Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: In modern theory of finance, the so-called First and Second Fundamental Theorems of Asset
Pricing play an important role in pricing options with no-arbitrage. These theorems gives a
necessary and sufficient conditions for a market to have no-arbitrage and for a market to be
complete. An early version of the First Fundamental Theorem of Asset Pricing was proven
by Harrison and Kreps [30] in the case of a finite probability space. A more general version
was proven by Harrison and Pliska [31] in the case of a finite probability space and discrete
time. In the case of continuous time, Delbaen and Schachermayer [19] introduced a more
general concept of no-arbitrage called "No-Free Lunch With Vanishing Risk" (NFLVR),
and showed that for a locally-bounded semimartingale price process NFLVR is essentially
equivalent to the existence of an equivalent local martingale measure.
The goal of this thesis is to review the theory of arbitrage pricing and the extension of
this theory to include liquidity risk. At the current time, liquidity risk is a key challenge
faced by investors. Consequently there is a need to develop more realistic pricing models
that include liquidity risk. We present an approach to liquidity risk by Çetin, Jarrow and
Protter [10]. In to this approach the liquidity risk is embedded into the classical theory
of arbitrage pricing by having investors act as price takers, and assuming the existence
of a supply curve where prices depend on trade size. This framework assumes that the
quantity impact on the price transacted is momentary. Using trading strategies that are
both continuous and of finite variation allows one to avoid liquidity costs. Therefore, the
First and Second Fundamental Theorems of Asset Pricing and the Black-Scholes model
can be extended. / AFRIKAANSE OPSOMMING: In moderne finansiële teorie speel die sogenaamde Eerste en Tweede Fundamentele Stellings
van Bateprysbepaling ’n belangrike rol in die prysbepaling van opsies in arbitrage-vrye
markte. Hierdie stellings gee nodig en voldoende voorwaardes vir ’n mark om vry van
arbitrage te wees, en om volledig te wees. ’n Vroeë weergawe van die Eerste Fundamentele
Stelling was deur Harrison en Kreps [30] bewys in die geval van ’n eindige waarskynlikheidsruimte.
’n Meer algemene weergawe was daarna gepubliseer deur Harrison en Pliska
[31] in die geval van ’n eindige waarskynlikheidsruimte en diskrete tyd. In die geval van
kontinue tyd het Delbaen en Schachermayer [19] ’n meer algemene konsep van arbitragevryheid
ingelei, naamlik “No–Free–Lunch–With–Vanishing–Risk" (NFLVR), en aangetoon dat
vir lokaalbegrensde semimartingaalprysprosesse NFLVR min of meer ekwivalent is aan die
bestaan van ’n lokaal martingaalmaat.
Die doel van hierdie tesis is om ’n oorsig te gee van beide klassieke arbitrageprysteorie,
en ’n uitbreiding daarvan wat likideit in ag neem. Hedendaags is likiditeitsrisiko ’n
vooraanstaande uitdaging wat beleggers die hoof moet bied. Gevolglik is dit noodsaaklik
om meer realistiese modelle van prysbepaling wat ook likiditeitsrisiko insluit te ontwikkel.
Ons bespreek die benadering van Çetin, Jarrow en Protter [10], waar likiditeitsrisiko in
die klassieke arbitrageprysteorie ingesluit word deur die bestaan van ’n aanbodkromme
aan te neem, waar pryse afhanklik is van handelsgrootte. In hierdie raamwerk word aangeneem
dat die impak op die transaksieprys slegs tydelik is. Deur gebruik te maak van
handelingsstrategië wat beide kontinu en van eindige variasie is, is dit dan moontlik om
likiditeitskoste te vermy. Die Eerste en Tweede Fundamentele Stellings van Bateprysbepaling
en die Black–Scholes model kan dus uitgebrei word om likiditeitsrisiko in te sluit.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/79975 |
| Date | 03 1900 |
| Creators | El Ghandour, Laila |
| Contributors | Ouwehand, Peter, Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | 107 p. |
| Rights | Stellenbosch University |
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