Modelling of financial risk using forward-looking distributions derived from contingent claims

Abstract

English: In this thesis, we investigate several methods for extracting the forecast distribution from historical asset returns and market-quoted option prices. Typically, risk-neutral distributions, extracted from market quoted option prices, are considered biased estimates of the forecast distribution, and therefore need to be transformed into a real-world distribution. Transformation processes often require the use of historical data and restrictive assumptions on a representative investor. Alternatively, the recovery theorem provides a theoretically appealing method to recover the real-world distribution from the risk-neutral transition probability matrix without the use of historical returns. However, estimating the risk-neutral transition probability matrix has proven to be a challenging task, as it involves solving an ill-posed problem. Therefore, we propose a regularised multivariate Markov chain in the estimation of the risk-neutral transition probability matrix to obtain a more accurate real-world forecast distribution than obtained using the univariate model.

Comparative studies on the accuracy of real-world forecast distributions are scarce in the literature. Therefore, we further backtested and compared the accuracy of the extracted distributions on the South African Top 40 index, where we found that the forward-looking real-world distribution improved forecasting in certain situations. We also proposed a forward-looking mixture model of historical and option-implied distributions to improve forecasting. Furthermore, we implemented the extracted forecast distributions in determining safe retirement withdrawal rates. In our empirical study, we showed that the use of forward-looking distributions drastically improved the success in retirement withdrawal rates.

Sepedi: Mo theseseng ye, re nyakišiša mekgwa ye mmalwa ya go ntšha kabo ya ponelopele go tšwa dipoelong tša matlotlo tša histori le ditheko tša dikgetho tšeo di tsopotšwego mmarakeng. Ka tlwaelo, dikabo tše di hlokago dikotsi, tšeo di ntšhitšwego dithekong tša dikgetho tše di tsopotšwego mmarakeng, di tšewa bjalo ka dikakanyetšo tše sekametšego kabong ya ponelopele, ka fao di swanetše go fetošetšwa kabong ya lefase la kgonthe. Ditshepetšo tša phetogo, gantši di nyaka tšhomišo ya datha ya histori le dikakanyo tše di iletšago moemedi wa mmeeletši. Ka mo go fapanego, teorema ya go tsošološa e fana ka mokgwa wa theory wo o ipiletšago teoring wa go bušetša kabo ya lefase la kgonthe go tšwa go matrix wa kgonagalo ya phetogo woo o sa sekamelago kotsing ntle le tšhomišo ya dipoelo tša histori. Le ge go le bjale, go akanyetša mathrikse ya kgonagalo ya phetogo yeo e sa sekamelago kotsing go ipontšhitše e le mošomo wo boima, ka ge go akaretša go rarolla bothata bjo bo sa bewago gabotse. Ka fao, re šišinya molokoloko wo o beakantšwego wa go fetoga wa Markov wa mehutahuta wa kotsi ya magareng ya kgonego ya mathrikse wa go hwetša kabo ya ponelopele ya lefase la nnete ye e nepagetšego kudu go feta yeo e hweditšwego ka go šomiša mohlala wa wa papetšo o tee. Dinyakišišo tša go bapetša tša go nepagala ga dikabo tša ponelopele ya lefase la kgonthe di a hlaelela ka dingwalweng. Ka fao, re ile ra tšwela pele go dira diteko tša pušetšomorago le go bapetša go nepagala ga dikabo tše di ntšhitšwego go Dipalopalo tša Maemo a Godimo Aa bo 40 ka Afrika Borwa, moo re hweditšego gore kabo ya lefase ya kgonthe ya tebelelopele e kaonafaditše ponelopele maemong a itšego. Re ile ra šišinya gape mohlala wa mohlakanelwa wa tebelelopele ya dikabo tša histori le tšeo di akaretšwago ke kgetho go kaonafatša ponelopele. Godimo ga fao, re phethagaditše dikabo tša ponelopele tše di ntšhitšwego go laetša ditekanyo tša go ntšha tšhelete ka go rola modiro ka polokego. Ka dinyakišišong tša rena tša bohlatse, re bontšhitše gore tšhomišo ya dikabo tša tebelelopele di kaonafaditše kudu katlego ya ditekanyo tša go ntšha tšhelete ge go rolwa modiro.

Afrikaans: In hierdie proefskrif ondersoek ons verskeie metodes om die voorspelde verdeling uit historiese bate-opbrengste en markgekwoteerde opsiepryse te onttrek. Tipies word risiko-neutrale verdelings, onttrek uit markgekwoteerde opsiepryse, as bevooroordeelde ramings van die voorspelde verdeling beskou en moet dus in 'n werklike verdeling omskep word. Transformasieprosesse vereis dikwels die gebruik van historiese data en beperkende aannames oor 'n verteenwoordigende belegger. Alternatiewelik bied die herwinnigstelling 'n teoreties aantreklike metode om die werklike verspreiding uit die risiko-neutrale oorgangswaarskynlikheidsmatriks te herwin sonder die gebruik van historiese opbrengste. Die beraming van die risiko-neutrale oorgangswaarskynlikheidsmatriks het egter uitdagend geblyk, aangesien dit die oplossing van 'n swak geformuleerde probleem behels. Daarom stel ons 'n gereguleerde meerveranderlike Markov-ketting voor in die beraming van die risiko-neutrale oorgangswaarskynlikheidsmatriks om 'n meer akkurate werklike voorspellingsverspreiding te verkry as wat verkry word met behulp van die eenveranderlike model. Vergelykende studies oor die akkuraatheid van werklike voorspellingsverspreidings is skaars in die literatuur. Daarom het ons die akkuraatheid van die onttrekte verspreidings op die Suid-Afrikaanse Top 40-indeks verder teruggetoets en vergelyk, waar ons gevind het dat die vooruitskouende werklike verspreiding voorspelling in sekere situasies verbeter het. Ons het ook 'n vooruitskouende gemengde model van historiese en opsie-geïmpliseerde verspreidings voorgestel om voorspelling te verbeter. Verder het ons die onttrekte voorspellingsverspreidings geïmplementeer in die bepaling van veilige aftree-onttrekkingsyfers. In ons empiriese studie het ons getoon dat die gebruik van toekomsgerigte verspreidings die sukses in aftree-onttrekkingsyfers drasties verbeter het.

IsiZulu: Kulolu cwaningo, siphenya izindlela ezimbalwa zokukhipha isibikezelo sokusatshalaliswa okuvela ekubuyekezweni kwenzuzo yempahla yomlando kanye nezintengo zokukhetha ezibizwe emakethe. Ngokuvamile, ukusatshalaliswa okungenangozi nokungagqamisi uhlobo, okukhishwe kukukhetha okubizwe emakethe, kuthathwa njengezilinganiso ezichemile zokubikezela ukusatshalaliswa , ngakho-ke kudingeka ukuthi kuguqulelwe kukusatshalaliswa komhlaba wangempela. Kule thesisi, siphenya izindlela ezimbalwa zokukhipha isibikezelo sokusatshalaliswa okuvela ekubuyekezweni kwenzuzo yempahla yomlando kanye nezintengo zokukhetha ezibizwe emakethe. Okunye ukuthi, ithiyoremu yokubuyisa ihlinzeka ngendlela ekhangayo ngokomqondo ukuze kubuyiselwe ukusatshalaliswa komhlaba wangempela ukusuka ku-mathiksi yamathuba angabakhona enguquko engenangozi nokugqamisa uhlobo ngaphandle kokusetshenziswa kwezinzuzo zomlando. Kodwa-ke, ukulinganisa imethriksi yamathuba angabakhona enguquko engenangozi nokugqamisa uhlobo kufakazela ukuthi kuwumsebenzi oyinselelo, njengoba kubandakanya ukuxazulula inkinga ebekwe kabi. Ngakho-ke, siphakamisa iketanga elijwayelekile le-Markov elinhloboningi futhi elenziwe ngokomthetho ekulinganisweni kwemethriksi yamathuba angabakhona enguquko engenangozi nokugqamisa uhlobo ukuze kutholwe isibikezelo sokusatshalaliswa komhlaba wempela okunemba kakhudlwana kunaloko okutholwa kusetshenziswa izifanekiso eziwuhlobo olulodwa. Izifundo Eziqhathanisa ngokunemba kwesibikezelo sokusatshalaliswa komhlaba wangempela ziyingcosana emibhalweni. Ngakho-ke, siqhubekile nokuhlola saqhathanisa ukunemba kokusatshalaliswa okutholakele ku-Top 40 index yaseNingizimu Afrika, lapho sithole khona ukuthi ukusatshalaliswa okubheke phambili kwangempela kuthuthukise ukubikezela kwezinye izimo. Siphinde saphakamisa umfanekiso oyinhlanganisela obikezela ukusabalalisa okungokomlando kanye nokukhetha okuqondayo ukwenza ngcono ukubikezela. Ngaphezu kwalokho, sisebenzise isibikezelo sokusatshalaliswa esikhishwe ekunqumeni izilinganiso eziphephile zokudonsa umhlalaphansi. Ocwaningweni lwethu lwendlela yesayensi, sibonise ukuthi ukusetshenziswa kokubikezela ukusatshalaliswa kuyenza kangcono kakhulu impumelelo kuzilinganiso zokudonsa umhlalaphansi.