Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
Perturbation methods in derivatives pricing under stochastic volatility
Thesis (MSc)--Stellenbosch University, 2012.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | en_ZA |
| Published: |
Stellenbosch : Stellenbosch University
2012
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613957875826688 |
|---|---|
| access_status_str | Open Access |
| author | Kateregga, Michael |
| author2 | Ghomrasni, Raouf |
| author_browse | Ghomrasni, Raouf Kateregga, Michael |
| author_facet | Ghomrasni, Raouf Kateregga, Michael |
| author_sort | Kateregga, Michael |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2012. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/71708 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:44:24.378Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2012 |
| publishDateRange | 2012 |
| publishDateSort | 2012 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/71708 Perturbation methods in derivatives pricing under stochastic volatility Kateregga, Michael Ghomrasni, Raouf Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Perturbation (Mathematics) Derivative securities -- Pricing Stochastic volatility Ergodic Markov process Lipschitz stochastic differential equation Dissertations -- Mathematics Theses -- Mathematics Thesis (MSc)--Stellenbosch University, 2012. ENGLISH ABSTRACT: This work employs perturbation techniques to price and hedge financial derivatives in a stochastic volatility framework. Fouque et al. [44] model volatility as a function of two processes operating on different time-scales. One process is responsible for the fast-fluctuating feature of volatility and corresponds to the slow time-scale and the second is for slowfluctuations or fast time-scale. The former is an Ergodic Markov process and the latter is a strong solution to a Lipschitz stochastic differential equation. This work mainly involves modelling, analysis and estimation techniques, exploiting the concept of mean reversion of volatility. The approach used is robust in the sense that it does not assume a specific volatility model. Using singular and regular perturbation techniques on the resulting PDE a first-order price correction to Black-Scholes option pricing model is derived. Vital groupings of market parameters are identified and their estimation from market data is extremely efficient and stable. The implied volatility is expressed as a linear (affine) function of log-moneyness-tomaturity ratio, and can be easily calibrated by estimating the grouped market parameters from the observed implied volatility surface. Importantly, the same grouped parameters can be used to price other complex derivatives beyond the European and American options, which include Barrier, Asian, Basket and Forward options. However, this semi-analytic perturbative approach is effective for longer maturities and unstable when pricing is done close to maturity. As a result a more accurate technique, the decomposition pricing approach that gives explicit analytic first- and second-order pricing and implied volatility formulae is discussed as one of the current alternatives. Here, the method is only employed for European options but an extension to other options could be an idea for further research. The only requirements for this method are integrability and regularity of the stochastic volatility process. Corrections to [3] remarkable work are discussed here. AFRIKAANSE OPSOMMING: Hierdie werk gebruik steuringstegnieke om finansiële afgeleide instrumente in ’n stogastiese wisselvalligheid raamwerk te prys en te verskans. Fouque et al. [44] gemodelleer wisselvalligheid as ’n funksie van twee prosesse wat op verskillende tyd-skale werk. Een proses is verantwoordelik vir die vinnig-wisselende eienskap van die wisselvalligheid en stem ooreen met die stadiger tyd-skaal en die tweede is vir stadig-wisselende fluktuasies of ’n vinniger tyd-skaal. Die voormalige is ’n Ergodiese-Markov-proses en die laasgenoemde is ’n sterk oplossing vir ’n Lipschitz stogastiese differensiaalvergelyking. Hierdie werk behels hoofsaaklik modellering, analise en skattingstegnieke, wat die konsep van terugkeer to die gemiddelde van die wisseling gebruik. Die benadering wat gebruik word is rubuust in die sin dat dit nie ’n aanname van ’n spesifieke wisselvalligheid model maak nie. Deur singulêre en reëlmatige steuringstegnieke te gebruik op die PDV kan ’n eerste-orde pryskorreksie aan die Black-Scholes opsie-waardasiemodel afgelei word. Belangrike groeperings van mark parameters is geïdentifiseer en hul geskatte waardes van mark data is uiters doeltreffend en stabiel. Die geïmpliseerde onbestendigheid word uitgedruk as ’n lineêre (affiene) funksie van die log-geldkarakter-tot-verval verhouding, en kan maklik gekalibreer word deur gegroepeerde mark parameters te beraam van die waargenome geïmpliseerde wisselvalligheids vlak. Wat belangrik is, is dat dieselfde gegroepeerde parameters gebruik kan word om ander komplekse afgeleide instrumente buite die Europese en Amerikaanse opsies te prys, dié sluit in Barrier, Asiatiese, Basket en Stuur opsies. Hierdie semi-analitiese steurings benadering is effektief vir langer termyne en onstabiel wanneer pryse naby aan die vervaldatum beraam word. As gevolg hiervan is ’n meer akkurate tegniek, die ontbinding prys benadering wat eksplisiete analitiese eerste- en tweede-orde pryse en geïmpliseerde wisselvalligheid formules gee as een van die huidige alternatiewe bespreek. Hier word slegs die metode vir Europese opsies gebruik, maar ’n uitbreiding na ander opsies kan’n idee vir verdere navorsing wees. Die enigste vereistes vir hierdie metode is integreerbaarheid en reëlmatigheid van die stogastiese wisselvalligheid proses. Korreksies tot [3] se noemenswaardige werk word ook hier bespreek. 2012-11-08T13:42:18Z 2012-12-12T08:09:09Z 2012-11-08T13:42:18Z 2012-12-12T08:09:09Z 2012-12 Thesis http://hdl.handle.net/10019.1/71708 en_ZA Stellenbosch University 140 p. application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Perturbation (Mathematics) Derivative securities -- Pricing Stochastic volatility Ergodic Markov process Lipschitz stochastic differential equation Dissertations -- Mathematics Theses -- Mathematics Kateregga, Michael Perturbation methods in derivatives pricing under stochastic volatility |
| title | Perturbation methods in derivatives pricing under stochastic volatility |
| title_full | Perturbation methods in derivatives pricing under stochastic volatility |
| title_fullStr | Perturbation methods in derivatives pricing under stochastic volatility |
| title_full_unstemmed | Perturbation methods in derivatives pricing under stochastic volatility |
| title_short | Perturbation methods in derivatives pricing under stochastic volatility |
| title_sort | perturbation methods in derivatives pricing under stochastic volatility |
| topic | Perturbation (Mathematics) Derivative securities -- Pricing Stochastic volatility Ergodic Markov process Lipschitz stochastic differential equation Dissertations -- Mathematics Theses -- Mathematics |
| url | http://hdl.handle.net/10019.1/71708 |
| work_keys_str_mv | AT katereggamichael perturbationmethodsinderivativespricingunderstochasticvolatility |