Original title: Spojité modely trhu se stochastickou volatilitou
Translated title: Continuous market models with stochastic volatility
Authors: Petrovič, Martin ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee)
Document type: Master’s theses
Year: 2018
Language: slo
Abstract: [eng] [cze]
Vilela Mendes et al. (2015), based on the discovery of long-range dependence in the volatility of stock returns, proposed a stochastic volatility continuous mar- ket model where the volatility is given as a transform of the fractional Brownian motion (fBm) and studied its No-Arbitrage and completeness properties under va- rious assumptions. We investigate the possibility of generalization of their results from fBm to a wider class of Hermite processes. We have reworked and completed the proofs of the propositions in the cited article. Under the assumption of indepen- dence of the stock price and volatility driving processes the model is arbitrage-free. However, apart from a case of a special relation between the drift and the volatility, the model is proved to be incomplete. Under a different assumption that there is only one source of randomness in the model and the volatility driving process is bounded, the model is arbitrage-free and complete. All the above results apply to any Hermite process driving the volatility. 1 Vilela Mendes a kol. (2015) na základe zistenia, že volatilita výnosov akcií vykazuje dlhú pamäť, navrhujú spojitý trhový model so stochastickou volatilitou, ktorá je daná ako transformácia frakcionálneho Brownovho pohybu (fBm), a za rôznych podmienok skúmajú, či pripúšťa arbitráž a je úplný. Nás zaujíma, či ich závery možno z fBm zovšeobecniť na širšiu triedu hermitovských procesov. Prepracovali a doplnili sme dôkazy tvrdení z citovaného článku. Za predpokladu nezávislosti riadiacich procesov ceny a volatility model nepripúšťa arbitráž, avšak okrem prípadu špeciálneho vzťahu medzi driftom a volatilitou sme dokázali, že nie je úplný. Za iných predpokladov, keď v modeli je len jeden zdroj šumu a proces riadiaci volatilitu ohraničíme, model nepripúšťa arbitráž a je úplný. Všetky tieto výsledky ostávajú v platnosti, ak volatilitu riadi ľubovoľný hermitovský proces. 1
Keywords: equivalent martingale measure (EMM); fractional Brownian motion; fundamental theorems of asset pricing; Hermite processes; Stochastic volatility; ekvivalentná martingalová miera (EMM); frakcionálny Brownov pohyb; fundamentálne vety oceňovania; hermitovské procesy; Stochastická volatilita

Institution: Charles University Faculties (theses) (web)
Document availability information: Available in the Charles University Digital Repository.
Original record: http://hdl.handle.net/20.500.11956/101099

Permalink: http://www.nusl.cz/ntk/nusl-387019


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Universities and colleges > Public universities > Charles University > Charles University Faculties (theses)
Academic theses (ETDs) > Master’s theses