Zaštita od rizika u nepotpunim diskretnim modelima financijskih tržišta

master's thesis

• Title (english): Risk protection in incomplete discrete models of financial markets
• Author: Neva Nakić
• Mentor: Vanja Wagner (mentor)
• Committee member: Vanja Wagner (predsjednik povjerenstva)
• Committee member: Zvonimir Bujanović (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-12-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu predstavljamo jedan od pristupa problemu zaštite od rizika u nepotpunim diskretnim modelima financijskih tržišta; u središtu je minimizacija kvadratne greške zaštite. U prvom poglavlju uvodimo osnovne pojmove kako bismo definirali model financijskog tržišta u diskretnom vremenu. U drugom poglavlju iznosimo lokalnu verziju problema minimizacije kvadratne greške. Problem promatramo kao proces sekvencijalne regresije te dolazimo do postupka rekurzivne konstrukcije strategije minimizacije lokalnog rizika. Rješenje problema također uključuje ortogonalnu dekompoziciju danog slučajnog zahtjeva. Nadalje, u trećem poglavlju proučavamo minimalne martingalne mjere. Važno svojstvo takvih ekvivalentnih martingalnih mjera jest da se vrijednost portfelja generirana strategijom minimizacije lokalnog rizika može opisati kao martingal uvjetnih očekivanja danog slučajnog zahtjeva. Također, uvodimo pojam preostalog uvjetnog rizika. Na kraju, u četvrtom poglavlju promatramo problem zaštite od rizika s obzirom na optimalnu varijancu gdje želimo minimizirati globalnu kvadratnu grešku. Pokazujemo da se rješenja lokalnog i globalnog problema podudaraju ako je ishodišna mjera martingalna.

Abstract (english)

In this thesis, we present one of the approaches to the problem of hedging in incomplete financial market models in discrete time; the focus is on minimizing the quadratic hedging error. In the first chapter, we introduce fundamental concepts used to define the financial market model in discrete time. In the second chapter, we present a local version of the minimization problem. We view the problem as a sequential regression procedure and we develop a process of recursive construction of a locally risk-minimizing strategy. The solution to the problem also involves an orthogonal decomposition of a given contingent claim. Furthermore, in the third chapter, we study the minimal martingale measures. An important characteristic of such equivalent martingale measures is that the value process generated by a locally risk-minimizing strategy can be described as the martingale of conditional expectations of the given contingent claim. We also introduce the concept of the remaining conditional risk. Finally, in the fourth chapter, we analyze the problem of variance-optimal hedging where we try to minimize the global quadratic hedging error. We show that the solutions to the local and global problem coincide if the underlying measure is itself a martingale measure.

• Keywords: diskretni modeli financijskih tržišta; minimizaciae kvadratne greške; sekvencijalna regresija; minimalna martingalna mjera; minimizacija lokalnog rizika
• Keywords (english): discrete models of financial markets; minimization of square errors; sequential regression; minimum martingale measure; minimizing local risk
• Language: croatian
• URN:NBN: urn:nbn:hr:217:017084
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-12-15 12:33:30