This thesis studies the problems of semistatic trading strategies in a discrete-time financial market, where stocks are traded dynamically and European options at maturity are traded statically. First, we show that pointwise limits of semistatic trading strategies are again semistatic strategies. The analysis is carried out in full generality for a two-period model, and under a probabilistic condition for multi-period, multi-stock models. Our result contrasts with a counterexample of Acciaio, Larsson and Schachermayer, and shows that their observation is due to a failure of integrability rather than instability of the semistatic form. Mathematically, our results relate to the decomposability of functions as studied in the context of Schrödinger bridges. Second, we study the so-called martingale Schrödinger bridge 𝑄⁎ in a two-period financial market; that is, the minimal-entropy martingale measure among all models calibrated to option prices. This minimization is shown to be in duality with an exponential utility maximization over semistatic portfolios. Under a technical condition on the physical measure 𝑃, we show that an optimal portfolio exists and provides an explicit solution for 𝑄⁎. Specifically, we exhibit a dense subset of calibrated martingale measures with particular properties to show that the portfolio in question has a well-defined and integrable option position.

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Most of the notations and terminological conventions used in this thesis are Statistical. The aim in risk management is to describe the risk factors present in time series. In order to group these risk factors, one needs to distinguish between different stochastic processes and put them into different classes. The risk factors discussed in this thesis are fat tails and mean reversion. The presence of these risk factors fist need to be found in the historical dataset. I will refer to the historical dataset as the original dataset. The Ljung- Box-Pierce test will be used in this thesis to determine if the distribution of the original dataset has mean reversion or no mean reversion. / Mathematical Sciences / M.Sc. (Applied Mathematics)

Thesis (MSc)--Stellenbosch University / ENGLISH ABSTRACT : In this thesis, we first review the traditional pre-credit crunch approach that considers a single curve to consistently price all instruments. We review the theoretical pricing framework and introduce pricing formulas for plain vanilla interest rate derivatives. We then review the curve construction methodologies (bootstrapping and global methods) to build an interest rate curve using the instruments described previously as inputs. Second, we extend this work in the modern post-credit framework. Third, we review the calibration of the SABR model. Finally we present applications that use interest rate curves and SABR model: stripping implied volatilities, transforming the market observed smile (given quotes for standard tenors) to non-standard tenors (or inversely) and calibrating the market volatility smile coherently with the new market evidences. / AFRIKAANSE OPSOMMING : Geen Afrikaanse opsomming geskikbaar nie

This thesis contains a discussion of four problems arising from the application of stochastic differential equations and real option theory to investment decision problems in a continuous-time framework. It is based on four papers written jointly with the author’s supervisor. In the first problem, we study an evolutionary stock market model in a continuous-time framework where uncertainty in dividends is produced by a single Wiener process. The model is an adaptation to a continuous-time framework of a discrete evolutionary stock market model developed by Evstigneev, Hens and Schenk-Hoppé (2006). We consider the case of fix-mix strategies and derive the stochastic differential equations which determine the evolution of the wealth processes of the various market players. The wealth dynamics for various initial set-ups of the market are simulated. In the second problem, we apply an entry-exit model in real option theory to study concessionary agreements between a private company and a state government to run a privatised business or project. The private company can choose the time to enter into the agreement and can also choose the time to exit the agreement if the project becomes unprofitable. An early termination of the agreement by the company might mean that it has to pay a penalty fee to the government. Optimal times for the company to enter and exit the agreement are calculated. The dynamics of the project are assumed to follow either a geometric mean reversion process or geometric Brownian motion. A comparative analysis is provided. Particular emphasis is given to the role of uncertainty and how uncertainty affects the average time that the concessionary agreement is active. The effect of uncertainty is studied by using Monte Carlo simulation. In the third problem, we study numerical methods for solving stochastic optimal control problems which are linear in the control. In particular, we investigate methods based on spline functions for solving the two-point boundary value problems that arise from the method of dynamic programming. In the general case, where only the value function and its first derivative are guaranteed to be continuous, piecewise quadratic polynomials are used in the solution. However, under certain conditions, the continuity of the second derivative is also guaranteed. In this case, piecewise cubic polynomials are used in the solution. We show how the computational time and memory requirements of the solution algorithm can be improved by effectively reducing the dimension of the problem. Numerical examples which demonstrate the effectiveness of our method are provided. Lastly, we study the situation where, by partial privatisation, a government gives a private company the opportunity to invest in a government-owned business. After payment of an initial instalment cost, the private company’s investments are assumed to be flexible within a range [0, k] while the investment in the business continues. We model the problem in a real option framework and use a geometric mean reversion process to describe the dynamics of the business. We use the method of dynamic programming to determine the optimal time for the private company to enter and pay the initial instalment cost as well as the optimal dynamic investment strategy that it follows afterwards. Since an analytic solution cannot be obtained for the dynamic programming equations, we use quadratic splines to obtain a numerical solution. Finally we determine the optimal degree of privatisation in our model from the perspective of the government.

published_or_final_version / abstract / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

This study develops empirical models for comprehensive inflation effects on stock returns in the Japanese economic and financial framework. Basically these models deal with the two kinds of wealth effects and inflation risk premia. The wealth transfers are related to a tax system and other institutional constraints while the wealth-size effect is based on the more fundamental stock price determinant of the flows of earnings. The inflation risk premia are the additionally required part of returns due to relative uncertainties in common stock investment under unstable inflation. Based on the stock valuation theory and on the efficiency of stock markets, it is found that the net effect of wealth transfers appears in ex post stock returns if expected inflation shifts from one level to another. However, the effect of the inflation-caused wealth transfers will not appear in stock returns even in varying inflation if positive and negative wealth transfers are perfectly offset. The test result supports this offset. As the result of testing inflation risk premia, the stock market tends to compensate the premia of unstable inflation for investors. Finally the wealth-size effect relates anticipated real activity to inflation in monetary sector behaviors as well as anticipated real activity to stock returns in real sector behaviors. The intermediate variable to transmit inflation to stock returns is real activity. In this context, inflation is just the proxy for real activity which essentially determines firm values. Empirically the wealth-size effect is supported with the inverse relationship between inflation and real stock returns. For Japanese economy, however, the wealth-size effect is not explained by the standard theory of capital investment. Real activity is correlated to (profit) returns on existing capital but not related to corporate capital investment. Capital investment is independent of other real sector variables as well as inflation. The result is attributed to governmental policy and controls for corporate investment. Thus, the obtained relationship between stock returns and inflation includes less practical implication in investment behaviors.

Submitted in partial fulfilment of the requirements for the degree of Masters of Management in Finance and Investments In the Faculty of Commerce, Law and Management University of the Witwatersrand, Wits Business School, 2016 / This paper presents a first time application of a variant of the concepts underpinning the Fed Model, amalgamated with the Bond-Stock Earnings Yield Differential, by applying it to the dividend yields of REIT indices. This modification is termed the yield gap, quantitatively constructed and adapted in this paper as the Reverse REIT-Bond Yield Gap. This metric is then used as the variable of interest in a multivariate Markov regime switching model framework, along with a set of three regressors. The REIT indices trailing dividend yield and associated metrics are the FTSE/EPRA NAREIT series. All data are from Bloomberg Terminals. This paper examines 11 markets, of which the EMEs are classified as Brazil, Mexico, Turkey and South Africa, whereas the advanced market counterparts are Australia, France, Japan, the Netherlands, Singapore, the United Kingdom, and the United States. The time-frame spans the period June 2013 until November 2015 for the EMEs, whilst their advanced market counterparts time-span covers the period November 2009 until November 2015. This paper encompasses a tri-fold research objective, and aims to accomplish them in a scientifically-based, objective and coherent fashion. Specifically, the purpose is in an attempt to gauge the reasons underlying EMEs observed anomalies entailing reverse REIT-Bond yield gaps, whereby their tenyear nominal government bonds out-yield their trailing dividend yields on their associated REIT indices; what drives fluctuations in this metric; and whether or not profitable tactical asset allocation strategies can be formulated to exploit any arbitrage mispricing opportunities. The Markov models were unable to generate clear-cut, definitive reasons regarding why EMEs experience this anomaly. Objectives two and three were achieved, except for France and Mexico. The third objective was also met. The REIT-Bond Yield Gaps static conditions have high probabilities of continuing in the same direction and magnitude into the future. In retrospection, the results suggest that by positioning an investment strategy, taking cognisance of the chain of economic events that are likely to occur following static REIT-Bond Yield Gaps, then investors, portfolio rebalancing and risk management techniques, hedging, targeted, tactical and strategic asset allocation strategies could be formulated to exploit any potential arbitrage profits. The REIT-Bond Yield Gaps are considered highly contentious, yet encompasses the potential for significant reward. The Fed Model insinuates that EME REIT markets are overvalued relative to their respective government bonds, whereas their advanced market counterparts exhibit the opposite phenomenon. / XL2018

Submitted as a Requirement of the Master of Management (Finance and Investment Management) University of the Witwatersrand Business School Johannesburg / The mid 2000’s saw the materialization of research into the financial engineering field of high frequency trading. It is arguable that the most prominent model to emerge from the research has been pairs trading. This idea can be extended to allow for more than two assets in a modelling method now known as statistical arbitrage. The research identifies a collection of assets with a deterministic component; it then follows a multiple linear regression to exploit persistent mispricings among these assets. Further, multiple linear regression metrics are used to identify the analytic form of the trading rule and to validate the performance of the model. The first part of model constructs combinations of assets which contain a significant predictable component by co-integration, the second part builds a predictive models for the dynamics of the mispricing using statistical model. The success of the model is demonstrated with reference to a statistical analysis of 5-minute closing prices on the Johannesburg Stock Exchange (JSE) TOP40 Index and the constituent shares of the JSE TOP40 Index. / MT2017

Thesis (M.M. (Finance & Investment)--University of the Witwatersrand, Faculty of Commerce, Law and Management, Wits Business School, 2016 / Purpose - The paper provides an investigation of the relationship of macroeconomic risk factors and REITs. The study considers the conditional volatilities of macroeconomic variables on the excess returns and conditional variance of excess returns in developing and developed markets and provides a comparison thereof. Methodology approach - The study employs three-step approach estimation in the methodology (Principal Component Analysis, GARCH (1,1) and GMM) to estimate the asset pricing model. The preliminary study indicated that there are only two developing economies (Bulgaria and South Africa), as defined by National Association of Real Estate Investment Trust (NAREIT), with REIT indices. We additionally included the United States as the developed economy. Findings – Our results indicate that the real economy and business cycles (proxied by GDP growth rate and industrial production index), price stability (proxied by the GDP deflator), exchange rates and interest rates do not explain developing country REIT returns represented by Bulgaria and South Africa, as well as in developed markets, represented by the US. However unlike the developing markets, changes in industrial production and inflation are important variables that affect the conditional variance of REIT returns in the US. / GR2018