Vrednotenje evropskih opcij s procesi razvejanja : delo diplomskega seminarja

Đekanović, Stefan (author), Bernik, Janez (mentor)

Proces razvejanja je slučajni proces, ki opisuje razvoj populacije. Začetki teorije procesov razvejanja segajo v drugo polovico 19. stoletja, ko sta Francis Galton in Henry William Watson reševala problem o verjetnosti izumrtja posameznega priimka. Kasneje se je teorija razširila na področje biologije, fizike, kemije in drugih ved. Najbolj enostaven tip procesov razvejanja je Galton-Watsonov proces, ki predpostavi, da so objekti v generaciji med seboj neodvisni, imajo enako porazdelitev števila potomcev, ki jo označimo s slučajno spremenljivko $X$, in generirajo sebi enake potomce. V teoriji procesov razvejanja nas zanima predvsem porazdelitev števila vseh objektov $Z_n$ v n-ti generaciji in verjetnost izumrtja procesa. Izkaže se, da pri pogoju $E(X)\leq 1$ proces skoraj gotovo izumre, pri pogoju $E(X)>1$ pa proces preživi z neko pozitivno verjetnostjo. \\ Leta 1996 Thomas Wake Epps uporabi Galton-Watsonov proces $(Z_n)_{n\in \mathbb{N}_0}$, ki ima Poissonov proces $(N_t)_{t\geq 0}$ kot subordinator, za modeliranje cen delnic. Pod pogoji podobnimi kot pri izpeljavi znane Black-Scholesove formule izpeljemo natančno formulo za premijo evropske nakupne opcije s pomočjo naključno indeksiranega procesa $(Z_{N_t})_{t\geq 0}$.

• Language: Slovenian
• Keywords: Proces razvejanja, Galton-Watsonov proces, rodovna funkcija, naključno indeksiran proces razvejanja, evropska opcija, Black-Scholesova formula
• Work type: Bachelor thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-120194
• UDC: 519.2
• COBISS.SI-ID: 58554627
• Publication date in RUL: 17.09.2020

Secondary language

• Language: English
• Title: European options pricing with branching processes

Abstract

Branching process is a stochastic process which describes development of a population. The beginnings of the theory of branching processes date back to the second half of the 19th century when Francis Galton and Henry William Watson tried to solve the problem of the probability of family name extinction. Later the theory spread to the fields of biology, physics, chemistry and other sciences. The most simple type of branching processes is the Galton-Watson process which assumes that the objects in the generation are independent of each other, have the same offspring distribution denoted by a random variable $X$ and they produce offsprings which are the same type as their parents. In the theory of branching processes we are mainly interested in the distribution of the number of all objects $Z_n$ in the nth generation and the probability of the process extinction. It turns out that under the condition $E(X)\leq 1$ the process almost surely becomes extinct and under the condition $E(X)>1$ the process survives with some positive probability. \\ In 1996 Thomas Wake Epps used the Galton-Watson process $(Z_n)_{n\in \mathbb{N}_0}$ with the Poisson process $(N_t)_{t\geq 0}$ as a subordinator to model stock prices. Under conditions similiar to the derivation of the well-known Black-Scholes formula we derive the exact formula for the premium of the European call option using a randomly indexed process $(Z_{N_t})_{t\geq 0}$.

• Keywords: Branching processes, Galton-Watson process, probability generating function, randomly indexed branching process, European option, Black-Scholes formula