Calculation aspects of the European Rebalanced Basket Option using Monte Carlo methods

Van der Merwe, Carel Johannes (Stellenbosch : University of Stellenbosch, 2010-12)

Thesis (MComm (Statistics and Actuarial Science)--University of Stellenbosch, 2010.

Thesis

ENGLISH ABSTRACT: Life insurance and pension funds offer a wide range of products that are invested in a mix of assets. These portfolios (II), underlying the products, are rebalanced back to predetermined fixed proportions on a regular basis. This is done by selling the better performing assets and buying the worse performing assets. Life insurance or pension fund contracts can offer the client a minimum payout guarantee on the contract by charging them an extra premium (a). This problem can be changed to that of the pricing of a put option with underlying . It forms a liability for the insurance firm, and therefore needs to be managed in terms of risks as well. This can be done by studying the option’s sensitivities. In this thesis the premium and sensitivities of this put option are calculated, using different Monte Carlo methods, in order to find the most efficient method. Using general Monte Carlo methods, a simplistic pricing method is found which is refined by applying mathematical techniques so that the computational time is reduced significantly. After considering Antithetic Variables, Control Variates and Latin Hypercube Sampling as variance reduction techniques, option prices as Control Variates prove to reduce the error of the refined method most efficiently. This is improved by considering different Quasi-Monte Carlo techniques, namely Halton, Faure, normal Sobol’ and other randomised Sobol’ sequences. Owen and Faure-Tezuke type randomised Sobol’ sequences improved the convergence of the estimator the most efficiently. Furthermore, the best methods between Pathwise Derivatives Estimates and Finite Difference Approximations for estimating sensitivities of this option are found. Therefore by using the refined pricing method with option prices as Control Variates together with Owen and Faure-Tezuke type randomised Sobol’ sequences as a Quasi-Monte Carlo method, more efficient methods to price this option (compared to simplistic Monte Carlo methods) are obtained. In addition, more efficient sensitivity estimators are obtained to help manage risks.

AFRIKAANSE OPSOMMING: Lewensversekering en pensioenfondse bied die mark ’n wye reeks produkte wat belê word in ’n mengsel van bates. Hierdie portefeuljes (II), onderliggend aan die produkte, word op ’n gereelde basis terug herbalanseer volgens voorafbepaalde vaste proporsies. Dit word gedoen deur bates wat beter opbrengste gehad het te verkoop, en bates met swakker opbrengste aan te koop. Lewensversekeringof pensioenfondskontrakte kan ’n kliënt ’n verdere minimum uitbetaling aan die einde van die kontrak waarborg deur ’n ekstra premie (a) op die kontrak te vra. Die probleem kan verander word na die prysing van ’n verkoopopsie met onderliggende bate . Hierdie vorm deel van die versekeringsmaatskappy se laste en moet dus ook bestuur word in terme van sy risiko’s. Dit kan gedoen word deur die opsie se sensitiwiteite te bestudeer. In hierdie tesis word die premie en sensitiwiteite van die verkoopopsie met behulp van verskillende Monte Carlo metodes bereken, om sodoende die effektiefste metode te vind. Deur die gebruik van algemene Monte Carlo metodes word ’n simplistiese prysingsmetode, wat verfyn is met behulp van wiskundige tegnieke wat die berekeningstyd wesenlik verminder, gevind. Nadat Antitetiese Veranderlikes, Kontrole Variate en Latynse Hiperkubus Steekproefneming as variansiereduksietegnieke oorweeg is, word gevind dat die verfynde metode se fout die effektiefste verminder met behulp van opsiepryse as Kontrole Variate. Dit word verbeter deur verskillende Quasi-Monte Carlo tegnieke, naamlik Halton, Faure, normale Sobol’ en ander verewekansigde Sobol’ reekse, te vergelyk. Die Owen en Faure-Tezuke tipe verewekansigde Sobol’ reeks verbeter die konvergensie van die beramer die effektiefste. Verder is die beste metode tussen Baanafhanklike Afgeleide Beramers en Eindige Differensie Benaderings om die sensitiwiteit vir die opsie te bepaal, ook gevind. Deur dus die verfynde prysingsmetode met opsiepryse as Kontrole Variate, saam met Owen en Faure-Tezuke tipe verewekansigde Sobol’ reekse as ’n Quasi-Monte Carlo metode te gebruik, word meer effektiewe metodes om die opsie te prys, gevind (in vergelyking met simplistiese Monte Carlo metodes). Verder is meer effektiewe sensitiwiteitsberamers as voorheen gevind wat gebruik kan word om risiko’s te help bestuur.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/5190
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