21 research outputs found
Optimal Strategies for Round-Trip Pairs Trading Under Geometric Brownian Motions
This paper is concerned with an optimal strategy for simultaneously trading a
pair of stocks. The idea of pairs trading is to monitor their price movements
and compare their relative strength over time. A pairs trade is triggered by
the divergence of their prices and consists of a pair of positions to short the
strong stock and to long the weak one. Such a strategy bets on the reversal of
their price strengths. A round-trip trading strategy refers to opening and
closing such a pair of security positions. Typical pairs-trading models usually
assume a difference of the stock prices satisfies a mean-reversion equation.
However, we consider the optimal pairs-trading problem by allowing the stock
prices to follow general geometric Brownian motions. The objective is to trade
the pairs over time to maximize an overall return with a fixed commission cost
for each transaction. Initially, we allow the initial pairs position to be
either long or flat. We then consider the problem when the initial pairs
position may be long, flat, or short. In each case, the optimal policy is
characterized by threshold curves obtained by solving the associated HJB
equations.Comment: 47 pages, 5 figure
Pairs Trading: An Optimal Selling Rule with Constraints
The focus of this paper is on identifying the most effective selling strategy
for pairs trading of stocks. In pairs trading, a long position is held in one
stock while a short position is held in another. The goal is to determine the
optimal time to sell the long position and repurchase the short position in
order to close the pairs position. The paper presents an optimal pairs-trading
selling rule with trading constraints. In particular, the underlying stock
prices evolve according to a two dimensional geometric Brownian motion and the
trading permission process is given in terms of a two-state {trading allowed,
trading not allowed} Markov chain. It is shown that the optimal policy can be
determined by a threshold curve which is obtained by solving the associated HJB
equations (quasi-variational inequalities). A closed form solution is obtained.
A verification theorem is provided. Numerical experiments are also reported to
demonstrate the optimal policies and value functions
The Kohn-Laplacian and Cauchy-Szeg\"{o} projection on Model Domains
We study the Kohn-Laplacian and its fundamental solution on some model
domains in , and further discuss the explicit kernel of the
Cauchy-Szeg\"o projections on these model domains using the real analysis
method. We further show that these Cauchy-Szeg\"o kernels are
Calder\'on-Zygmund kernels under the suitable quasi-metric
Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions in a Complete Pseudohermitian manifold
I will introduce the basic notion of pseudhermitian manifold first and derive the sub-gradient estimate for positive pseudoharmonic functions in a complete pseudohermitian (2n + 1)-manifold (M, J, θ) which satisfies the CR sub-Laplacian comparison property. It is served as the CR analogue of Yau’s gradient estimate. Secondly, we obtain the Bishop-type sub-Laplacian comparison theorem in a class of complete noncompact pseudohermitian manifolds. Finally we will show the natural analogue of Liouville-type theo- rems for the sub-Laplacian in a complete pseudohermitian manifold of vanishing pseudohermitian torsion tensors and nonnegative pseudohermitian Ricci curvature tensors. This a joint project with Shu-Cheng Chang and Ting-Jung Kuo of National Taiwan University.Non UBCUnreviewedAuthor affiliation: University of GeorgiaFacult
A Threshold Type Policy for Trading a Mean-Reverting Asset with Fixed Transaction Costs
A mean-reverting model is often used to capture asset price movements fluctuating around its equilibrium. A common strategy trading such mean-reverting asset is to buy low and sell high. However, determining these key levels in practice is extremely challenging. In this paper, we study the optimal trading of such mean-reverting asset with a fixed transaction (commission and slippage) cost. In particular, we focus on a threshold type policy and develop a method that is easy to implement in practice. We formulate the optimal trading problem in terms of a sequence of optimal stopping times. We follow a dynamic programming approach and obtain the value functions by solving the associated HJB equations. The optimal threshold levels can be found by solving a set of quasi-algebraic equations. In addition, a verification theorem is provided together with sufficient conditions. Finally, a numerical example is given to illustrate our results. We note that a complete treatment of this problem was done recently by Leung and associates. Nevertheless, our work was done independently and focuses more on developing necessary optimality conditions