Stochastic approximation methods and applications in financial optimization problems
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Optimizations play an increasingly indispensable role in financial decisions and financial models. Many problems in mathematical finance, such as asset allocation, trading strategy, and derivative pricing, are now routinely and efficiently approached using optimization. Not until recently have stochastic approximation methods been applied to solve optimization problems in finance. This dissertation is concerned with stochastic approximation algorithms and their appli- cations in financial optimization problems. The first part of this dissertation concerns trading a mean-reverting asset. The strategy is to determine a low price to buy and a high price to sell so that the expected return is maximized. Slippage cost is imposed on each transaction. Our effort is devoted to developing a recursive stochastic approximation type algorithm to esti- mate the desired selling and buying prices. In the second part of this dissertation we consider the trailing stop strategy. Trailing stops are often used in stock trading to limit the maximum of a possible loss and to lock in a profit. We develop stochastic approximation algorithms to estimate the optimal trailing stop percentage. A modification using projection is devel- oped to ensure that the approximation sequence constructed stays in a reasonable range. In both parts, we also study the convergence and the rate of convergence. Simulations and real market data are used to demonstrate the performance of the proposed algorithms. The advantage of using stochastic approximation in stock trading is that the underlying asset is model free. Only observed stock prices are required, so it can be performed on line to provide guidelines for stock trading. Other than in stock trading, stochastic approximation methods can also be used in parameter estimations. In the last part of this dissertation, we consider a regime switching option pricing model. The underlying stock price evolves according to two geometric Brownian motions coupled by a continuous-time finite state Markov chain. Recur- sive stochastic approximation algorithms are developed to estimate the implied volatility. Convergence of the algorithm is obtained and the rate of convergence is also ascertained. Then real market data are used to compare our algorithms with other schemes.