Određivanje cijena izvedenica Monte Carlo simulacijama

master's thesis

• Title (english): Pricing of derivatives of Monte Carlo simulations
• Author: Sara Rubčić
• Mentor: Azra Tafro (mentor)
• Committee member: Azra Tafro (predsjednik povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Committee member: Franka Miriam Brückler (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-12-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES

Abstract

Monte Carlo simulacije su skup algoritama koji pomoću velikog broja izračuna i ponavljanja predviđaju ponašanje složenih slučajnih sustava. Pomoću njih procjenjujemo vrijednosti neke nepoznate varijable koristeći principe statističkog zaključivanja. Monte Carlo metode najčešće se koriste za rješavanje tri klase problema: integraciju, optimizaciju i generiranje slučajnih uzoraka iz vjerojatnosne distribucije. Problem kojim se bavimo u ovom radu je određivanje cijena izvedenica korištenjem Monte Carlo simulacija. Kako cijene izvedenica pod određenim pretpostavkama možemo računati kao očekivanje, procjenjujemo ih Monte Carlo integracijom. Koristimo Black-Scholes-Mertonov model financijskog tržišta u kojem cijene dionica modeliramo geometrijskim Brownovim gibanjem. Definiramo europsku call opciju, azijsku opciju i opciju izbora te pomoću Monte Carlo simulacija računamo cijene tih izvedenica. Kako bismo analizirali rizike na financijskim tržištima, definiramo Grke. Za računanje Grka Monte Carlo simulacijama koriste se posebne metode. Ukratko smo objasnili metodu izračuna po putu i pomoću nje simulirali izračun delta zaštite za portfelj koji se sastoji od europskih call opcija.

Abstract (english)

Monte Carlo simulations are a set of algorithms which use a large number of calculations and repetitions to predict the behavior of complex random systems. We use them to estimate the values of an unknown variable using the principles of statistical inference. Monte Carlo methods are most commonly used to solve three classes of problems: integration, optimization and generation of random samples from a probability distribution. The problem we address in this thesis is the pricing of the derivatives using Monte Carlo simulations. As the prices of derivatives under certain assumptions can be calculated as an expectation, we estimate them by Monte Carlo integration. We use the Black-Scholes- Merton financial market model in which we model stock prices by geometric Brownian motion. We define the European call option, the Asian option and the Chooser option, and using Monte Carlo simulations we calculate the prices of these derivatives. To analyze the risks in the financial markets, we define the Greeks. Special methods are used to calculate the Greeks using Monte Carlo simulations. We briefly explain the pathwise method and use it to simulate the calculation of delta hedging for a portfolio consisting of European call options.

• Keywords: Monte Carlo metode; složeni slučajni sustav; Black-Scholes-Mertonov model financijskog tržišta; Brownovo gibanje
• Keywords (english): Monte Carlo methods; complex random systems; Black-Scholes- Merton financial market model; Brownian motion
• Language: croatian
• URN:NBN: urn:nbn:hr:217:801847
• 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: 2021-02-24 12:31:22