Ocenění opcí na index PX se stochastickou volatilitou a časově závislou očekávanou bezrizikovou úrokovou sazbou

Filip Štěrba

Doctoral thesis

Valuation of PX Index Options with NGARCH Volatility and Time Dependent Expected Risk Free Rate

Abstract:

Hlavním cílem této práce je navrhnout způsob prvotního ocenění opcí na index PX. Práce uvolňuje původní předpoklady Black-Scholesovy formule o konstantní krátkodobé úrokové míře a o konstantní volatilitě. Namísto toho se předpokládá, že krátkodobá bezriziková úroková sazba následuje za předpokladu platnosti hypotézy očekávání očekávanou trajektorii úrokových sazeb a volatilita logaritmických výnosů indexu PX se řídí NGARCH-mean procesem. Výsledné ocenění vede k zpětně dopočteným tvarům imlikované volatility, které jsou v souladu s tvary implikované volatility, které můžeme běžně pozorovat na rozvinutých trzích.

Abstract:

The main purpose of this thesis is to propose the valuation method of PX index options. PX index consists of blue chip stocks traded on Prague Stock Exchange. There are traded a few futures contracts on PX index on Prague Stock Exchange. However, the options on PX index are traded neither on Prague Stock Exchange nor on the OTC market. It is reasonable to think that it is only question of time when the trading of these options will emerge and thus, it is highly relevant subject of research to propose the method for valuation of these options. The traditional Merton's approach for valuation of equity index options assumes constant volatility and constant risk free rate. This results in serious mispricing which can be easily seen when we compare market prices and Merton formula derived prices. Instead, this thesis releases the assumptions of constant risk free rate and constant volatility. Firstly, it is assumed that that the risk free rate is time dependent function based on current market expectations and secondly it is assumed that the volatility of underlying asset follows NGARCH-mean process. For the purpose of former, the validity of pure expectation theory assumption is made. This enables to employ the instantaneous forward rate curve estimation procedure. For the purpose of the latter, the locally risk-neutral valuation relationship is applied. The assumption of NGARCH-mean process is essential in an effort to capture usually observed patterns of volatility (volatility skews) whereas the assumption of time dependent risk free rate still moves the valuation option model closer to the reality. The author derives the expected path of risk free rate and estimates the parameters of NGARCH process. Subsequently, the empirical martingale Monte Carlo simulation is used to price the PX options with different moneyness and with different times to maturity. It is shown that this proposed model results in volatility pattern which is usually observed on developed markets and the author's results are in line with similar empirical studies testing the GARCH Option Pricing Theory. The author concludes that proposed valuation method superiors original Merton's model and thus is more appropriate for primary valuation of PX options.

Keywords

ocenění, stochastická volatilita, opce

Keywords

volatility skew, NGARCH, options

Language used: Czech

Date on which the thesis was submitted / produced: 6. 10. 2004

Thesis defence

• Date of defence: 12. 3. 2009
• Supervisor: Jiří Málek
• Reader: Jan Kodera, Jiří Hnilica

Vysoká škola ekonomická v Praze

Doctoral programme / field:
Finance a účetnictví / Finance