Completeness of Financial Market Discrete Time Models

Abstract

U ovom diplomskom radu cilj nam je bio analizirati potpunost modela financijskog tržišta u diskretnom vremenu. Kako bismo uopće mogli definirati potpunost matematičkog modela financijskog tržišta u diskretnom vremenu, prvo smo opisati financijsko tržište u diskretnom vremenu, financijske instrumente (osnovne i izvedene) kojima se trguje na financijskom trži- štu, te njihove cijene i povrate. Pritom smo od važnijih pojmova definirali pojmove kao što su portfelj, europska call i put opcija, samofinancirajuća i dopustiva strategija trgovanja, vjerojatnost neutralna na rizik, te arbitraža i martingal. Zatim smo opisali neke jednoperiodne i višeperiodne modele u diskretnom vremenu, koji koriste dva financijska instrumenta, od kojih je jedan nerizičan (novac u domaćoj valuti), a drugi rizičan (dionica). Nakon toga definirali smo slučajni zahtjev, te što treba biti zadovoljeno da bi taj slučajni zhatjev bio dostižan. Na temelju dobivenih rezultata definirali smo potpun model financijskog tržišta u diskretnom vremenu kao model tržišta bez arbitraže na kojemu je svaki slučajni zahtjev dostižan. Za svaki opisani model iznijeli smo neke zahtjeve koje ti modeli trebaju ispunjavati da bi bili potpuni.This thesis aim was to analyse the completeness of financial market discrete time models. In order to even be able to define the completeness of financial market discrete time models, we first described the financial market in discrete time, financial instruments (basic and derived) that are traded on financial markets, and its prices and returns. In doing so, we of the important terms defined terms such as the portfolio, a European call and put options, self-financing and admissible trading strategy, the risk-neutral measure, arbitrage and martingale. Then we described some one-period and multi-period models in discrete time, using two financial instruments, one of which is risk free (money in local currency), and the other risk (stocks). Then we defined a contingent claim, and what needs to be satisfied that this contingent claim was admissible. Based on the obtained results, we have defined a complete model of the fi- nancial market in discrete time as a model market without arbitrage in which each contingent claim is admissible. For each described model we have put forward some demands that these models must fulfill in order to be complete