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|Title:||Applications of Stochastic Control Theory in The Trading of Stocks and Futures|
|Department:||Mathematics and Statistics|
|Abstract:||In this thesis, we apply stochastic control methodology to analyze the trading of stocks and futures dynamically. The trader's quest for profits is formulated as an optimal control problem with finite horizon, where the objective is to maximize the expected utility of wealth at the end of the horizon, and the optimized trading strategy is given by the optimal control. Based on various well-established stochastic models of these financial securities, we derive the stochastic differential equations that describe the price dynamics in each model, formulate the utility maximization problems, analyze the associated Hamilton-Jacobi-Bellman (HJB) equations, and solve for the trading strategies in closed form. Numerical examples based on securities traded in the US markets are presented for all models. Specifically, we investigate pairs trading with cointegrated stock pairs under the Duan-Pliska model, and with volatility index futures where the spot index is modeled as a Central Tendency Ornstein-Uhlenbeck process. We also study optimal trading of commodity futures under the two-factor Schwartz model, and under a more general $n$-factor Cortazar and Naranjo model. Given the closed-form expressions for the optimal strategies, the value functions, and the wealth processes, we see directly the dependence of the optimal positions on different model parameters, and therefore we can quantify the impact of varying parameter and coefficient values. Qualitatively, we will see, in line with intuition, that the optimal positions in general decrease in magnitude as the volatilities in the underlying factors increase. In all cases we find that the magnitudes of the optimal positions are inversely proportional to the degree of risk aversion, as expected. In the pairs trading cases where two stocks or two futures contracts are traded, the optimal positions are of opposite signs, corresponding to a long and a short position, where the quantities are given explicitly by the closed-form formulae. Based on parameters calibrated from VIX futures historical data, we find that traders should take bigger positions in the long end of the futures curve. In the WTI oil futures trading example, we see that the optimal positions are insensitive to time to maturity. We also find that the certainty equivalent for trading two contracts simultaneously is significantly greater than that derived from trading only a single contract, regardless of the maturity of either of the single contract. Analogous result holds for the more general $n$-factor model as well.|
|Appears in Collections:||Open Access Dissertations and Theses|
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