OpenUCT :: Browsing by Department "Division of Actuarial Science"

Browsing by Department "Division of Actuarial Science"

Now showing 1 - 20 of 113

Open Access
Accelerated Adjoint Algorithmic Differentiation with Applications in Finance
(2017) De Beer, Jarred; Ouwehand, Peter; Kuttel, Michelle Mary
Adjoint Differentiation's (AD) ability to calculate Greeks efficiently and to machine precision while scaling in constant time to the number of input variables is attractive for calibration and hedging where frequent calculations are required. Algorithmic adjoint differentiation tools automatically generates derivative code and provide interesting challenges in both Computer Science and Mathematics. In this dissertation we focus on a manual implementation with particular emphasis on parallel processing using Graphics Processing Units (GPUs) to accelerate run times. Adjoint differentiation is applied to a Call on Max rainbow option with 3 underlying assets in a Monte Carlo environment. Assets are driven by the Heston stochastic volatility model and implemented using the Milstein discretisation scheme with truncation. The price is calculated along with Deltas and Vegas for each asset, at a total of 6 sensitivities. The application achieves favourable levels of parallelism on all three dimensions implemented by the GPU: Instruction Level Parallelism (ILP), Thread level parallelism (TLP), and Single Instruction Multiple Data (SIMD). We estimate the forward pass of the Milstein discretisation contains an ILP of 3.57 which is between the average range of 2-4. Monte Carlo simulations are embarrassingly parallel and are capable of achieving a high level of concurrency. However, in this context a single kernel running at low occupancy can perform better with a combination of Shared memory, vectorized data structures and a high register count per thread. Run time on the Intel Xeon CPU with 501 760 paths and 360 time steps takes 48.801 seconds. The GT950 Maxwell GPU completed in 0.115 seconds, achieving an 422⇥ speedup and a throughput of 13 million paths per second. The K40 is capable of achieving better performance.
Open Access
Accurate estimation of risk when constructing efficient portfolios for the capital asset pricing model
(2010) Zwane, Samkelo Sifiso; Clark, Allan; Troskie, Casper G
In this paper, we investigate the behaviour of the efficient frontier and optimal portfolio of the Troskie-Hossain Capital Asset Pricing Model (TrosHos CAPM) and Sharpe Capital Asset Pricing Model (Sharpe CAPM) when the covariance structure of the residuals is correlated under the Markowitz formulation. By building in the dynamic time series models: AR, GARCH and AR/GARCH we were able to model the autocorrelation and heteroskedasticity of the residuals.
Open Access
Adjoint Venture: Fast Greeks with Adjoint Algorithmic Differentiation
(2017) McPetrie, Christopher Lindsay; McWalter, Thomas
This dissertation seeks to discuss the adjoint approach to solving affine recursion problems (ARPs) in the context of computing sensitivities of financial instruments. It is shown how, by moving from an intuitive 'forward' approach to solving a recursion to an 'adjoint' approach, one might dramatically increase the computational efficiency of algorithms employed to compute sensitivities via the pathwise derivatives approach in a Monte Carlo setting. Examples are illustrated within the context of the Libor Market Model. Furthermore, these ideas are extended to the paradigm of Adjoint Algorithmic Differentiation, and it is illustrated how the use of sophisticated techniques within this space can further improve the ease of use and efficiency of sensitivity calculations.
Open Access
Alternative distributions in the Black-Litterman model of asset allocation
(2011) Mbofana, Stewart; Becker, Ronald
In this thesis we replace the normal distribution assumption in the calculation of the prior equilibrium returns used in the model with a more general distribution which captures the skewness and fat tails exhibited by stock data. We consider the á stable distributions as an alternative distribution to the normal distribution. Consequently we also consider alternative measures of risk, the Value at Risk and the Conditional Value at Risk other than the variance used in the normal case.
Open Access
Analysing the structure and nature of medical scheme benefit design in South Africa
(2015) Kaplan, Josh Tana; Ramjee, Shivani
This dissertation intends to shed light on open-membership medical scheme benefit design in South Africa. This will be done by analysing the benefit design of 118 benefit options, so as to provide an overview of the structure and nature of the benefit offerings available in the market in 2014. In addition, affordability of these benefit options was analysed in order to identify whether or not there exist connections between the benefits on offer and the price of cover. This paper will argue that at present, the large number of benefit options available in the market, the lack of standardisation between benefit options, together with the mosaic of confusing terminology employed in scheme brochures, creates a highly complex environment that hampers consumer decision making. However, this implicit complexity was found to be necessary owing to the incomplete regulatory environment surrounding medical schemes. The findings of this investigation show that benefit design requires significant attention in order to facilitate equitable access to cover in South Africa.
Open Access
Analysis of CDO tranche valuation and the 2008 credit crisis
(2013) Muzenda, Nevison; Becker, Ronald
The causes of the 2008 financial crisis were wide ranging. Some financial commentators have suggested there were significant inadequacies in the models used to price complex derivatives such as synthetic Collaterilised Debt Obligations (CDOs). We discuss the technical properties of CDOs and the modeling approaches used by CDO traders and the watchdog credit rating agencies. We look at how the pricing models fared before and during the financial crisis. Comparing our model prices to market synthetic CDO prices, we investigate how well these pricing models captured the underlying financial risks of trading in CDOs.
Open Access
Analysis of equity and interest rate returns in South Africa under the context of jump diffusion processes
(2015) Mongwe, Wilson Tsakane
Over the last few decades, there has been vast interest in the modelling of asset returns using jump diffusion processes. This was in part as a result of the realisation that the standard diffusion processes, which do not allow for jumps, were not able to capture the stylized facts that return distributions are leptokurtic and have heavy tails. Although jump diffusion models have been identified as being useful to capture these stylized facts, there has not been consensus as to how these jump diffusion models should be calibrated. This dissertation tackles this calibration issue by considering the basic jump diffusion model of Merton (197G) applied to South African equity and interest rate market data. As there is little access to frequently updated volatility surfaces and option price data in South Africa, the calibration methods that are used in this dissertation are those that require historical returns data only. The methods used are the standard Maximum Likelihood Estimation (MLE) approach, the likelihood profiling method of Honore (1998), the Method of Moments Estimation (MME) technique and the Expectation Maximisation (EM) algorithm. The calibration methods are applied to both simulated and empirical returns data. The simulation and empirical studies show that the standard MLE approach sometimes produces estimators which are not reliable as they are biased and have wide confidence intervals. This is because the likelihood function required for the implementation of the MLE method is not bounded. In the simulation studies, the MME approach produces results which do not make statistical sense, such as negative variances, and is thus not used in the empirical analysis. The best method for calibrating the jump diffusion model to the empirical data is chosen by comparing the width of the bootstrap confidence intervals of the estimators produced by the methods. The empirical analysis indicates that the best method for calibrating equity returns is the EM approach and the best method for calibrating interest rate returns is the likelihood profiling method of Honore (1998).
Open Access
Analysis of fertility estimates in Zimbabwe : A comparison of the census and DHS data
(2014) Madari, Zvikomborero T R; Moultrie, Tom
Analysis of census data is important to uncover new insights as well as highlight where improvements in future data collection are required. The study provides an assessment of the fertility estimates derived from census data in comparison to those derived from the Zimbabwe Demographic and Health Surveys. Robust methods are used to estimate fertility levels and to identify the trends in fertility in Zimbabwe. Fertility decline in Zimbabwe is observed to have started in the early 1980s. The greatest level of decline occurred between the 1980s and the mid-1990s. In more recent years fertility in Zimbabwe has stalled at roughly four children per woman. Using projected parity progression ratios fertility decline has been observed to be in part a result of parity limitation, as fewer women progress to higher parities. A comparison of the census and Zimbabwe Demographic and Health Survey fertility measures show that for the same cohort of women, the measures of fertility are strongly congruent. While there are problems with census data, it has been shown that using robust estimation the census fertility estimates are comparable to those from the Demographic and Health Surveys.
Open Access
Beta, size and value effects on the JSE Securities Exchange, 1994-2007
(2010) Strugnell, Dave; Gilbert, Evan; Kruger, Ryan
Open Access
Bismut–Elworthy–Li formula for subordinated Brownian motion applied to hedging financial derivatives
(Taylor and Francis, 2017-09-27) Kateregga, Michael; Mataramvura, Sure; Taylor, David
he objective of the paper is to extend the results in Fournié, Lasry, Lions, Lebuchoux, and Touzi (1999), Cass and Fritz (2007) for continuous processes to jump processes based on the Bismut–Elworthy–Li (BEL) formula in Elworthy and Li (1994). We construct a jump process using a subordinated Brownian motion where the subordinator is an inverse 훼-stable process (Lt )t≥0 with (0, 1]. The results are derived using Malliavin integration by parts formula. We derive representation formulas for computing financial Greeks and show that in the event when Lt ≡ t, we retrieve the results in Fournié et al. (1999). The purpose is to by-pass the derivative of an (irregular) pay-off function in a jump-type market by introducing a weight term in form of an integral with respect to subordinated Brownian motion. Using MonteCarlo techniques, we estimate financial Greeks for a digital option and show that the BEL formula still performs better for a discontinuous pay-off in a jump asset model setting and that the finite-difference methods are better for continuous pay-offs in a similar setting. In summary, the motivation and contribution of this paper demonstrates that the Malliavin integration by parts representation formula holds for subordinated Brownian motion and, this representation is useful in developing simple and tractable hedging strategies (the Greeks) in jump-type derivatives market as opposed to more complex jump models.
Open Access
Bootstrapping the OIS curve in a South African bank
(2017) Van Heeswijk, Dirk; Mahomed, Obeid
The financial crisis in 2007 highlighted the credit and liquidity risk present in interbank (LIBOR) rates, and resulted in changes to the pricing and valuation of financial instruments. The shift to Overnight Indexed Swap (OIS) discounting and multi-curve framework led to changes in the construction of interest rate zero curves, with the OIS curve being central to this methodology. Developed markets, such as the European (EUR), were able to adopt this framework due to the existence of a liquid OIS market. In the case of the South African (ZAR) market, the lack of such tradeable instruments poses the issue of how to construct or infer the OIS curve. Jakarasi et al. (2015) proposed a method to infer the OIS curve through the statistical relationship between SAFEX ROD and 3M JIBAR. The extension of the statistical relationship used by Jakarasi et al. (2015) to more statistically rigorous models, capable of capturing more information relating to the relationship between the rates, arises from the expected cointegrating relationship exhibited between rates. This dissertation investigates the implementation of such statistical models to infer the OIS curve in the ZAR market.
Open Access
Calibrating the LIBOR market model to swaptions with an extension for illiquidity in South Africa
(2016) Moodliyar, Leenesh; Taylor, David
The popularity of the LIBOR Market Model (LMM) in interest rate modelling is a result of its consistency with market practice of pricing interest rate derivatives. In the context of a life insurance company, the LMM is calibrated to swaptions as they are actively traded for a wide variety of maturities and they serve as the natural hedge instruments for many of the long dated maturity products with embedded options. Before calibrating the model we extend the calibration process to address the issue of illiquidity in the South African swaption market. The swaption surface used in calibrating the model is generated with market implied quotes for the hedgeable component and thereafter using historical volatilities for the unhedgeable or illiquid component. Rebonato's 3 parameter correlation function proposed by Rebonato (2005) provides the best fit to historical data. We assume a general piecewise constant parameterisation for the instantaneous forward rate volatilities. These volatilities are then determined analytically using the Rectangular Cascade Calibration Algorithm from Brigo and Morini (2006). The calibration generates a stable volatility term structure with the instantaneous forward rate volatilities being positive and real. Through an extension of the calibration we are able to capture the benefits of a pure replication component and accommodate a large unhedgeable component in the price faced by life insurance companies in South Africa.
Open Access
Child mortality in South Africa - we have lost touch
(2007) Bradshaw, Debbie; Dorrington, Rob
Reducing child mortality is, appropriately, one of the eight Millennium Development Goals (MDGs) for reducing poverty and inequality in the world. The target for this goal is to reduce child mortality by 2015 by two-thirds of the rate in 1990. Monitoring child mortality rates, however, is posing a challenge for low- and middle-income countries. Estimates that many countries, particularly in Africa, use to track progress in meeting this goal have to be extrapolated from earlier empirical data, since there are no up-to-date data. Despite great strides that have been made in improving population health statistics, South Africa is unfortunately no exception. The most recent reasonably reliable estimates of child mortality for South Africa are for the mid-1990s, in other words 10 years out of date.
Open Access
A comparative analysis of non-linear techniques in South African stock selection
(2015) Hutheram, Nikhil Arnaidas; Bosman, Petrus
Forecasting stock performance has long been one of the primary objectives of financial practitioners. Literature has shown that the classical linear approach to modelling the interactions among company-specific factors and its stock market re- turns in time have become less suited for capturing the movements of the stock market. Hence, attempts to predict the performance of a stock have become associated with additional layers of complexity. This has led to the adoption of non-linear approaches to forecast stock performance. This dissertation explores the performance of some non-linear models in the South African market. These were classification and regression trees (CART), logistic regression and a random forest approach com- pared against a linear regression model. Moreover, a hybrid model between CART and logistic regression was considered. The models fell into two categories (i.e., static and dynamic models). Using a set of classification and portfolio performance metrics it was found that that a dynamic modelling approach outperformed a static approach. Overall, the logistic and linear regression models dominated in terms of performance against the tree-based models and hybrid approaches. The results also demonstrated that a hybrid approach offered an improvement over a stand-alone CART.
Open Access
Completeness of death registration in Cape Town and its health districts, 1996-2004
(2007) Zinyakatira, Nesbert; Dorrington, Rob
It is important for health planners to have timeous and accurate data on deaths. The Department of Home Affairs is responsible for the registration of deaths and the City of Cape Town has a well-established system of collating the death statistics based on vital registration, but the completeness of the death registration has not been assessed previously. The completeness was assessed for the City of Cape Town by comparing their statistics with an estimate based on data obtained from adult deaths reported in the 2001 census. A second approach assessed the trend in completeness between 1996 and 2004 by identifying three rates of mortality considered to be stable over time (non-lung and non-oesophageal cancers, the 10-14 age group and the 60+ age group) and inspecting to observe whether there was any trend apparent over time. Since deaths in most cases are under reported, and the under reporting usually differs in completeness between children and adults, child deaths from the ASSA model projection assuming that they are more complete were compared with the child deaths from the vital registration between 1996 and 2004 to check for completeness of the child vital registration data in Cape Town and its eight health districts The results show high levels of completeness in the adult deaths for Cape Town as a whole in 2001, around 95 per cent, but varying levels in the health districts. The completeness of reporting of male deaths in Cape Town declines with age, whilst completeness for females is fairly level with respect to age, with similar trends being observed in the health districts. Completeness of child (0 -4) death registration averaged around 60 per cent, about 35 per cent lower than the completeness of adult deaths in Cape Town. Cape Town as a whole and most of its health districts revealed two levels of completeness in the registration of deaths, 1996-1999 and 2001-2004 with 2000 sometimes consistent with the first and sometimes with the second period or different from either period in some of the health districts. In conclusion, the completeness estimates obtained are more rigorous from 2001 onwards suggesting that they can be reliably used to monitor trends in the levels of mortality in the city of Cape Town.
Open Access
Compositional observations and their role in regression
(2018) Spracklen, Callum
This dissertation examines the properties and some of the uses of compositional data. It gives a brief history of the distinction between 'normal' data and compositions, as well as the various methods of analysing compositional data. It is mainly concerned with performing regression analysis including compositions. In order to model a composition it is necessary to understand the nature of compositions and how to use standard statistical tools with them. This dissertation describes the simplex and several functions which can be performed in it, as well as introducing several useful covariate structures for compositional samples after transformation. It also introduces the transformations between the simplex and unconstrained real space. The dissertation concludes with four examples of regression analyses involving compositions.
Open Access
Constructing volatility surfaces for managed funds
(2014) Brinkman, Trevor Joseph; Van Rooyen, Marchand; Innocenzi, Paolo
In this dissertation, a methodology is developed for constructing a volatility surface for a managed fund by extending the work of Bakshi et al. (2003) and Taylor (2014). The power utility assumption (with constant relative risk aversion for a specific maturity) and historical returns series data are used for the identified factors in influencing the return of the fund and the fund itself. The coefficient of relative risk aversion for a specific maturity and market is estimated from quoted option prices on a market index. This is used in combination with the identified factors and fund return series to estimate the risk-neutral skewness of the fund. An optimisation procedure is then used to determine the volatility smile of the fund for a specific maturity. Thereafter, the volatility surface of the fund is constructed by repeating each step for different maturities. Although this methodology produces sensible results, the optimisation routine used is sensitive to initial values and constraints.
Open Access
The cost of using misspecified models to exercise and hedge American options on coupon bearing bonds
(2016) Welihockyj, Alexander; Silverman, Searle; McWalter, Thomas
This dissertation investigates the cost of using single-factor models to exercise and hedge American options on South African coupon bearing bonds, when the simulated market term structure is driven by a two-factor model. Even if the single factor models are re-calibrated on a daily basis to the term structure, we find that the exercise and hedge strategies can be suboptimal and incur large losses. There is a vast body of research suggesting that real market term structures are in actual fact driven by multiple factors, so suboptimal losses can be largely reduced by simply employing a well-specified multi-factor model.
Open Access
Currency trios - using geometric concepts to visualise and interpret relationships between currencies
(2016) Davidson, Abby; Mahomed, Obeid; Polakow, Daniel; Van de Linde, Gideon
A currency trio is a set of three currencies and their respective exchange rates, which have a relationship fixed by a triangular arbitrage condition. This condition forms the basis for the derivation of a geometric interpretation of the relationships between the exchange rates. In the geometric framework, the three currencies in a currency trio are represented by a triangle, where each of the vertices represents a currency. The volatilities of the exchange rates are represented by the lengths of the sides joining the respective currencies and the cosine of each angle represents the correlation between the two exchange rates depicted by the angle's adjacent sides. The geometric approach is particularly useful when dealing with implied data as it allows the calculation of implied correlation using implied volatility. This is valuable as implied volatility is frequently quoted in the foreign exchange market; whereas, implied correlation is not directly quoted and is more difficult to extract from market data. This dissertation aims to thoroughly investigate the geometric framework and use it to visualise and interpret the relationships between currencies in a currency trio. The analysis will initially look at currency trios with realised spot data before moving on to implied data. In the implied data context, the framework will be used to extract and evaluate implied correlation estimates using implied volatility data extracted from the foreign exchange market. The framework will be extended to investigate whether an illiquid option can be proxy hedged using options on the two other currencies in a currency trio. Finally, the findings will be discussed and the feasibility of the applications of the framework will be considered.
Open Access
Deriving traditional reproductive regimes to explain subnational fertility differentials in Zambia
(2008) Wotela, Kambidima; Moultrie, Tom
This thesis applies multivariate statistical techniques to six data sets to account for past and present-day features underlying ethnic fertility differentials in Zambia.