A derivation of the the black-scholes equation using martingales
- Authors: Nyarko , Ebenezer Narh
- Date: 2018
- Subjects: Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/14572 , vital:40016
- Description: This work focuses on the application of stochastic differential equations, with martingales, in finance. The emphasis is on the derivation of the Black-Scholes model for the valuation of options. A theoretical framework in stochastic analysis, together with Itô calculus (Kiyoshi Itô), is explored. The Girsanov Theorem is applied in order to transform a modelled stochastic equation based, on predetermined stock and bond prices, into equivalent martingale measures. A replication strategy is then adopted to solve the two equations analytically, by finding the natural logarithm of the expectation of the solution to the stochastic models. We finally compute the resulting solution based on a standard, normal distribution to get the desired outcome of the Black-Scholes model.
- Full Text:
- Date Issued: 2018
Above ground woody community attributes, biomass and carbon stocks along a rainfall gradient in the savannas of the central lowveld, South Africa
- Authors: Shackleton, Charlie M , Scholes, Robert J
- Date: 2011
- Language: English
- Type: Article
- Identifier: vital:6658 , http://hdl.handle.net/10962/d1007079
- Description: Enumeration of carbon stocks at benchmark sites is a necessary activity in assessing the potential carbon sequestration and possible generation of credits through restoration of intensively impacted sites. However, there is a lack of empirical studies throughout much of the savannas of sub-Saharan Africa, including South Africa. We report an estimation of species specific and site biomass and carbon stocks, and general vegetation structural attributes from three protected areas along a rainfall gradient in the central lowveld, South Africa. Estimates of biomass and carbon stocks were effected through destructive sampling to establish locally derived allometric equations. There was a gradient of increasing woody density, height of the canopy, number of species, density of regenerative stems and a greater proportion of stems in small size classes from the arid locality to the mesic locality, with the semi-arid locality being intermediate. The proportion of spinescent species decreased with increasing rainfall. The mesic locality was significantly more woody than either the arid or semi-arid sites, having double the biomass, four times the density and 40% higher basal area. Above ground carbon pools were also higher; carbon stocks were approximately 9 t/ha for the arid and semi-arid sites and 18 t/ha for the mesic site.
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- Date Issued: 2011
An analysis of the Libor and Swap market models for pricing interest-rate derivatives
- Authors: Mutengwa, Tafadzwa Isaac
- Date: 2012
- Subjects: LIBOR market model , Monte Carlo method , Interest rates -- Mathematical models , Derivative securities
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5573 , http://hdl.handle.net/10962/d1005535
- Description: This thesis focuses on the non-arbitrage (fair) pricing of interest rate derivatives, in particular caplets and swaptions using the LIBOR market model (LMM) developed by Brace, Gatarek, and Musiela (1997) and Swap market model (SMM) developed Jamshidan (1997), respectively. Today, in most financial markets, interest rate derivatives are priced using the renowned Black-Scholes formula developed by Black and Scholes (1973). We present new pricing models for caplets and swaptions, which can be implemented in the financial market other than the Black-Scholes model. We theoretically construct these "new market models" and then test their practical aspects. We show that the dynamics of the LMM imply a pricing formula for caplets that has the same structure as the Black-Scholes pricing formula for a caplet that is used by market practitioners. For the SMM we also theoretically construct an arbitrage-free interest rate model that implies a pricing formula for swaptions that has the same structure as the Black-Scholes pricing formula for swaptions. We empirically compare the pricing performance of the LMM against the Black-Scholes for pricing caplets using Monte Carlo methods.
- Full Text:
- Date Issued: 2012
Studying a Tumor Growth Partial Differential Equation via the Black–Scholes Equation
- Authors: Winter Sinkala , Tembinkosi F. Nkalashe
- Date: 2020
- Language: English
- Type: Journal Article
- Identifier: http://hdl.handle.net/11260/4551 , vital:44132
- Full Text:
Impact of fire frequency on woody community structure and soil nutrients in the Kruger National Park
- Authors: Shackleton, Charlie M , Scholes, Robert J
- Date: 2000
- Subjects: To be catalogued
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/181874 , vital:43776 , xlink:href="https://doi.org/10.4102/koedoe.v43i1.210"
- Description: Although fire is recognised as an important determinant of the structure and function of South African savannas, there are few studies of long-term impacts. Controlled burning blocks of contrasting fire season and frequency have been maintained throughout the Kruger National Park for almost 50 years. This paper reports on a quantitative study of the Satara plots to determine the long-term impacts of fire frequency on woody community structure and soil nutrients. Increasing fire frequency significantly decreased woody plant basal area, biomass, density, height, and mean stem circumference. The number of stems per plant and the proportion of regenerative stems increased with increasing fire frequency. Effects on species richness of woody plants were inconsistent. There were no significant differences attributable to fire frequency for any of the soil variables except organic matter and magnesium. Organic carbon was highest in the fire exclusion treatment and lowest in soils from plots burnt triennially. Magnesium levels were greatest in the annually burnt soils and least in the triennial plots.
- Full Text:
- Date Issued: 2000
On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation
- Authors: Winter Sinkala
- Date: 2020
- Language: English
- Type: Journal Article
- Identifier: http://hdl.handle.net/11260/4335 , vital:44106
- Full Text:
On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation
- Authors: Winter Sinkala
- Date: 2017
- Language: English
- Type: Research Article
- Identifier: http://hdl.handle.net/11260/5642 , vital:44619
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Letter from Edmond Robert Wodehouse, to Walter Jardine, 22 May 1864
- Authors: Wodehouse, Edmond Robert , Jardine, Walter
- Date: 1864-05-22 , 2021-09-29
- Subjects: Jardine, Walter , Botanical Gardens -- South Africa -- Grahamstown
- Type: text , letter
- Identifier: http://hdl.handle.net/10962/197654 , vital:45893 , Cory Library for Humanities Research, Rhodes University
- Description: Jardine Correspondence transcribed by Jean Kelly August 2021. Transcription: Graham’s Town. May 22 1864. I am much obliged to you, Walter, for the Disa Bulbs which arrived yesterday all safe, and with which Mr Tuck of the Botanical Gardens is much pleased. I remain Yours faithfully, Edmund R Wodehouse. ************************************************************************ Noted on scrap paper: Left Cape Town 22nd May 1867 Got to Grahamstown 1st June Bot Gardens 4th June. Walter Jardine.
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The Southern African Program on Ecosystem Change and Society: an emergent community of practice
- Authors: Biggs, Reinette , Reyers, Belinda , Blanchard, Ryan , Clements, Hayley S , Cockburn, Jessica , Cumming, Graeme S , Cundill, Georgina , de Vos, Alta , Dziba, Luthando , Esler, Karen J , Fabricius, Christo , Hamann, Maike , Henriksson, Rebecka , Kotschy, Karen , Lindborg, Regina , Luvuno, Linda , Masterson, Vanessa A , Nel, Jeanne L , O'Farrell, Patrick , Palmer, Carolyn G , Pereira, Laura , Pollard, Sharon , Preiser, Rika , Roux, Dirk J , Scholes, Robert J , Selomane, Odirlwe , Shackleton, Charlie M , Shackleton, Sheona , Sitas, Nadia , Slingsby, Jasper A , Spierenburg, Marja , Tengö, Maria
- Date: 2023
- Subjects: To be catalogued
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/401330 , vital:69726 , xlink:href="https://doi.org/10.1080/26395916.2022.2150317"
- Description: Sustainability-focused research networks and communities of practice have emerged as a key response and strategy to build capacity and knowledge to support transformation towards more sustainable, just and equitable futures. This paper synthesises insights from the development of a community of practice on social-ecological systems (SES) research in southern Africa over the past decade, linked to the international Programme on Ecosystem Change and Society (PECS). This community consists of a network of researchers who carry out place-based SES research in the southern African region. They interact through various cross-cutting working groups and also host a variety of public colloquia and student and practitioner training events. Known as the Southern African Program on Ecosystem Change and Society (SAPECS), its core objectives are to: (1) derive new approaches and empirical insights on SES dynamics in the southern African context; (2) have a tangible impact by mainstreaming knowledge into policy and practice; and (3) grow the community of practice engaged in SES research and governance, including researchers, students and practitioners. This paper reflects on experiences in building the SAPECS community, with the aim of supporting the development of similar networks elsewhere in the world, particularly in the Global South.
- Full Text:
- Date Issued: 2023
Condolence letter to Sheila Burnett from Robert Reynolds Burnett
- Date: 1994-09-26
- Subjects: Burnett, Bill Bendyshe -- 1917-1994 Condolence notes Burnett family
- Type: Image
- Identifier: http://hdl.handle.net/10962/32008 , vital:24002 , This image is held at the Cory Library for Humanities Research at Rhodes University. For further information contact cory@ru.ac.za. The digitisation of this image was made possible through a generous grant received from the Andrew W. Mellon Foundation 2014-2017. , MS 20 0052
- Description: This collection contains the letters of condolences, addressed to Sheila Burnett on the death of her husband Bill Bendyshe Burnett, who died on the 23rd of August 1994.
- Full Text:
- Date Issued: 1994-09-26
Woodlands or wastelands: Examining the value of South Africa's woodlands paper
- Authors: Shackleton, Charlie M , Willis, C B , Scholes, Robert J
- Date: 2001
- Subjects: To be catalogued
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/182158 , vital:43805 , xlink:href="https://hdl.handle.net/10520/EJC33869"
- Description: The savanna woodlands are the largest biome in the country, constituting one-third of South Africa. They are also home to one-quarter of the population, with 70 % of the former homelands being in the savanna biome. Consequently, they have the potential to make a marked contribution to the national economy, both in the formal and informal sector. They are also valuable on a national scale in terms of the ecosystem services they provide, such as carbon storage, biodiversity and water yield. Until recently the real and potential values attached to South Africa's savannas have not been recognised in policy fora and government institutions. Recent policy changes, especially the National Forestry Action Programme and the National Forest Act, have attempted to remedy this situation. These policies have not filtered down to land owners and managers, nor have they resulted in a redirection of government resources. Thus, large areas remain subject to unsustainable use. It is necessary that the true value of woodlands be determined and acknowledged as a stimulus to government agencies, the private sector and local users to use this valuable resource sustainably.
- Full Text:
- Date Issued: 2001
Petrographic and geochemical characterisation of the hangingwall and the footwall rocks (the Dipeta and R.A.T. stratigraphic units) to the Kinsevere and Nambulwa copper ore deposits of the Lufilian Arc, southern Democratic Republic of Congo
- Authors: Nkulu, Robert Kankomba
- Date: 2020
- Subjects: Petrogenesis -- Congo (Democratic Republic) , Analytical geochemistry -- Congo (Democratic Republic) , Copper ores -- Congo (Democratic Republic) , Ore deposits -- Congo (Democratic Republic) , Katangan Sequence , Geological mapping -- Congo (Democratic Republic) , Central African Copperbelt (Congo and Zambia) , Lufilian Arc , Neoproterozoic Katangan R.A.T. (Roches Argilo Talqueuse) Subgroup , Dipeta Subgroup
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/142772 , vital:38115
- Description: The Kinsevere and Nambulwa copper deposits in the Democratic Republic of Congo (D.R.C.) are set in the eastern side of the Neoproterozoic Katanga Supergroup, forming the Lufilian Arc, resulting from a cratonic collision between the Congo and the Kalahari Cratons (ca.620-570_Ma). The Katanga Supergroup was deposited in an extensional rift setting with a sedimentary thickness succession ranging between 7 to 10 km, sub-divided into: − the Roan, the Nguba and the Kundelungu Groups. The stratigraphic column of the Roan Group consists of the R.A.T. (Roche Argilo Talqueuse), the Mines, the Dipeta and the Mwashya Subgroups. Three major deformation phases have been described characterised by complex multiphase tectonics related to a curved superposition of folded, thrust and sheared blocks. The rocks of the R.A.T., Mines and Dipeta Subgroups are recognised as blocks that occur within a stratiform to discordant and diapiritic megabreccia. The blocks were rafted upward with salt tectonics, resulting in the juxtaposition with the hangingwall and the footwall terranes. Therefore, in that context it has been found that the Dipeta may appear overlying the R.A.T. Subgroup through the unconformity decollement surface of heterogeneous breccia. The petrographic observations made of the R.A.T. and Dipeta samples indicates in both units the presence of detrital quartz and feldspar that have been altered and replaced by sericite and muscovite minerals. Gypsum is intimately associated with magnesite, showing an evaporitic environment domain, while magnesite is common as alteration phase both in the R.A.T. and Dipeta Subgroups. Pyrophyllite has been observed in the Dipeta, resulting from reaction of silica with the Kaolinite at low temperature. Accessory detrital minerals include zircon, as well as xenotime intergrown with altered Fe-Ti-oxide hematite, forming complex textures with disseminated Ti-oxides both in R.A.T. and Dipeta units. Major and trace element geochemistry indicates that the Dipeta is more dolomitic and magnesite while the R.A.T. is clay-rich. The Ti2O value of Dipeta and R.A.T samples is relatively low, ranging between 0.36 and 0.69 wt.% respectively, which suggest highly evolved felsic material in the protolith. This is consistent with interpretation based on the Al2O3/TiO2 ratio, which ranges between 18 and 23 for the R.A.T. and Dipeta respectively, indicating an intermediate to felsic granitoids as the protolith of R.A.T. and Dipeta siltstones. The Ti/Zr ratio of R.A.T. and Dipeta samples of less than 10, while, the higher La/Sc ratio of between 2.6 and 5.5 (for the R.A.T. and Dipeta respectively) indicate that both the R.A.T. and Dipeta are active continental and passive margin tectonic setting. Based on the geochemical variation with depth across the R.A.T. and Dipeta and their contact zone, a geochemical fingerprinting suggests that the ratio TiO2/Al2O3 appears to be useful and could be considered as a stratigraphic geochemical maker able to discriminate the R.A.T. and the Dipeta Subgroups during the geological mapping.
- Full Text:
- Date Issued: 2020
Social-ecological change: insights from the Southern African Program on Ecosystem Change and Society
- Authors: Biggs, Reinette , Clements, Hayley S , Cumming, Graeme S , Cundill, Georgina , de Vos, Alta , Hamann, Maike , Luvuno, Linda , Roux, Dirk J , Selomane, Odirlwe , Blanchard, Ryan , Cockburn, Jessica , Dziba, Luthando , Esler, Karen J , Fabricius, Christo , Henriksson, Rebecka , Kotschy, Karen , Lindborg, Regina , Masterson, Vanessa A , Nel, Jeanne L , O'Farrell, Patrick , Palmer, Carolyn G , Pereira, Laura , Pollard, Sharon , Preiser, Rika , Scholes, Robert J , Shackleton, Charlie M , Shackleton, Sheona , Sitas, Nadia , Slingsby, Jasper A , Spierenburg, Maria , Tengö, Maria , Reyers, Belinda
- Date: 2022
- Subjects: To be catalogued
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/399817 , vital:69561 , xlink:href="https://doi.org/10.1080/26395916.2022.2097478"
- Description: Social-ecological systems (SES) research has emerged as an important area of sustainability science, informing and supporting pressing issues of transformation towards more sustainable, just and equitable futures. To date, much SES research has been done in or from the Global North, where the challenges and contexts for supporting sustainability transformations are substantially different from the Global South. This paper synthesises emerging insights on SES dynamics that can inform actions and advance research to support sustainability transformations specifically in the southern African context. The paper draws on work linked to members of the Southern African Program on Ecosystem Change and Society (SAPECS), a leading SES research network in the region, synthesizing key insights with respect to the five core themes of SAPECS: (i) transdisciplinary and engaged research, (ii) ecosystem services and human well-being, (iii) governance institutions and management practices, (iv) spatial relationships and cross-scale connections, and (v) regime shifts, traps and transformations. For each theme, we focus on insights that are particularly novel, interesting or important in the southern African context, and reflect on key research gaps and emerging frontiers for SES research in the region going forward. Such place-based insights are important for understanding the variation in SES dynamics around the world, and are crucial for informing a context-sensitive global agenda to foster sustainability transformations at local to global scales.
- Full Text:
- Date Issued: 2022
The effects of short-duration overnight kraaling on herbaceous vegetation and soils in mesic grassland
- Authors: Mgwali, Nompendulo
- Date: 2023-10-13
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/424721 , vital:72178
- Description: Land degradation is widespread in communal rangelands in the grassland biome of South Africa, and often attributed to overstocking and lack of coordinated management. Excessive pressure on the herbaceous component has contributed to the uncontrolled spread of opportunistic invasive alien woody species e.g. Acacia mearnsii in many degraded areas, resulting in significant loss of ecosystem service capacity, along with soil and land productivity. Short-duration overnight kraaling has been suggested as a tool for restoring degraded rangelands. Recent studies in semi-arid savannas and shrublands have reported increased grass cover, soil nutrients and palatability and concluded that short-duration kraaling is a low-cost and effective way of restoring degraded rangelands using livestock. However, the response of different plant functional types and communities to such intense livestock impact may vary depending on local context. This study used twelve paired kraal and control sites to investigate the effects of short-duration (7-24 days) overnight kraaling of livestock on herbaceous vegetation and soils in a mesic grassland. The study area is generally considered to be overgrazed but has considerable variation in grass composition and basal cover. Sites included relatively intact natural grassland and sites where wattle infestations had been cleared and where mostly bare ground remained. I tested the hypotheses that overnight kraaling would result in (1) increased basal cover due to introduction of grass seed and stimulation of germination through hoof action, (2) increased infiltration due to hoof action, and (3) increased soil nutrients and organic matter due to dung and urine deposition. I also hypothesized that factors such as a site’s initial grass cover, its slope, the occurrence and amount of rainfall before and during kraaling, and the kraaling intensity (number of livestock and duration of the kraaling event) would influence the magnitude and direction of the kraaling effect. The effect of kraaling on vegetation was strongly dependent on initial condition. Kraaling increased basal cover of grasses when sites had low initial basal cover, but decreased basal cover if initial values were over 50%. Infiltration increased if kraaling took place during or after rain but decreased if kraaling took place when soils were dry. Kraaling increased soil P and K. In mesic grasslands, short-duration overnight kraaling is promising as a tool for rehabilitating degraded sites but should be avoided where the grass sward is relatively intact. I recommend that the suitability of kraaling be further evaluated per vegetation type and local context. , Thesis (MSc) -- Faculty of Science, Botany, 2023
- Full Text:
- Date Issued: 2023-10-13
Cytogenetic studies of Pseudobarbus and selected Barbus (Pisces: Cyprinidae) of southern Africa
- Authors: Naran, Daksha
- Date: 1997
- Subjects: Cyprinidae
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5252 , http://hdl.handle.net/10962/d1005095 , Cyprinidae
- Description: The aim of this study was to explore the karyology of temperate southern African barbine species in order to determine the extent and significance of polyploidy within the species. The study presents an optimised in vivo karyological protocol for the small barbines. The analysis of chromosomal data was explored using two approaches; measured (quantitative) and visual (qualitative). The karyology of 16 species of Barbus and six species of Pseudobarbus (Pisces: Cyprinidae) is reported. The study represents an almost complete (22/23 species) karyological survey of the temperate barbines. Chromosome number, arm numbers (NF) are recorded and karyotype morphology described for all species. Nucleolar organiser regions (NOR)s of 18 species have been examined by silver staining. Seven different AgNOR phenotypes occur among the species examined. Chromosome complements of the different species indicate at least three ploidy levels; diploid, tetraploid and hexaploid. Higher ploidy levels are represented in 78% (18/23) of species within the temperate fauna. This finding is remarkable, in the light of global proportions of cyprinid karyotypes, and provides an opportunity to shed further light on the evolution of polyploidy in barbine cyprinids. Results show that morphologically defined species complexes have distinct karyotypes, and most probably form cohesive phyletic clades. A phylogenetic tree was constructed using traditional morphological characters and tested against karyological data. The results suggests that similar ploidy states do not necessarily indicate close relationship between species. This supports a hypothesis of polyphyly for the African barbine cyprinids and also provides independent support for species complexes such as the "chubbyhead barb group" and the Pseudobarbus lineage. The broader implications of karyological findings are discussed within the context of African barbines and recommendations for further cytogenetic research are provided.
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- Date Issued: 1997
Woodlands in South Africa and the national forests act
- Authors: Shackleton, Charlie M
- Date: 2000
- Subjects: To be catalogued
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/182065 , vital:43796 , xlink:href="https://doi.org/10.1080/10295925.2000.9631252"
- Description: The development of the National Forestry Action Programme (NFAP) and promulgation of the National Forests Act (NFA) have established a new vision for the care, management and distribution of benefits from South Africa's woodlands. The Department of water Affairs and Forestry is mandated with ensuring this vision is put into practice. However, historically the Department has had little to do with woodlands, and suffers from a lack of capacity and expertise, a situation which it readily acknowledges. Additionally, the legal definition of a woodland within the NFA is problematic. Within this context, this paper examines oft cited definitions of woodlands and seeks to find an appropriate one for the South African context. It then briefly reviews the major classifications of woodland types at a national scale, as the minimum basis for homogenous reporting units for which monitoring of the success of the NFAP and the NFA should be pursued. Finally, a brief description of moist/dystrophic and arid/eutrophic woodlands is presented.
- Full Text:
- Date Issued: 2000
Spatial and temporal heterogeneity of plant flammability
- Authors: Calitz, Wynand
- Date: 2018
- Subjects: Forest plants , Forest ecology Plants -- Adaptation
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10948/23500 , vital:30572
- Description: Heterogeneity in the natural environment has led to plants adapting traits to fit a niche. Within natural systems, fire has been a major driver of vegetation across the globe. Flammability has been suggested by many to be a trait adapted to reoccurring fire events. Much of the literature on flammability is theoretical and little has been done to cement it as a functional trait. In this thesis, I explore flammability across a wide range of plant species from both fire-free and -driven communities and compare observed results to other functional traits to identify if any link may exist between fire and being flammable. I further focus on the impact moisture availability may have on flammability as fire regimes are often correlated to rainfall season. Lastly, I investigate how different vegetation types respond to seasonal climate in terms of flammability. As a functional trait, flammability does correlate with traits associated with fire-driven system (e.g. small leaves in dense twig matrix). However, fire associated traits were also observed in fire-free species (Forest, Thicket, and Nama-Karoo), and in some cases traits not associated to fire proved flammable (e.g. large leaves on trees). I find that at a regional scale, plant moisture only correlates to flammability when rainfall amounts are well above or below average (e.g. drought). I also identify that species have different responses to moisture fluctuations and that inherent or accidental responses may influence observed flammability (e.g. trichomes). Lastly, I note that some biomes indicate strong association to season or climate (Fynbos and Thicket), while others indicate plasticity towards weather with species having different responses (Grassland). The study is the first to present estimates of flammability across a large number of species sampled at different times of the year. Future research will have to approach flammability as a meta-analysis by experimenting on different scales, particularly temporal and spatial scales.
- Full Text:
- Date Issued: 2018
Spatial and temporal heterogeneity of plant flammability
- Identifier: http://vital.seals.ac.za8080/10948/26318 , http://hdl.handle.net/10948/23495 , vital:30570
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Group portrait of Cecil John Rhodes, Robert Dundas Graham and one other unidentified man, possibly fellow students
- Authors: Grey Brothers Photographers
- Date: 1877
- Subjects: Rhodes, Cecil, 1853-1902 -- Photographs , Graham, Robert Dundas -- Photographs
- Type: Image
- Identifier: http://hdl.handle.net/10962/20224 , vital:22836 , PIC/M 22 , This image is held at the Cory Library for Humanities Research at Rhodes University. For further information contact cory@ru.ac.za. The digitisation of this image was made possible through a generous grant received from the Andrew W. Mellon Foundation 2014-2017.
- Full Text: false
- Date Issued: 1877
Analytic pricing of American put options
- Authors: Glover, Elistan Nicholas
- Date: 2009
- Subjects: Options (Finance) -- Prices -- Mathematical models , Derivative securities -- Prices -- Mathematical models , Finance -- Mathematical models , Martingales (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5566 , http://hdl.handle.net/10962/d1002804 , Options (Finance) -- Prices -- Mathematical models , Derivative securities -- Prices -- Mathematical models , Finance -- Mathematical models , Martingales (Mathematics)
- Description: American options are the most commonly traded financial derivatives in the market. Pricing these options fairly, so as to avoid arbitrage, is of paramount importance. Closed form solutions for American put options cannot be utilised in practice and so numerical techniques are employed. This thesis looks at the work done by other researchers to find an analytic solution to the American put option pricing problem and suggests a practical method, that uses Monte Carlo simulation, to approximate the American put option price. The theory behind option pricing is first discussed using a discrete model. Once the concepts of arbitrage-free pricing and hedging have been dealt with, this model is extended to a continuous-time setting. Martingale theory is introduced to put the option pricing theory in a more formal framework. The construction of a hedging portfolio is discussed in detail and it is shown how financial derivatives are priced according to a unique riskneutral probability measure. Black-Scholes model is discussed and utilised to find closed form solutions to European style options. American options are discussed in detail and it is shown that under certain conditions, American style options can be solved according to closed form solutions. Various numerical techniques are presented to approximate the true American put option price. Chief among these methods is the Richardson extrapolation on a sequence of Bermudan options method that was developed by Geske and Johnson. This model is extended to a Repeated-Richardson extrapolation technique. Finally, a Monte Carlo simulation is used to approximate Bermudan put options. These values are then extrapolated to approximate the price of an American put option. The use of extrapolation techniques was hampered by the presence of non-uniform convergence of the Bermudan put option sequence. When convergence was uniform, the approximations were accurate up to a few cents difference.
- Full Text:
- Date Issued: 2009