Podrazumijevana volatilnost

master's thesis

• Author: Blanka Horvatić
• Mentor: Nela Bosner (mentor)
• Committee member: Nela Bosner (predsjednik povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-04-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U diplomskom radu bavimo se temom podrazumijevane volatilnosti i njenom primjenom u financijama. U uvodnom poglavlju su pregledi modela i njihovih osnovnih značajki potrebnih za daljnji rad, npr. Black-Scholesov model, te Cox-Ross-Rubinsteinov model binomnih stabala. Prvo poglavlje daje kratki uvod u podrazumijevanu volatilnost, procijenjenu volatilnost cijene vrijednosnih papira, najčešće korištenu kod određivanja cijena opcija. Kako bi se odredila, potrebno je riješiti Black-Scholesovu formulu koristeći uočene cijene opcija za konstantnu volatilnost \(\sigma\). Značajka podrazumijevane volatilnosti je njena promjena obzirom na cijene izvršenja, takozvani "smile". Na kraju poglavlja koriste se dvije varijante glavnih komponenti. To su analiza glavnih komponenti za određivanje glavnih komponenti kod kretanja volatilnosti obzirom na vrijeme do dospijeća, te opća analiza glavnih komponenti koja omogućuje istovremeno modeliranje podrazumijevane volatilnosti obzirom na vrijeme dospijeća i cijenu izvršenja. Zatim se uvodi pojam podrazumijevanih binomnih stabala (IBT), metode koja ne ignorira "smile" efekt. Empirijske činjenica da podrazumijevana volatilnost pada s cijenom izvršenja, te raste s vremenom do dospijeća opcija bolje je opisana ovom strukturom nego standardnim CRR stablom. Osnovna upotreba IBT je u zaštiti od rizika te određivanju podrazumijevane vjerojatnosne distribucije (eng. state price density, SPD). Dana su dva algoritma za konstrukciju IBT-a. Prvo se upoznajemo s algoritmom Dermana i Kanija, a potom s onim Barlea i Cakicia. U posljednjem poglavlju govorimo o SPD-u te načinima određivanja istog. Osnovna ideja je da je cijena vrijednosnog papira jednaka očekivanoj sadašnjoj vrijednosti budućih isplata, a pritom je očekivanje uzeto obzirom na vjerojatnost neutralnu na rizik. Govorimo o nearbitražnom pristupu određivanja SPD-a. Prednost je što nema pretpostavki na kretanje vezane imovine. Prvo dajemo ideje za poluparametarsku procjenu funkcije cijene call opcije te potom predlažemo načine određivanja SPD-a i prezentiramo primjerom poluparametarsku procjenu istog.

Abstract (english)

This master’s thesis deals with implied volatility and its usage in financial mathematics. First, in introduction we give some basic information about models used in this paper, e.g. the Black and Scholes model and Cox-Ross-Rubinstein model of binomial tree. First section gives introduction to implied volatility, estimated volatility of a security’s price, most commonly used when pricing options. To derive implied volatilities the Black and Scholes formula is solved for the constant volatility parameter \(\sigma\) using observed option prices. It appears to be non flat across strikes, a stylized fact which has been called ”smile” effect. In the end of this chapter we use two variants of principal components, standard principal component analysis to identify the dominant factor components driving term structure movements of options, and the common principal components approach that enables modeling not only term structure movements of implied volatilities but the dynamics of the ”smile” as well. Then we introduce the implied binomial tree (IBT) method which constructs a numerical procedure consistent with the volatility smile. The empirical fact that the market implied volatilities decrease with the strike level, and increase with the time to maturity of options is better reflected by this construction than standard CRR trees. The basic purpose of the IBT is its use in hedging and calculations of implied probability distributions (or state price density (SPD)), and volatility surfaces. Two algorithms for construction of IBT for a liquid European option are given. Firstly, we follow the Derman and Kani algorithm, discuss its possible shortcomings, and then present the Barle and Cakici construction. In the last chapter we talk about state price density and ways of obtaining it. The central idea is that the price of a financial security is equal to the expected net present value of its future payoffs under the risk-neutral probability density function. This method constitutes a no-arbitrage approach to recover the SPD. No assumption on the underlying asset dynamics are required. First, we propose ways for semiparametric estimation of the call pricing function and the necessary steps to recover the SPD. Then, in the end, an numerical example is given.

• Keywords: volatilnost; Black-Scholesov model; Cox-Ross-Rubinsteinov model; podrazumijeva binomna stabla; IBT; smile efekt; podrazumijevana vjerojatnosna distribucija; SPD; algoritam Dermana i Kanija; algoritam Barlea i Cakicia
• Keywords (english): volatility; Black and Scholes model; Cox-Ross-Rubinstein model; implied binomial tree; IBT; volatility smile; implied probability distributions; state price density; SPD; Derman and Kani algorithm; Barle and Cakici construction
• Language: croatian
• URN:NBN: urn:nbn:hr:217:911996
• 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: 2018-08-30 11:19:55