Theoretical study of technical analysis indicators.

Summary The objective of my thesis is to study mathematically a volatility breakout indicator widely used by practitioners in the trading room. The Bollinger Bands indicator belongs to the family of so-called technical analysis methods and is therefore based exclusively on the recent history of the price considered and a principle deduced from past market observations, independently of any mathematical model. My work consists in studying the performance of this indicator in a universe governed by stochastic differential equations (Black-Scholes) whose diffusion coefficient changes its value at an unknown and unobservable random time, for a practitioner wishing to maximize an objective function (for example, a certain expected utility of the portfolio value at a certain maturity). In the framework of the model, the Bollinger indicator can be interpreted as an estimator of the time of the next break. In the case of small volatilities, we show that the behavior of the density of the indicator depends on the volatility, which makes it possible to detect, for a large enough volatility ratio, the volatility regime in which the indicator's distribution is located. Also, in the case of high volatilities, we show by an approach via the Laplace transform, that the asymptotic behavior of the indicator's distribution tails depends on the volatility. This makes it possible to detect the change in the large volatilities. Then, we are interested in a comparative study between the Bollinger indicator and the classical estimator of the quadratic variation for the detection of change in volatility. Finally, we study the optimal portfolio management which is described by a non-standard stochastic problem in the sense that the admissible controls are constrained to be functionals of the observed prices. We solve this control problem by drawing on the work of Pham and Jiao to decompose the initial portfolio allocation problem into a post-breakdown management problem and a pre-breakdown problem, and each of these problems is solved by the dynamic programming method. Thus, a verification theorem is proved for this stochastic control problem.