modelu, která
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