Optimal Pairs-Trading Strategy And The Well-Posedeness Of Linear Forward-Backward Stochastic Differential Equations

This thesis is mainly concerned with two kinds of stochastic control problem,one of which is a pairs trading strategy under geometric Brownian motion,and the oth-er is an optimal pricing policy under a Markov chain model.We also investigate the well-posedness of the linear fully-coupled Forward-Backward Stochastic Differential E-quations(FBSDEs)and one kind of Ordinary Differential Equations(ODEs).A linear transformation method is introduced for these equations in which situations the well-posedness is hard to get.New FBSDEs after transformation exist a unique solution by some criteria of the non-degenerate transformation matrix.Pairs trading was initially introduced by Bamberger and followed by Tartaglia's quantitative group at Morgan Stanley in the 1980s.The main idea of pairs trading strategy is about simultaneously trading a pair of stocks.Pairs trading involves tracking the price movements of these two stocks and comparing their relative price strengths.A pairs trade is triggered when their prices diverge and consists of a short position in the strong stock and a long position in the weak one.The idea is to bet on the reversal of their price strength.In practice,pairs trading is attractive because of its 'market neutral' nature in the sense that it can be profitable under any market trend conditions.In practice,many systems appear to be switching between some certain states in a long period.We introduce a Markov chain model to investigate these features.Compared to the classical diffusion model,the advantages of Markov chain model are obvious in the following perspectives.From a modelling point of view,the Markov chain model may evolve slowly and the dynamic can be captured by finite-state Markov chains.On the other hand,from a practical view point,Markov chain model has been applied in derivatives pricing,portfolio selection and their applications.In connection with dynamic programming principle,the corresponding Hamilton-Jacobi-Bellman(HJB)equations are of the first order,which are easier to analyze than those under traditional Brownian motion based models.In addition,it is closely related to the traditional geometric Brownian motion models when the jump rate are large.Inspired by previous studies,this thesis concentrates on studying the problems above,and attempts to apply the relevant theoretical results to solve some practical problems.The structure of this thesis is as follows:In Chapter 1,we investigate the research background and illustrate the main contributions of each chapter.In Chapter 2,we study the optimal pairs trading strategy with cutting losses.S-tock prices are considered to follow geometric Brownian motion without mean-reverting assumption.We consider an optimal pairs trading rule in which a pairs(long-short)position consists of a long position of one stock and a short position of the other.The objective is to open(buy)and close(sell)the pairs positions sequentially to maximize a discounted reward function.To limit downside risk of the pairs position,we impose a hard cut loss level.Any existing position will be automatically closed upon entering the cut loss region.Following a dynamic programming approach,we establish the as-sociated HJB equations(variational inequalities)for the value functions.Given a cut loss level,we show that the corresponding optimal stopping times can be determined by three threshold curves.We also provide examples and examine the dependence of these threshold levels on some parameters.We also show backtest results using historical data.The third chapter is concerned with optimal inventory-price coordination policy under a continuous review inventory model.An inventory-price coordination policy is about dynamical price adjustment over time to maximize an overall reward function.Instead of treating the pricing under diffusion models,we study the problem under a two-state Markov chain model.This is motivated by the consideration that product demand in practice often evolves slowly and the dynamic can be captured by finite-state Markov chains.In particular,we consider a price dependent state process and that allows continuous price adjustment.The objective is to dynamically and continuously vary the price to maximize a reward function.Following a dynamic programming approach,we establish the associated HJB equations and the threshold type optimal pricing control.These threshold levels can be obtained by solving algebra equations.Such structural property is desirable from a practical viewpoint because it provides deeper insights into the problem such as dependence on various parameters and typically generalize the applications.In Chapter 4,we study the well-posedness of linear fully coupled Forward-Backward Stochastic Differential Equations(FBSDEs).Firstly,we generalize the classical mono-tonicity conditions into linear FBSDEs.Inspired by the unified approach,we prove that the monotonicity conditions can be deduced into some special cases of the Unified Ap-proach.An example of FBSDEs which does not satisfy the monotonicity conditions can be dealt with by the Unified Approach.We also note that for some linear FBSDEs the traditional monotonicity conditions even the Unified Approach can notbe applied direct-ly.And then,one kind of linear transformation method is introduced for the cases which can not meet the requirement of the monotonicity conditions and the Unified Approach.By virtue of the non-degeneracy matrix we provide a main result to guarantee the new FBSDEs after transformation satisfy the monotonicity conditions,and consequently get the well-posedness of the former linear FBSDEs.Also,an example is presented to illus-trate it.This kind of linear transformation method develops the well-posedness theory of FBSDEs and has powerful application in some research areas such as optimal control,PDEs theory,and mathematical finance.The fifth chapter is concerned with the well-posedness for a class of two-point boundary value problems associated with ordinary differential equations(ODEs).For the constant-coefficient cases,a regular decoupling field method was introduced to get the existence and uniqueness of solution for ODEs.It is proved that monotonicity condi-tions are initially a special case of the regular decoupling field method.For the functional coefficient cases,by virtue of the boundary of coefficients,we generalize the upper lower bound equation to get the regulation of decoupling field.In additon,the linear transfor-mation method can be used to handle the situation where the monotonicity conditions and regular decoupling field method cannot be directly applied.These two methods overall develop the well-posedness theory of two-point boundary value problems which has potential applications in optimal control and partial differential equation theory.In the last chapter,we conclude this thesis and present some potential research directions.