### Browsing by Author "Mahomed, Obeid"

- ItemOpen AccessA Comparison Between Break-Even Volatility and Deep Hedging For Option Pricing(2022) Claassen, Quintin; Mahomed, ObeidThe Black-Scholes (1973) closed-form option pricing approach is underpinned by numerous well-known assumptions (see (Taleb, 1997, pg.110-111) or (Wilmott, 1998, ch.19)), where much attention has been paid in particular to the assumption of constant volatility, which does not hold in practice (Yalincak, 2012). The standard in industry is to use various volatility estimation and parameterisation techniques when pricing to more closely recover the market-implied volatility skew. One such technique is to use Break-Even Volatility (BEV), the method of retrospectively solving for the volatility which sets the hedging profit and loss at option maturity to zero (conditional on a single, or set of, stock price paths). However, using BEV still means pricing using existing model frameworks (and using the assumptions which come with them). The new paradigm of Deep Hedging (DH) (as explored by Buehler et al. (2019)), ie. using deep neural networks to solve for optimal option prices (and the respective parameters needed to hedge these options at discrete time steps), has allowed the market-maker to go ‘modelfree', in the sense of being able to price without any prior assumptions about stock price dynamics (which are needed in the traditional closed-form pricing approach). Using simulated stock price data of various model dynamics, we first investigate whether DH is more successful than BEV in recovering the model implied volatility surface. We find both to perform reasonably well for time-homogeneous models, but DH struggles to recover correct results for time in-homogeneous models. Thereafter, we analyse the impact of incorporating risk-aversion for both approaches only for time-homogeneous models. We find both methods to produce pricing results inline with varying risk aversion levels. We note the simple architecture of our DHNN as a potential point of departure for more complex neural networks.
- ItemOpen AccessAn Application of Deep Hedging in Pricing and Hedging Caplets on the Prime Lending Rate(2022) Patel, Keyur; Mahomed, ObeidDerivatives in South Africa are traded via an exchange, such as the JSE's derivatives markets, or over-the-counter (OTC). This dissertation focuses on the pricing and hedging of caplets written on the South African prime lending rate. In a complete market, caplets can be continuously hedged with zero risk. However, in the particular case of caplets written on the prime lending rate, market completeness ceases to exist. This is because the prime lending rate is a benchmark for retail lending and is not tradeable, in general. Since parametric models may not be specified and calibrated for such incomplete markets, the aim of this dissertation is to consider the deep hedging approach of Buehler et al. (2019) for pricing and hedging such a derivative. First, a model dependent approach is taken to set a benchmark level of performance. This approach is derived using techniques outlined in West (2008) which rely heavily on interest rate pairs being cointegrated to use the market standard Black (1976) model. Thereafter, the deep hedging approach is considered in which a neural network is set up and used to price and hedge the caplets. The deep hedging approach performs at least as well as the model dependent approach. Furthermore, the deep hedging approach can also be used to recover a volatility skew which is in fact, needed as an input in the model dependent approach. The approach has certain downsides to it: a rich set of historical data is required and it is more time consuming to conduct than the model dependent approach. The deep hedging approach, in this specific implementation, also has a limitation that only one hedge instrument is used. When this limitation is also applied to the model dependent approach, the deep hedging approach performs better in all cases. Therefore, deep hedging proves to be a sufficient alternative to pricing and hedging caplets on the prime lending rate in an incomplete market setting.
- ItemOpen AccessAn Application of Generative Adversarial Networks to One-Dimensional Value-at-Risk(2024) Swallow, Rachel; Mahomed, ObeidA generative adversarial network (GAN) is an implicit generative model made up of two neural networks. This minor dissertation applies GANs to recover target statistical distributions. GANs have a distinctive training architecture designed to create examples that reproduce target data samples. These models have been applied successfully in high-dimensional domains such as natural image generation and processing. Much less research has been reported on applications with low dimensional distributions, where properties of GANs may be better identified and understood. One such area in finance is the use of GANs for estimating value-at-risk (VaR). Through this financial application, this dissertation introduces readers to the concepts and practical implementations of GAN variants to generate one-dimensional portfolio returns over a single period. Large portions of the discussions should be accessible to anyone who has an entry-level statistics course. It is aimed at data science or finance students looking to better their understanding of GANs and the potential of these models for other financial applications. Five GAN loss variants are introduced and three of these models are practically implemented to estimate VaR. The GAN estimates are compared to more traditional VaR estimation techniques and all models are backtested. Most GAN models trained in this dissertation are able to capture key features of each of the distributions, however these models do not outperform historical VaR estimates.
- ItemOpen AccessApplication of Volatility Targeting Strategies within a Black-Scholes Framework(2019) Vakaloudis, Dmitri; Mahomed, ObeidThe traditional Black-Scholes (BS) model relies heavily on the assumption that underlying returns are normally distributed. In reality however there is a large amount of evidence to suggest that this assumption is weak and that actual return distributions are non-Gaussian. This dissertation looks at algorithmically generating a Volatility Targeting Strategy (VTS) which can be used as an underlying asset. The rationale here is that since the VTS has a constant prespecified level of volatility, its returns should be normally distributed, thus tending closer to an underlying that adheres to the assumptions of BS.
- ItemOpen AccessBootstrapping the OIS curve in a South African bank(2017) Van Heeswijk, Dirk; Mahomed, ObeidThe 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.
- ItemOpen AccessBreak-Even Volatility(2019) Mitoulis, Nicolas; Taylor, David; Mahomed, ObeidA profit or loss (P&L) of a dynamically hedged option depends on the implied volatility used to price the option and implement the hedges. Break-even volatility is a method of solving for the volatility which yields no profit or loss based on replicating the hedging procedure of an option on a historical share price time series. This dissertation investigates the traditional break-even volatility method on simulated data, how the break-even formula is derived and details the implementation with reference to MATLAB. We extend the methodology to the Heston model by changing the reference model in the hedging process. Resultantly, the need to employ characteristic function pricing methods arises to calculate the Heston model sensitivities. The break-even volatility solution is then found by means of an optimisation of the continuously delta hedged P&L over the Heston model parameters.
- ItemOpen AccessBreak-even volatility for caps, floors and swaptions(2019) Cresswell, Wade; Mahomed, ObeidThis dissertation investigates break-even volatility in the context of the South African interest rate market. Introduced by Dupire (2006), break-even volatility is a retrospective measure defined as the volatility that ensures the profit or loss from a delta hedged option position is zero. Break-even volatility sheds light on the inner structure of the market and is a promising investigatory tool. Insurance houses in South Africa are interested in modelling long-dated interest rate derivatives embedded within their liabilities. In pursuit of this goal, some are currently calibrating the Lognormal Forward-LIBOR Market Model to market prices. They rarely directly trade in said derivatives, but merely delta hedge their risk daily. In this case, break-even volatility surfaces become more relevant than recovering market prices (which incorporate the banks risk premium and profit margin) as it should better represent the historical cost of replicating the option under consideration. This dissertation ultimately assesses the use of the Lognormal Forward-LIBOR Market Model in the South African interest rate market using break-even volatility. It is found that several caps and swaptions are trading at volatilities that differ significantly from their break-even volatility estimates. Furthermore, through an investigation into the calibration of the Lognormal Forward-LIBOR Market Model to break-even volatilities, an argument that the underlying dynamics of the model are incompatible with that of the South African interest rate market is developed.
- ItemOpen AccessCurrency trios - using geometric concepts to visualise and interpret relationships between currencies(2016) Davidson, Abby; Mahomed, Obeid; Polakow, Daniel; Van de Linde, GideonA 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.
- ItemOpen AccessThe detection of phase transitions in the South African market(2016) Van Gysen, Michael; Mahomed, Obeid; Bosman, PetrusThis dissertation details the performance of two specific trading strategies which are based on the Johansen-Ledoit-Sornette (JLS) model. Both positive and negative bubbles are modelled as a log-periodic power law (LPPL) ending in a finite time singularity. The stock prices of the constituents of the FTSE/JSE Top40 index are taken as inputs to the JLS model from 3 June 2003 to 31 August 2015. It is shown that for certain time horizons into the past, the JLS based trading strategies significantly outperform random trading strategies. However this result is highly dependent on how far the model looks into the past, and if the model is calibrating to positive or negative bubbles. The lack of research with regards to the "stylized facts" of the JLS model, specifically relating to the time horizon and type of bubble, poses a significant hurdle in correctly identifying a LPPL structure in stock prices. These core features of the JLS model were developed from a number of positive bubbles that built up over many years. The results suggest that these features may not apply over all time horizons, and for both types of bubbles.
- ItemOpen AccessEstimating Long Term Equity Implied Volatility(2019) Crawford, Danielle Ana; Mahomed, ObeidEstimating and extrapolating long term equity implied volatilities is of importance in the investment and insurance industry, where ’long term’ refers to periods of ten to thirty years. Market-consistent calibration is difficult to perform in the South African market due to lack of long term liquid tradable derivatives. In this case, practitioners have to estimate the implied volatility surface across a range of expiries and moneyness levels. A detailed evaluation is performed for different estimation techniques to assess the strengths and weaknesses of each of the models. The estimation techniques considered include statistical and time-series techniques, non-parametric techniques and three potential methods which use the local volatility model.
- ItemOpen AccessFlexible risk-based portfolio optimisation(2020) Landman, Jayson; Mahomed, Obeid; Flint, EmlynThe purpose of this study is to present and test a general framework for risk-based investing. It permits various risk-based portfolios such as the global minimum variance, equal risk contribution and equal weight portfolios. The framework also allows for different estimation techniques to be used in finding the portfolios. The design of the study is to collate the existing research on risk-based investing, to analyse some modern methods to reduce estimation risk, to incorporate them in a single coherent framework, and to test the result with South African equity data. The techniques to reduce estimation risk draw from the usual mean-variance and risk-based optimisation literature. The techniques include regime switching, quantile regression, regularisation and subset resampling. In the South African experiment, risk-based portfolios materially outperformed the market weight portfolio out-of-sample using a Sharpe ratio measure. Additionally, the global minimum variance portfolio performed better than other risk-based portfolios. Given the long estimation window, no estimation techniques consistently outperformed the application of sample estimators only.
- ItemOpen AccessKVA in Black Scholes Pricing(2019) Pavlou, Petro; Ouwehand, Peter; Mahomed, ObeidThe post 2007-financial crisis era has led to renewed zeal in quantifying market incompleteness when pricing contingent claims. This quantification exercise is necessary in maintaining a stable and sustainable banking operation and thus the XVAs have emerged as the metrics for market incompleteness. This dissertation focuses solely on the capital valuation adjustment (KVA) and aims to use the definition of KVA as set out by Albanese et al. (2016) in an investigation of different numerical techniques for calculating KVA. A single equity forward is considered first, followed by an equity option and then portfolios of options on two underlying assets, with the dissertation ending by considering a practical example on discrete delta and vega-delta hedging an index option. The numerical approaches explored are the binomial tree method and a combination of the crude and quasi-Monte Carlo method.
- ItemOpen AccessModelling Equities with a Stochastic Volatility Jump Diffusion(2018) Gorven, Matthew; Mahomed, Obeid; Taylor, DavidThe Bates model provides a parsimonious fit to implied volatility surfaces, and its usefulness in developed markets is well documented. However, there is a lack of research assessing its applicability to developing markets. Additionally, research surrounding its usefulness for hedging long term liabilities is limited, despite its frequent use for this purpose. This dissertation dissects the dynamics of the Bates model into the Heston and Merton models in order to separately examine the effects of stochastic volatility and jumps. Challenges surrounding application of this model are investigated through an evaluation of risk-neutral calibration and simulation methods. The model’s ability to fit the implied volatility surfaces from the JSE Top 40 equity index is analysed. Lastly, an evaluation of the model’s delta and vega hedging performance is presented by comparing it to the hedge performance of other commonly used models.
- ItemMetadata onlyModelling illiquid volatility skews(2014) Crowther, Servaas Marcus; Mahomed, Obeid; Taylor, DavidMost markets trade liquidly in options on the market index, in fact they often trade at a wide range of strike levels. Thus, using the Black-Scholes model, we can obtain the implied volatilities at the various strike levels, forming the associated implied volatility skew of the respective market under consideration. This, however, is not always feasible when it comes to the individual stocks within the market, as single stock options trade a lot less frequently. This dissertation makes use of data from the Eurozone, in particular we consider the Euro Stoxx 50 market index and its underlying constituents. Options written on the Euro Stoxx 50 and its constituents are highly liquid, and volatility skews are obtained for the market as well as for most of the single stocks within the market. I then artificially created 3 cases of illiquid markets, each with increasing degrees of sparseness mimicking various possible realities. Using principal component analysis, this dissertation aims to find an appropriate model for relating the volatility skew of the index to that of single stocks within the market in order to fill gaps in the data of the skews of the individual stocks. Results indicate that simpler models perform similarly in all scenarios of sparse- ness whereas the performance of more complex models decrease as the data becomes sparser. This indicates that basic relationships can be formed between the index and single stocks in cases with relatively low levels of trade in the market but more accurate estimates are more difficult to achieve. However, if we use the skew data, as is, as an input to the models, their performance remains by and far the same using the full data set and using monthly information. This is encouraging, as it means we can fill gaps in the individual stocks' skew data with as good a fit as if we modeled with a full set of data.
- ItemOpen AccessModelling probabilities of corporate default(2019) Van Jaarsveldt, Cole; Mahomed, ObeidThis dissertation follows, scrupulously, the probability of default model used by the National University of Singapore Risk Management Institute (NUS-RMI). Any deviations or omissions are noted with reasons related to the scope of this study on modelling probabilities of corporate default of South African firms. Using our model, we simulate defaults and subsequently, infer parameters using classical statistical frequentist likelihood estimation and one-world-view pseudo-likelihood estimation. We improve the initial estimates from our pseudo-likelihood estimation by using Sequential Monte Carlo techniques and pseudo-Bayesian inference. With these techniques, we significantly improve upon our original parameter estimates. The increase in accuracy is most significant when using few samples which mimics real world data availability
- ItemOpen AccessModelling Term and Inflation Risk Premia in the South African Bond Market(2022) van Schaik, Luke; Mahomed, Obeid; Ismail, RezaA variety of approaches has been used to estimate the term premium of bond yields. Early attempts include linear regression models, such as those of Fama and Bliss (1987) and Cochrane and Piazzesi (2005), but these have been shown to be inconsistent and lacking in robustness (Kim and Orphanides (2007)). Affine term structure models developed by Duffie and Kan (1996) and extended by Duffee (2002) provide a more sophisticated framework for modelling bond yields and term premia, with improved results over the aforementioned regressions. However, parameters of these models have historically been estimated using maximum likelihood methods which are computationally inefficient and have been shown to have problems in finding the global maximum of the likelihood function (Hamilton and Wu (2012), Adrian et al. (2013)). The framework and estimation procedure of Adrian, Crump and Moench, or ACM, addresses the above problems by using ordinary least-square regressions exclusively to estimate the parameters of an affine term structure model (Adrian et al. (2013)). This dissertation applies the ACM procedure to South African zero-coupon nominal bond yields to estimate the term premium embedded in these yields. Performance of the ACM procedure is tested under Monte Carlo simulation and under applications to the smoothed United States nominal bond yield curves of Gurkaynak et al. (2006) and bootstrapped nominal bond yield curves from the South African market. Dynamics of the level, slope and curvature components of the US yield curves are compared to each other and to the estimated term premia. Results show that the ACM procedure generates very accurate fits to observed yield curves, but has some trouble capturing idiosyncratic features of the South African yield curve. Further, magnitudes of the term premium estimate are shown to be affected by the choice of time series of yield curves. Despite these limitations, the ACM procedure is shown to be a fast estimation procedure which generates term premium dynamics consistent with other approaches.
- ItemOpen AccessModelling the South African Inter-Bank Interest Rate Market using a Log-Normal Rational Pricing Kernel Model(2019) Hammond, Graeme; Taylor, David; Mahomed, ObeidThis dissertation examines the performance of two log-normal rational pricing kernel models and their calibration to the South African Inter-bank interest rate market. We investigate using Monte-Carlo simulation to price caps, floors and swaptions. Model-performance for both models was tested on single-strikes and entire volatility surfaces. Our results show that a one-factor model cannot reproduce the volatility smile present in the caps/floor market but can reproduce the at-the money swaption volatility surface. The two-factor model produces a better calibration to the volatility smile and captures most of the characteristics of the volatility surface.
- ItemOpen AccessQuantifying Model Risk in Option Pricing and Value-at-Risk Models(2019) Ngwenza, Dumisani; Mahomed, Obeid; Ouwehand, PeterFinancial practitioners use models in order to price, hedge and measure risk. These models are reliant on assumptions and are prone to ”model risk”. Increased innovation in complex financial products has lead to increased risk exposure and has spurred research into understanding model risk and its underlying factors. This dissertation quantifies model risk inherent in Value-at-Risk (VaR) on a variety of portfolios comprised of European options written on the ALSI futures index across various maturities. The European options under consideration will be modelled using the Black-Scholes, Heston and Variance-Gamma models.
- ItemOpen AccessA risk-budgeting framework for the combination of factor equity portfolios(2016) Wegener, Fergus; Mahomed, ObeidThis dissertation examines a risk-budgeting approach to the construction of factor equity portfolios, proposed by de Carvalho et al. (2014). The approach begins with the construction of active-weighted portfolios with exposure to factors that historically have been linked to excess returns in the market. These factor portfolios are then combined using a risk-budgeting approach. Implied stock-level returns are then estimated using this combined active allocation, and a further optimisation allows for the incorporation of specific investor constraints. The framework constitutes a risk-based approach to portfolio construction in the sense that no direct estimation of expected stock returns is required, but is dependent on a robust estimation of the covariance structure of stock returns. The framework is first evaluated in the context of a simulation study. This section provided confirmation for the risk model estimation methodology used, as well as insight into the intricacies of the framework, in an environment where the underlying structure of data was known. The framework is useful for investors who wish to combine a set of active portfolios, by controlling the allocation of risk, and understanding the exposure of the final portfolio to each of the factor portfolio components. Based on the findings of the simulation study and a back-test of the framework on JSE data, it was found that at the risk-budgeting juncture, the level of prior information imposed (with regard to the performance of factor portfolios) has a significant impact on the performance of final portfolios. In addition, the application of investor constraints, such as long-only and absolute weight limits, ultimately hinder the investor's ability to retain the views taken on in the factor portfolio components. Furthermore, due to significant discrepancies in ex-ante and ex-post tracking error risk measurement, the use of alternative, or adjusted, risk measures is recommended.
- ItemOpen AccessSouth African Inflation Modelling Under the HJM Framework(2022) Rizzo, Massimo; Mahomed, ObeidInflation modelling is typically done following an econometric approach, however this results in models being constructed that are not consistent with the observable bond market and as such they cannot be used in hedging market instruments or in pricing inflation-linked derivatives. Jarrow and Yildirim (2003) were one of the first to propose a framework under which nominal and real forward rates and an inflation index could be jointly modelled in a consistent manner, based on the Heath-Jarrow-Morton (HJM) framework as first developed by Heath et al. (1992). They showed that under this framework it is possible to recover observed nominal and inflation-linked bond prices, hedge these instruments, and price related inflation-linked derivatives. A shortfall of this framework however, as critiqued by Mercurio (2005) and Belgrade et al. (2004), is that it depends entirely on non observable parameters. As such, estimating the parameters of a model constructed under this framework is non-trivial. This dissertation applies the approach detailed by Jarrow and Yildirim (2003) to construct a model that fits the South African context, and makes use of the Kalman filter, as originally documented by Kalman (1960), to overcome the issues that arise in parameter estimation. Using the model constructed, forecasts of future inflation in South Africa are produced.