Konvergencija cijena opcija s barijerom u binomnom modelu

master's thesis

• Author: Domagoj Demeterfi
• Mentor: Zoran Vondraček (mentor)
• Committee member: Zoran Vondraček (predsjednik povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-04-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Opcija s barijerom je egzotična opcija čija isplata ovisi o tome je li cijena temeljne imovine na koju je ta opcija napisana prešla određenu unaprijed dogovorenu razinu (barijeru) u određenom vremenskom razdoblju. U ovom diplomskom radu određujemo cijene europskih call opcija s barijerom u binomnom modelu s \(n\) perioda (CRR modelu) te proučavamo brzinu konvergencije tih cijena u odgovarajuće cijene iz Black-Scholes-Mertonovog modela kada broj perioda \(n\) teži u beskonačnost. Pokazujemo da su, općenito, greške reda \(1/\sqrt{n}\) te dajemo eksplicitne formule za koeficijente uz \(1/\sqrt{n}\) i \(1/n\) u asimptotskom razvoju greške (koji ovise o poziciji barijere i cijene izvršenja u binomnoj rešetci). Nadalje, pokazujemo kako poznavanje eksplicitnih formula za te koeficijente omogućuju da se prikladnim odabirom broja perioda \(n\) ubrza konvergencija, tj. da je za prigodno odabrane \(n\) greška reda \(1/n\). Na kraju dobivene rezultate ilustriramo i provjeravamo numeričkim primjerima.

Abstract (english)

Barrier option is an exotic option whose payoff depends upon underlying asset's price reaching a certain preset barrier level during a certain period of time. In this Master's thesis we price European barrier call options in \(n\)-period binomial model (CRR model) and study the rate of convergence of these prices to the prices given by the Black-Scholes-Merton model as the number of periods \(n\) tends to infinity. We show that, in general, the error is of order \(1/\sqrt{n}\) and give explicit formulas for the coefficients of \(1/\sqrt{n}\) and \(1/n\) in the asymptotic expansion of the error (which depend on the positions of the barrier and strike in the binomial lattice). Furthermore, we show how knowing explicit formulas for these coefficients enables us to speed up the convergence by choosing \(n\) in an appropriate way, i.e. we show that for appropriately chosen \(n\) error is of order \(1/n\). Lastly, we illustrate and verify results with numerical examples.

• Keywords: opcija s barijerom; CRR model; Black-Scholes-Mertonov model
• Keywords (english): barrier option; CRR model; Black-Scholes-Merton model
• Language: croatian
• URN:NBN: urn:nbn:hr:217:153094
• 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: 2019-07-18 11:37:22