Dissertations (Masters) M&S
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Item Pricing barrier options when the dynamics of the prices are driven by the Mean Reverting Process(University of Dar es Salaam, 2013) Komunte, MasoudThis dissertation considers a problem of pricing barrier options when the dynamics of the asset prices (X(t)) are driven by the mean reverting process, the market/asset price X(t) is obtain from mean reversion model and a Black-Scholes PDE model for pricing barrier options under mean reversion model is obtained upon using It ˆ o formula. Through the Homotopy Analysis Method (HAM) the price of the chosen barrier option (upandout European call) that satisfies the Black-Scholes PDE model was determined. Thus, through HAM we can determine approximated prices of barrier options when the dynamics of the prices are driven by the mean reverting process (Liao, 2004). Lastly the analysis is conducted to observe the behaviour of the option price when value of one parameter increases while the value of the other two parameters remain constant. The analysis shows that the option price tends to increase with the increase of the value of the parameter for the case of volatility and degree of mean reversion while for interest rate the option price decreases when interest rate increases. In all cases it is observed that early exercise is better than late exercise to owner of the option since the option price tends to decrease as time increases also to minimize risk owner of the option should exercise the option when the volatility of the market become large. It is recommended that in future, areas of interest for research related to this study are; first, finding the option price by using direct integration after obtaining a reflection principle which is useful in determining the joint distribution of the It ˆ o integral and secondly, finding the price which is a closed form solution by using Laplace transform.Item Optimal portfolio management when stocks are driven by Mean-Reverting Processes(University of Dar es Salaam, 2012) Mbigili, Lusungu JuliusIn this work, we present and solve the problem of portfolio optimization within the context of continuous-time stochastic model of financial variables. We consider an investment problem where an investor has two assets, namely, risk-free assets (e.g. bonds) and risky assets (e.g. stocks) to invest on and tries to maximize the expected utility of the wealth at some future time. The evolution of the risk-free asset is described deterministically while the dynamics of the risky asset is described by the geometric mean reversion (GMR) model. The controlled wealth stochastic deferential equation (SDE) and the portfolio problem are formulated. The portfolio optimization problem is then successfully formulated and solved with the help of the theory of stochastic control technique where the dynamic programming principle (DPP) and the HJB theory were used. We obtained very interesting results which are the solution of the non-linear second order partial deferential equation and the optimal policy which is the optimal control strategy for the investment process. So far we have considered utility functions which are members of hyperbolic absolute risk aversion (HARA) family, called power and exponential utility. In both cases, the optimal control (investment strategy) has explicit form and is wealth dependent, in the sense that, as the investor becomes more rich, the less he invests on the risky assets. Linearization of the logarithmic term in the portfolio problem was necessary to be undertaken for making the work of obtaining the explicit form of the optimal control much simple than it was expected.Item Probabilistic weather forecasting using Bayesian Model averaging: the case of Sagcot Regions(Mzumbe University, 2018) Joel, EmmanuelOver the past decade Tanzania has experienced spontaneous population increase (1.556 mil annual). But the number is estimated to further increase by 2050 to 2.982 mil annual, thus Tanzania is estimated to have population of 137 million people in 2050 (UN, 2015). The fast growing population is mainly depending on rainfed agriculture, which contribute 29 percent of the country GDP and providing employment to 65.5 percent of Tanzanians (Deloitte, 2016). The diversity in climatic and weather activities has posed a challenge in rainfed agriculture especially on when to plant seeds. Therefore, in order to promote agricultural activities, stable and reliable weather information are crucial in order for production to match with population increase. This study explores the challenge facing the Numerical Weather Prediction (NWP) namely WRF-ARW, by creating the system of equation (ensembles) from WRF-ARW resulting from the use of different initial conditions. Ensemble allow for probabilistic forecast to take the form of predictive probability function (PDF). But, raw ensemble forecast system are finite hence they only capture some of the uncertainty of the NWP. Thus, this study used Bayesian Model Averaging (BMA) methods of post processing ensemble forecast to maximize the sharpness of the parameter and calibration. The findings show BMA method successively removes most of under disersion showed by raw ensembles. Thus, calibrated and sharp results of BMA approach resolves a number of the weaknesses of the ensemble forecasts including their under dispersion and the discrepancy between forecasts and observations. Therefore, BMA can be used to attain higher consistency in the probabilistic forecasts of an operational model.