Heath-Jarrow-Morton (HJM) okvir

master's thesis

• Title (english): Heath-Jarrow-Morton (HJM) framework
• Author: Kristina Jurin
• Mentor: Vanja Wagner (mentor)
• Committee member: Vanja Wagner (predsjednik povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj rad predstavlja pregled metodologije za određivanje cijene slučajnih zahtjeva prema ročnoj strukturi kamatne stope. Uzimajući u obzir početnu krivulju forward kamatne stope i mehanizam koji opisuje kako ona fluktuira, razvijamo model arbitražnog određivanja cijena koji daje procjene vrijednosti slučajnih zahtjeva koji ne ovise eksplicitno o tržišnim cijenama rizika. U prvom poglavlju formulirali smo temeljne postavke HJM pristupa te smo pronašli formulu za promjenu cijene obveznice. U drugom poglavlju pronašli smo dva dovoljna uvjeta za nepostojanje mogućnosti arbitraže za sve obveznice različitih dospijeća te smo zaključili da su ta dva uvjeta ekvivalentna. U trećem poglavlju pretpostavili smo da je volatilnost σ forward kamatne stope deterministička funkcija te smo pokazali kako tržišne cijene rizika u potpunosti proizlaze iz arbitražnih vrijednosti izvedenica osjetljivih na kamatnu stopu. U četvrtom poglavlju dobili smo izraz zatvorenog oblika za arbitražnu cijenu europske opcije na obveznice uz pretpostavku da su volatilnosti ograničene. Također smo iskazali eksplicitnu formulu za arbitražnu cijenu call opcije na dionicu, te dokazali da vrijedi formula za put-call paritet.

Abstract (english)

This thesis presents an overview of the methodology for pricing contingent claims on the term structure of interest rates. Given an initial forward rate curve and a mechanism which describes how it fluctuates, we develop an arbitrage pricing model which yields contingent claim valuations which do not explicitly depend on the market prices for risk. In the first chapter, we formulated the basic postulates of the HJM approach and we derived the formula for the dynamics of the bond price. In the second chapter, we present two sufficient conditions for the absence of arbitrage for all bonds of different maturities, and we conclude that these two conditions are equivalent. In the third chapter, we assumed that the volatility σ of the forward rate is a deterministic function and we showed that the market prices for risk are derived from arbitrage values of interest rate-sensitive derivatives. In the fourth chapter, we obtained a closed-form expression for the arbitrage price of a European bond option under the assumption that volatilities are bounded. We also presented an explicit formula for the arbitrage price of a call option on a stock and proved that the formula for the put-call parity is valid.

• Keywords: ročna struktura kamatne stope; forward kamatna stopa; model arbitražnog određivanja cijena; HJM pristup; volatilnost σ forward kamatne stope
• Keywords (english): term structure of the interest rate; forward interest rate; arbitrage pricing model; HJM approach; volatility σ of the forward rate
• Language: croatian
• URN:NBN: urn:nbn:hr:217:686866
• 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: 2022-12-16 12:38:51