Theses and Dissertations (Actuarial Science)

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Applying stochastic volatility models in the risk-neutral and real-world probability measures
(University of Pretoria, 2023) Mare, Eben; alexilevendis@gmail.com; Levendis, Alexis Jacques
Stochastic volatility models have become immensely popular since their introduction in 1993 by Heston. This is because their dynamics are more consistent with market behaviour compared to the standard Black-Scholes model. More specifically, stochastic volatility models can somewhat capture the asymmetric distribution often observed in daily equity returns. Numerous extensions to the stochastic volatility model of Heston have since been proposed, including jumps and stochastic interest rates. Due to their complex dynamics, numerical methods such as Monte Carlo simulation, the fast Fourier transform (FFT), and the efficient method of moments (EMM) are often required to calibrate and implement stochastic volatility models. In this thesis, we explore the application of stochastic volatility models to a variety of problems for which research is still in its infancy phase. We consider the pricing of embedded derivatives in the South African life insurance industry given the illiquid derivatives market; the pricing of rainbow and spread options that depend on two underlying assets; the calibration of stochastic volatility models with jumps to historical equity returns; and the use of stochastic volatility models in static hedging. Our findings suggest that stochastic interest rates are the dominant risk driver when pricing long-dated contingent claims; the FFT significantly outperforms Monte Carlo simulation in terms of efficiency; jumps are an important factor required to explain daily equity returns; and static hedging is a simple and effective way to replicate vanilla and exotic options.
Time-varying volatility models and indices : a GARCH option pricing approach
(University of Pretoria, 2022) Mare, Eben; venter.pierre7@gmail.com; Venter, Pierre Johan
In this thesis, the generalised autoregressive conditional heteroskedasticity (GARCH) option pricing model is applied to illiquid markets, volatility indices and in a modern derivative pricing framework. Chapter 2 provides empirical support for the use of a volatility index to obtain a more accurate GARCH option pricing model (applied to the South African equity market). In Chapter 3, the analysis (GARCH option pricing and volatility indices) is extended to FX markets. Empirical results show that asymmetry is an important factor to consider when modelling FX volatility indices. The aim of Chapter 4 is to quantify the effect of asymmetry in the cryptocurrency market. Furthermore, the accuracy of the GARCH option pricing model applied to cryptocurrencies is also considered. Results indicate that the GARCH option pricing model produces reasonable price discovery, and that asymmetric effects are not significant when pricing cryptocurrency options. Chapter 5 focuses on the construction of a cryptocurrency volatility index, the models in Chapter 4 are used as a basis. The term structure of the GARCH generated volatility indices are consistent with expectations. Furthermore, short term volatility tends to increase when large jumps occur in the underlying asset. In Chapter 6, the Heston–Nandi futures option pricing model is applied to Bitcoin (BTC) futures options. The model prices are compared to market prices to give an indication of the pricing performance. In addition, a multivariate Bitcoin futures option pricing methodology based on a multivatiate GARCH model is developed. The empirical results show that a symmetric model is a better fit when applied to Bitcoin futures returns, and also produces more accurate option prices compared to market prices for two out of three expiry dates considered. Chapter 7 focuses on the pricing of volatility index options respectively. In Chapter 7, the GARCH option pricing model is applied to the Standard and Poor's 500 (S&P500) Volatility Index (VIX) option pricing. The different GARCH models are fitted to VIX futures returns. The results show that the symmetric GARCH model with skewed Student-t errors is the best performing model, and that the GARCH option pricing model provides reasonable price discovery when applied to the VIX. In Chapter 8, the standard Black model and Heston-Nandi futures options pricing model are applied to the hedging of VIX futures options. The hedge performance is compared based on the stability of the profit and loss distribution (P&L) of the hedged portfolio. Empirical results show that the Heston-Nandi futures option pricing model is more reliable when applied to hedging of VIX futures options. The focus of Chapter 9 is the application of the GARCH model to the pricing of collateralised options in the South African equity market. Symmetric GARCH and nonlinear asymmetric GARCH (AGARCH) models are considered. The models are used to price fully collateralised and zero collateral options (European, Asian, and lookback options). The effect of collateral is illustrated by the difference between zero collateral and fully collateralised option price surfaces. Finally, the effect of asymmetry is shown by the difference between the symmetric and asymmetric GARCH option price surfaces. Finally, a closed-form expression for a collateralised European option in the presence of counterparty credit risk and stochastic volatility is derived in Chapter 10. The model is applied to S&P500 index options. The option prices obtained are consistent with expectations, default risky options are cheaper than options with no counterparty credit risk, and fully collateralised options are more expensive when compared to zero collateral options. The effect of correlation is tested by plotting the default risky at-the-money (ATM) option price for different levels of correlation. The results indicate that correlation has an insignificant impact when pricing using the calibrated parameters.
Modelling of financial risk using forward-looking distributions derived from contingent claims
(University of Pretoria, 2022) Mare, Eben; vvanappel@gmail.com; Van Appel, Vaughan
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.
Improving our understanding of the equal weighted portfolio
(University of Pretoria, 2022) Mare, Eben; byran.taljaard@gmail.com; Taljaard, Byran H.
This thesis analyses the performance of the equal weighted portfolio using an approach from stochastic portfolio theory. This framework allows for the decomposition of the relative performance of the equal weighted portfolio into four main parts; the change in the concentration of the cap weighted portfolio, the excess return generated by a diversification benefit, the difference in dividend rates, and a term called the leakage effect. In general equal weighted portfolios do outperform their cap weighted portfolio counterparts, although with varying degrees across different countries. In South Africa, for example, high levels of leakage over the past ten years and increasing concentration have led to poor relative performance of the equal weighted portfolio. In other countries such as the United Kingdom and Japan, equal weighted portfolios have done very well, with high levels of diversification benefits and low levels of leakage. Two models are presented in an attempt to reduce the relative drawdowns of the equal weighted portfolio and to blend the two weights (equal and cap) in an optimal manner. These models appear to do well in markets where the equal weighted portfolio has poor performance and large relative drawdowns.
The effect of observation errors on parameter estimates applied to seismic hazard and insurance risk modelling
(University of Pretoria, 2014-04-30) Kijko, Andrzej; Beyers, Frederik Johannes Conradie; conrad.beyers@up.ac.za; Pretorius, Samantha
The research attempts to resolve which method of estimation is the most consistent for the parameters of the earthquake model, and how these different methods of estimation, as well as other changes, in the earthquake model parameters affect the damage estimates for a specific area. The research also investigates different methods of parameter estimation in the context of the log-linear relationship characterised by the Gutenberg-Richter relation. Traditional methods are compared to those methods that take uncertainty in the underlying data into account. Alternative methods based on Bayesian statistics are investigated briefly. The efficiency of the feasible methods is investigated by comparing the results for a large number of synthetic earthquake catalogues for which the parameters are known and errors have been incorporated into each observation. In the second part of the study, the effects of changes in key parameters of the earthquake model on damage estimates are investigated. This includes an investigation of the different methods of estimation and their effect on the damage estimates. It is found that parameter estimates are affected by observation errors. If errors are not included in the method of estimation, the estimate is subject to bias. The nature of the errors determines the level of bias. It is concluded that uncertainty in the data used in earthquake parameter estimates is largely a function of the quality of the data that is available. The inaccuracy of parameter estimates depends on the nature of the errors that are present in the data. In turn, the nature of the errors in an earthquake catalogue depends on the method of compilation of the catalogue and can vary from being negligible, for single source catalogues for an area with a sophisticated seismograph network, to fairly impactful, for historical earthquake catalogues that predate seismograph networks. Probabilistic seismic risk assessment is used as a catastrophe modelling tool to circumvent the problem of scarce loss data in areas of low seismicity and is applied in this study for the greater Cape Town region in South Africa. The results of the risk assessment demonstrate that seemingly small changes in underlying earthquake parameters as a result of the incorporation of errors can lead to significant changes in loss estimates for buildings in an area of low seismicity.
Investigating the threshold of event detection with application to earthquake and operational risk theory
(University of Pretoria, 2013) Kijko, Andrzej; Beyers, Frederik Johannes Conradie; cjdw@imiginet.co.za; De Witt, Corn e Juan
This study provides systematic analysis of points of structural change in probability distributions. In observed frequency data of earthquakes, such a threshold exists due to the non-detection of events below a certain magnitude. By examining the factors in uencing the operational risk exposure of institutions, a similar threshold is hypothesized to exist in operational loss data. In both elds of study, this threshold is termed the threshold of completeness, above which 100% of events are de- tected. External factors have caused this level of completeness to shift over time for earthquake data. The level of complete recording in uences the volume of data that can be consistently incorporated in a study of seismic activity. Such data can be used by re-insurers and direct writers of catastrophe agreements who deal in seismic hazard. Historically, a variety of methods have been proposed by authors in an attempt to gauge the location of the threshold of completeness in earthquake data. This study aims to evaluate the e cacy of some of the most prominent methods under di ering assumptions regarding the incomplete portion of the data. Furthermore, a new threshold estimation scheme (MITC) is developed and tested against the prevailing methods. Additionally, earthquake data and the wealth of literature will aid in introductory analysis and assessing applicability of esti- mation techniques in the context of operational losses.
Artificial neural networks and their application to modelling South African market returns
(University of Pretoria, 2014) Beyers, Frederik Johannes Conradie; De Villiers, Johan Pieter ; mineonly.mattie@gmail.com; Smith, Matthew Lee
The modelling technique known as Artificial Neural Networks (ANNs) is investigated. ANNs have the ability to detect and project non-linear relationships between variables. Further, they can adapt in dynamically changing environments while providing accurate results. A method of constructing ANNs in order to form a forecasting system is presented here. Further, in many of the applications studies, ANNs are fitted using crude guesses as to the efficient input parameters. In this study detailed investigations into parameter estimates are performed. In addition, ANNs and traditional models (ARIMA, seasonal smoothing, geometric Brownian motion, etc.) are constructed to forecast monthly inflation and the average monthly return on the money, bond and equity markets in South Africa from 1975 to 2010. The ANNs constructed are done through an integrated and isolated approach. The performance of the traditional and ANN models are compared. No general conclusion, as to which model is superior for all the applications considered, can be made. This suggests that ANNs perform as well as traditional models when forecasting financial markets. Further, it is found that the money market and inflation are forecast efficiently through all the models, over a single month. As the forecast period extends to three months the money market favours the traditional model. However, a forecast period of twelve months leads to the preference of ANNs in the case of the money market. Neither technique can forecast the equity or bond market accurately, as these require additional explanatory variables to those considered. As the forecast period increased, the forecast accuracy decreased for all the models. The integrated ANNs, which allow interaction between the markets, do not lead to improved forecasts which indicates that the relationships between the markets have a limited effect on the future values of the markets. Hybrid models are constructed, trained and tested for the money market and inflation. They are found to add value to traditional models when forecasting inflation but not the money market. The sensitivity of the performance of ANNs and the traditional model to different subsets of the inflation data is tested. No statistical difference between the models is found. The implementation advantages of ANNs are also described.
Top-down stress testing of the largest full service South African banks
(University of Pretoria, 2020) Beyers, Frederik Johannes Conradie; dcuconradie@gmail.com; Conradie, Dirk Cornelis Uys
Banks are key to a well-functioning economy. Periods of economic stress could put banks and therefore the financial system at risk so regulators such as the Prudential Authority in South Africa need to know if banks are resilient to economic stress. A model that forecasts the impact of severe economic stress is developed using publicly available information. The model forecasts the credit losses, deposit volumes and other general equity movements of the biggest five full-service South African banks to assess capital and liquidity strain for any defined macroeconomic stress scenario over the next 3 years. The full-service banks being considered account for more than 90% of all bank lending in deposits in the market and therefore covers the vast majority of banking systemic risk in South Africa. It is shown that different macroeconomic factors affect these banks in different ways due to differences in the type of customers with deposits with each institution and differences in credit risk associated with various loan products. From an overall market perspective economic growth, lending levels, household debt levels and equity markets are the key drivers of deposit volumes. Credit risk in turn is primarily driven by interest rates, inflation and household debt to disposable income.
Modelling South African social unrest between 1997 and 2016
(University of Pretoria, 2019) Beyers, Frederik Johannes Conradie; Van Staden, Paul J.; Venter, Marli; sallyannesmart@gmail.com; Smart, Sally-Anne
Social unrest, terrorism and other forms of political violence events are highly unpredictable. These events are driven by human intent and intelligence, both of which are extremely difficult to model accurately. This has resulted in a scarcity of insurance products that cover these types of perils. Links have been found between the incidence of political violence and various economic and socioeconomic variables, but to date no relationships have been identified in South Africa. The aim of this study was to address this. Firstly, by identifying relationships between the incidence of social unrest events and economic and socio-economic variables in South Africa and secondly by using these interactions to model social unrest. Spearman’s rank correlation and trendline analysis were used to compare the direction and strength of the relationships that exist between protests and the economic and socio-economic variables. To gain additional insight with regards to South African protests, daily, monthly, quarterly and annual protest models were created. This was done using four different modelling techniques, namely univariate time series, linear regression, lagged regression and the VAR (1) model. The forecasting abilities of the models were analysed using both a one-step and n-step forecasting procedure. Variations in relationships for different types of protests were also considered for five different subcategories. Spearman’s rank correlation and trendline analysis showed that the relationships between protests and economic and socio-economic variables were sensitive to changes in data frequency and the use of either national or provincial data. The daily, monthly, quarterly and annual models all had power in explaining the variation that was observed in the protest data. The annual univariate model had the highest explanatory power (R2 = 0.8721) this was followed by the quarterly VAR (1) model (R2 = 0.8659), while the monthly lagged regression model had a R2 of 0.8138. The one-step forecasting procedure found that the monthly lagged regression model outperformed the monthly VAR (1) model in the short term. The converse was seen for the short-term performance of the quarterly models. In the long term, the VAR (1) model outperformed the other models. Limitations were identified within the lagged regression model’s forecasting abilities. As a model’s long-term forecasting ability is important in the insurance world, the VAR (1) model was deemed as the best modelling technique for South African social unrest. Further model limitations were identified when the subcategories of protests were considered. This study demonstrates that with the use of the applicable economic and socio-economic variables, social unrest events in South Africa can be modelled.
Quantitative topics in portfolio and risk management
(University of Pretoria, 2019) Maré, Eben; emlyn.flint@gmail.com; Flint, Emlyn James
The modern quantitative portfolio manager is the quintessential “jack of all trades”. Not only do they need to be an expert in the specific area of portfolio management, they also need to have a thorough understanding of the related areas of valuation, data processing, risk management and performance analysis. What this means practically is that quantitative portfolio managers are regularly faced with problems spanning the entire P − Q spectrum of quantitative finance. Spurred by this reality, the central research question motivating this thesis is exactly the core motivation behind every decision taken by a quantitative portfolio manager: What is the most efficient, practical method for constructing, managing and evaluating optimal multi-asset portfolios in dynamic, non-normal markets? In this thesis, we attempt to provide insight into this broad central research question by offering new perspectives and practical solutions to a selection of sub-problems that a quantitative portfolio manager would have to address in practice. In particular, this thesis is comprised of six essays that each tackle specific problems in the related areas of derivatives, return modelling, systematic trading strategies and portfolio construction.
An analysis of the determinants of sovereign credit ratings
(University of Pretoria, 2018) Beyers, Frederik Johannes Conradie; Seymore, Reyno; De Freitas, Allan; jackyang89125@yahoo.com; Yang, Jack Chao
The study aims to quantitatively assess the extent to which sovereign ratings could be explained by a set of economic variables. A wide variety of factors could potentially bias a credit rating agency’s decision. The analysis begins with replicating the results found in a seminal analysis by Cantor and Packer (1996). This analysis expanded by including more countries, dynamic over time and time lags. Multiple complementary statistical models and a Random Forest model are explored in this study. To ensure robustness of the model, out-sample-testing is applied. The results show that GNI per capita, GDP growth, total debt to GDP, inflation rate, default amount, default indicator, HDI, change in HDI and IMF indicator are statistically significant. It is observed that current account to GDP, GDP growth and inflation rate have a time-lagged effect on sovereign ratings. A further analysis by separating between developing and developed countries using the IMF indicator suggests that there is a discrepancy between developing countries ratings and developed country ratings. The model results also support the existence of subjective decisions or adjustments in sovereign risk assessment.
Network structure indirect losses and financial contagion in inhomogeneous stochastic interbank networks
(University of Pretoria, 2019) Beyers, Frederik Johannes Conradie; Van Zyl, A.J.; waltersnadine20@yahoo.com; Walters, Nadine Mari
We introduce new tiered bank network structures, allowing for many di erent bank sizes, and compare risk propagation in these structures with the well-known Erd˝os-R´enyi, assortative and disassortative structures. The simulations indicate that in the presence of market sentiment and liquidity e ects, the details of the structures in combination with the distribution of assets, the system’s interconnectedness and its size are crucially important in determining the risk of major capital loss in the network. In fact, even networks with similar levels of tiering can behave markedly di erent depending on these factors. In the absence of market sentiment and liquidity e ects, the di erences between the network structures is smaller. This highlights the importance of considering the network structure in conjunction with network characteristics, market sentiment and liquidity e ects. This implies that policy actions aimed at influencing a network’s characteristics must consider all aspects unique to that particular system and cannot follow a ‘one-size-fits-all’ approach. The framework is illustrated with an application using South African bank balance sheet data. Spikes in simulated assessments of systemic risk agree closely with spikes in documented subjective assessments of this risk. This indicates that network models can be useful for monitoring systemic risk levels. In a large network setting, the study then considers the fraction of nodes that default in stochastic, inhomogeneous financial networks following an initial shock to the system. Results for deterministic sequences of networks are generalized to stochastic networks to account for interbank lending relationships that change frequently. A general class of inhomogeneous stochastic networks is proposed for use in systemic risk research, and we illustrate how results that hold for Erd˝os-R´enyi networks can be generalized to the proposed network class. The network structure of a system is determined by interbank lending behaviour which may vary according to the relative sizes of the banks. We then use the results to illustrate how network structure influences the systemic risk inherent in large banking systems.
Hattendorff’s theorem and Thiele’s differential equation generalized
(University of Pretoria, 2007-02-20) Swart, Johan; upetd@ais.up.ac.za; Messerschmidt, Reinhardt
Hattendorff's theorem on the zero means and uncorrelatedness of losses in disjoint time periods on a life insurance policy is derived for payment streams, discount functions and time periods that are all stochastic. Thiele's differential equation, describing the development of life insurance policy reserves over the contract period, is derived for stochastic payment streams generated by point processes with intensities. The development follows that by Norberg. In pursuit of these aims, the basic properties of Lebesgue-Stieltjes integration are spelled out in detail. An axiomatic approach to the discounting of payment streams is presented, and a characterization in terms of the integral of a discount function is derived, again following the development by Norberg. The required concepts and tools from the theory of continuous time stochastic processes, in particular point processes, are surveyed.
Risk evaluation techniques in a general insurance environment
(University of Pretoria, 2005-11-01) Marx, G.L. (George Louis), 1954-; upetd@up.ac.za; Van den Heever, Rudolf Johannes
Please read the abstract in the section 00front of this document
Die wanbetaling van lewensversekering : ‘n analise van die redes, die gevolge vir die versekeraar, kliënt, bemarker en remediërende optrede (Afrikaans)
(University of Pretoria, 2011-04-07) De Wet, J.M.; Marx, G.L. (George Louis), 1954-; upetd@up.ac.za; Van der Walt, Phillipus Jacobus
AFRIKAANS: Die lewensversekeringsbedryf in Suid-Arrika is ‘n groot mobi1iseerder van langtermyn diskresionêre kapitaal en verleen beskerming teen finansiële ver1iese wat weens die intrede van persoonlike risiko’s voorkom. Die mate waarin fondse na die bedryf vloei toon dat die publiek lewensversekering wel belangrik ag en dat dit reeds ‘n algemene begrip en ‘n belangrike komponent van die uitgawes of kontantvloei van baie huishoudings vorm. Soos in die res van die wêreld. gaan die plaaslike bedryf ook gebuk onder die probleem dat k1iënte hulle versekering tot nadeel van al die belanghebbers ontydig beëindig. Daar bestaan nie ‘n bepaalde of dominante rede vir die wanbetaling van versekering nie, maar dit kan aan die sameloop van verskillende faktore toegeskryf word. Groot bedrae geld word op hierdie wyse oneffektief aangewend en nie benut waarvoor dit bedoel was nie. Uit die studie het dit duidclik geword dat die kliënte baie onkundig met betrekking tot persoonlike finansiële beplanning is waar ’n versekering ‘n besondere belangrike rol speel. Tydens die bemarkingsproses maak kliënte hoofsaaklik op die tussengangers se ervaring. kennis, betroubaarheid, integriteit ens., staat om van die “beste of toepas1ike advies" voorsien te word. Dit plaas lewensversekering in die kader van finansiële dienste en moet die kenmerke, te wete. nie-tasbaarheid. bederfbaarheid ens., deeglik in ag geneem word. K1iënte se reaktiewe optrede veroorsaak dat die meeste versekering aan hulle verkoop en nie gekoop word nie. Voorspruitend hieruit rus ‘n verpligting op versekeraars om toe te sien dat hulle tussengangers toegerus is om aan die kliënte se verwagtinge te voldoen. Indien nie, is die beskerming wat tans aan die kliënte via selfregulering, die Suid-Afrikaanse reg, ens., verleen word, so gebrekkig dat dit van nul en gener waarde is. Die belanghebbers se interafhanklikheid vereis dat elkeen ‘n verpligting het om toe te sien dat die diskresionêre kapitaal optimaal benut word. Die versekeraars se onherroeplike enIof onbeperkte afhanklikheid van kliënte en tussengangers noodsaak daadwerklike optrede deur hulle om die posisie te verbeter. Daar is egter geen beperking op versekeraars om aksie te neem nie, behalwe dit wat hulle self opgelê het. Die persepsie is dat die versekeraars poog om hulle verpligtinge teenoor die kliënte te systap. Die versekeraars kan die probleem op ‘n gefragmenteerde basis hanteer. Die gevaar is egter dat die simptome, in plaas van die probleem aangespreek word. Die ideaal is om die probleem holisties aan te spreek waarby die totale bemarkingsproses in heroorweging geneem behoort te word. Dit is egter twyfelagtig of statutêre intervensie enige oplossing gaan bied.
A framework for modelling losses arising from natural catastrophes in South Africa
(University of Pretoria, 2003-05-15) Marx, G.L. (George Louis), 1954-; roger.grobler@gmail.com; Grobler, Roger R.
Property insurance covers policyholders against losses arising out of a wide range of occurrences. Premiums are calculated by taking into account estimates of the frequency and the severity of the losses. Estimating the frequency and severity arising from claims caused by natural catastrophes is difficult, due to the relatively low frequency of natural catastrophes, and the unavailability of historical catastrophe claims data. The accumulation of a large number of claims in the geographical area affected by the catastrophe is of particular interest to insurers and reinsurers alike. This dissertation discusses the fundamental issues underlying the modelling insurance losses from natural catastrophes in South Africa. A suggestion is given of the key parameters that need to be taken into account, and a framework is given for models describing losses arising from floods, hail and tornadoes. Copyright

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