Mean‐Reverting Portfolios

Об Эффективных Численных Методах Решения Задач Энтропийно-Линейного Программирования

“…Интересно было бы сравнить предложенные в статье методы с другими известными численными методами решения задач ЭЛП [1,2,10,27,28]. Этому планируется посвятить отдельную публикацию.…”

“…Для более основательного знакомства с приложениями, в которых возникают задачи ЭЛП можно рекомендовать [3 -9]. Особенно интересно сопоставить предложенные в данной статье методы с методом балансировки [1] (говорят также методом Шелейховского, Брэгмана-Шелейховского [9], Синхорна [28]) применительно к задаче расчета матрицы корреспонденций по энтропийной модели [3,9,17]…”

A penalty decomposition algorithm with greedy improvement for mean‐reverting portfolios with sparsity and volatility constraints

“…Mean‐reverting portfolios play a significant role in mathematical finance (Cuturi and d'Aspremont, 2016; d'Aspremont et al., 2005; Zhao et al., 2019) due to their favorable predictability property. This appealing feature provides simple trading opportunities such that a trader can sell such portfolios when its price is higher than the expected mean and can buy such assets when the price of the corresponding portfolio is lower than the predicted mean.…”

“…Specifically, we consider the following optimization problem that aims to minimize the predictability notion introduced by Box and Tiao (1977) while ensuring sparsity and volatility: $$$$\begin{equation} (P): \qquad \min _{x \in \mathbb {R}^N} \ x^T M x \qquad \mbox{subject to}\; x^T A x\ge \phi , \quad x^T x=1, \quad \text{and}\quad \Vert x\Vert _0\le k, \end{equation}$$$$where $$M, A\in \mathbb {R}^{N\times N}$$ are symmetric and positive definite, ϕ is a positive number, $$\Vert \cdot \Vert _0$$ denotes the number of nonzero entries of a vector, and $$k\in \mathbb {N}$$ with $$k\ll N$$. Recently, several statistical proxies have been introduced to capture mean‐reversion property (Cuturi and d'Aspremont, 2016; d'Aspremont et al., 2005). For example, the following semidefinite program (SDP) relaxation‐based model is proposed (Cuturi and d'Aspremont, 2016) : $$$$\begin{equation} \min _{Y...$$…”

“…Recently, several statistical proxies have been introduced to capture mean‐reversion property (Cuturi and d'Aspremont, 2016; d'Aspremont et al., 2005). For example, the following semidefinite program (SDP) relaxation‐based model is proposed (Cuturi and d'Aspremont, 2016) : $$$$\begin{equation} \min _{Y\in \mathcal {S}^N} \quad \mbox{Tr}(MY)+\rho \Vert Y\Vert _1 \qquad \textrm {subject to} \; \mbox{Tr}(AY)\ge \phi , \quad \mbox{Tr}(Y)=1, \quad \text{and} \quad Y\succeq 0, \end{equation}$$$$where $$\rho >0$$ is a penalty parameter, and $$\Vert Y\Vert _1:=\sum _{i,j} |Y_{ij}|$$. The ℓ 1 ‐norm promotes the sparsity of the decision variable Y .…”