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This paper studies the problem of determining the optimal cut-off for pairs
trading rules. We consider two correlated assets whose spread is modelled by a
mean-reverting process with stochastic volatility, and the optimal pair trading
rule is formulated as an optimal switching problem between three regimes: flat
position (no holding stocks), long o...
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... − 1 is not empty in the form: − 1 = (0 , x − 1 ] for some x − 1 < 0 by Proposition 4.1, and S = ∅ . Such case arises when λ > ρε , and for L ≤ ( λ + ρε ) /μ , see 01 Remark 4.3(i). This is plotted in Figure 2. (iii) Both S − 1 and S 01 are empty. Such case arises when λ ≤ ρε , and for L ≤ ( ρε − λ ) /μ , see Remark 4.3(ii). This is plotted in Figure 3. Moreover, notice that in such case, we must have λ ≤ ρε by Lemma 4.2(2)(ii), and so by Proposition μL − 4.1, x ≥ 1 > 0, i.e. S = [ x , ∞ ...
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Citations
... [23] considered an optimal switching problem using the viscosity solution method. [20] considered the optimal switching problem with finite regimes. [26] dealt with a discontinuous coefficients problem with finite-regime switchings. ...
... Then by (20), (21), (22) and (23), we get ...
... Here the first inequality is derived by using (21), (22), (23) and u ≤ 1, and the second inequality is derived from (20). ...
... Our model is also similar to Ngo and Pham [8]. They consider the problem of determining the optimal cut-off of the pair trading rule for a three-state regime-switching model. ...
In this paper, we study a free boundary problem, which arises from an optimal trading problem of a stock whose price is driven by unobservable market status and noise processes. The free boundary problem is a variational inequality system of three functions with a degenerate operator. We prove that all the four switching free boundaries are no-overlapping, monotonic and C∞-smooth by the approximation method. We also completely determine their relative localities and provide the optimal trading strategies for the stock trading problem.
... The application of stochastic control to pairs trading originated in the work of Mudchanatongsuk et al. (2008). Further models of timeconsistent pairs trading strategies were developed recently in Chiu andWong (2011), Tourin andYan (2013), Ngo and Pham (2014), Lei and Xu (2015), and Leung and Li (2015). Besides, Hogan et al. (2004) provided a definition of the concept of statistical arbitrage and empirical measures to test the presence of opportunities. ...
We consider a pairs trading stochastic control problem with transaction costs and constraints on the gross market exposure, and propose a new monotone Finite Difference scheme approximating the viscosity solution of the Hamilton–Jacobi–Bellman equation characterizing the optimal trading strategies. Given a fixed time horizon and a portfolio of two cointegrated assets, the agent trades the spread between the two assets and the trading strategy is defined as the possibly negative portfolio weight maximizing the expected exponential utility derived from terminal wealth. Furthermore, trades incur transaction costs comprised of explicit transactions fees and commissions and the implicit cost due to slippage. These costs are modeled as a linear or square root function of the trading rate and respectively added or subtracted from the observable asset price at the time when a buy or a sell order enters the market. Our main contribution is the derivation of a robust approximation for the nonlinear transaction cost term in the Hamilton–Jacobi–Bellman equation. Finally, we combine our monotone Finite Difference scheme with a Monte Carlo sampling method to analyze the effects of transaction fees and slippage on the trading policies’ performance.
... Specifically, Leung and Li (2015) studies the optimal timing to open or close the position subject to fixed transaction costs and the effect of stop-loss level under the OU process by constructing the value function directly. Zhang and Zhang (2008), Song and Zhang (2013), and Ngo and Pham (2016) studied the optimal pairs trading rule that is based on optimal switching among two (buy and sell) or three (buy, sell, and flat) regimes with a fixed commission cost for each transaction, and solve the problem by finding viscosity solutions to the associated HJB equations (quasi-variational inequalities). Lei and Xu (2015) studied the optimal pairs trading rule of entering and exiting the asset market in finite horizon with proportional transaction cost for two convergent assets. ...
... Note that, Mudchanatongsuk et al. (2008) assumed no transaction cost and considered the strategy that always shorts one stock and longs the other in equal dollar amount, i.e., p(t)dL p (t) + q(t)dM q (t) = 0 or p(t)dM p (t) + q(t)dL q (t) = 0 at time t. Lei and Xu (2015) and Ngo and Pham (2016) considered a delta-neutral strategy that always long one share of a stock and short one share of the other stock, i.e., dy p (t) = −dy q (t) = 1 or dy p (t) = −dy q (t) = −1 at time t. Here, we also consider a delta-neutral strategy that requires the total of positive and negative delta of two assets is zero, hence it suggests that the number of shares of stock P bought (or sold) at time t are same as the number of shares of stock Q sold (or bought), i.e., ...
... Equation (8) implies that dy q (t) = −dy p (t) at any time t. Comparing to Lei and Xu (2015) and Ngo and Pham (2016), we remove the constraint dy p (t) = −dy q (t) = 1 or −1 and allow y p (t) = −y q (t) to be a control variable. Using Equations (5) and (8), the dynamics of g(t) in Equation (7) can be simplified as ...
Optimal trading strategies for pairs trading have been studied by models that try to find either optimal shares of stocks by assuming no transaction costs or optimal timing of trading fixed numbers of shares of stocks with transaction costs. To find optimal strategies that determine optimally both trade times and number of shares in a pairs trading process, we use a singular stochastic control approach to study an optimal pairs trading problem with proportional transaction costs. Assuming a cointegrated relationship for a pair of stock log-prices, we consider a portfolio optimization problem that involves dynamic trading strategies with proportional transaction costs. We show that the value function of the control problem is the unique viscosity solution of a nonlinear quasi-variational inequality, which is equivalent to a free boundary problem for the singular stochastic control value function. We then develop a discrete time dynamic programming algorithm to compute the transaction regions, and show the convergence of the discretization scheme. We illustrate our approach with numerical examples and discuss the impact of different parameters on transaction regions. We study the out-of-sample performance in an empirical study that consists of six pairs of U.S. stocks selected from different industry sectors, and demonstrate the efficiency of the optimal strategy.
... Then, Lei and Xu (2015) proposed a model for determining multiple entry and exit-points during a trading period, and illustrated their results with applications to dual-listed Chinese stocks. Ngo and Pham (2014) frames the pairs trading problem as a regime switching model between three regimes: flat positions, one long position on one asset and a short position on the other, and vice versa . Finally, Lintilhac (2014), Cartea and Jaimungal (2015) considered applications to portfolios of co-integrated assets, both generalizing the model in Tourin and Yan (2013). ...
We propose a pairs trading model that incorporates a time-varying volatility of the Constant Elasticity of Variance type. Our approach is based on stochastic control techniques; given a fixed time horizon and a portfolio of two co-integrated assets, we define the trading strategies as the portfolio weights maximizing the expected power utility from terminal wealth. We compute the optimal pairs strategies by using a Finite Difference method. Finally, we illustrate our results by conducting tests on historical market data at daily frequency. The parameters are estimated by the Generalized Method of Moments.
... Indeed, most studies in the literature consider the two-regime case. The exceptions are Pham et al. [21] and Ngo and Pham [17], who study problems with three regimes. The setting of their optimization problem in Pham et al. [21] includes transaction costs, denoted by g ij , which are incurred by switching from regime i to j, and they impose a (strict) triangular condition, g ik < g ij + g jk , j i, k, which prevents any simultaneous multiple switches. ...
... However, Ngo and Pham [17], Suzuki [26], and Suzuki [27] formally prohibit the direct switching from i to k in their transaction rules. Thus, to switch from i to k, the strategy requires two consecutive simultaneous switches by way of j, that is, i →j → k. ...
... Therefore, we need to consider the simultaneous switching theory more rigorously, which is the subject of this paper. Ngo and Pham [17] do not treat simultaneous switchings explicitly. Their way of solving their problem is rather ad hoc and specific to the particular problem, rather than being applicable to more general problems. ...
... Moreover, Cartea et al. [40] investigate the optimal execution problem with price effects for multiple cointegrated assets and uses the multivariate OU process to model the co-movements of asset midprices. Other than previous work focusing on the weights of self-financing strategies, there are studies focusing on when to buy or sell a predetermined position, referred to as optimal switching problems (see, e.g., [41,42]). ...
Continuous-time pairs-trading rules are often developed based on the diffusion limit of first-order autoregressive cointegration models. Empirical identification of cointegration effects is generally made according to discrete-time error correction representation of vector autoregressive (VAR(p)) processes. We show that the diffusion limit of a VAR(p) process appears as a stochastic delayed differential equation. Motivated by this, we investigate the dynamic portfolio problem under a class of path-dependent models embracing path-dependent cointegration models as special case. Under certain regular conditions, we prove the existence of the optimal strategy and show that it is related to a system of Riccati partial differential equations. The proof is developed by means of functional It\^{o}'s calculus. When the process satisfies cointegration conditions, our results lead to the optimal dynamic pairs-trading rule. Our numerical study shows that the path-dependent effect has significantly impact on the pairs-trading strategy.
... Indeed, most studies in the literature consider the two-regime case. The exceptions are Pham et al. [21] and Ngo and Pham [17], who study problems with three regimes. The setting of their optimization problem in Pham et al. [21] includes transaction costs, denoted by g ij , which are incurred by switching from regime i to j, and they impose a (strict) triangular condition, g ik < g ij + g jk , j i, k, which prevents any simultaneous multiple switches. ...
... However, Ngo and Pham [17], Suzuki [26], and Suzuki [27] formally prohibit the direct switching from i to k in their transaction rules. Thus, to switch from i to k, the strategy requires two consecutive simultaneous switches by way of j, that is, i →j → k. ...
... Therefore, we need to consider the simultaneous switching theory more rigorously, which is the subject of this paper. Ngo and Pham [17] do not treat simultaneous switchings explicitly. Their way of solving their problem is rather ad hoc and specific to the particular problem, rather than being applicable to more general problems. ...
Very few studies have explored the structure of optimal switching regimes. We extend the existing research on the infinite-horizon multiple-regime switching problem with an arbitrary number of switch options by replacing the linear running reward function with a quadratic function in the objective function. To make our analysis more rigorous, we establish the theoretical basis for the application of the simultaneous multiple-regime switches to the problem with the extended objective function, and provide the sufficient condition under which each certain separated region in the space includes, at most, one single connected optimal switching region, which determines the structure of the optimal switching regions, and we identify the structure of the optimal switching regions for the particular problem.
... Our model is also similar to Ngo and Pham [8]. They consider the problem of determining the optimal cut-off of the pair trading rule for a three-state regime-switching model. ...
In this paper, we study a free boundary problem, which arises from an optimal trading problem of a stock that is driven by a uncertain market status process. The free boundary problem is a variational inequality system of three functions with a degenerate operator. The main contribution of this paper is that we not only prove all the four switching free boundaries are no-overlapping, monotonic and -smooth, but also completely determine their relative localities and provide the optimal trading strategies for the stock trading problem.
... Although pairs-trading has been practiced by traders for some time, [1] and [2] pioneer the systematic study of this strategy using the concept of cointegration developed by [18]. Subsequent research includes dynamic trading strategies [11,12,31,28], transaction costs [26], dynamic hedging [33,9], robust pairs-trading rules [13], and consumption-investment problems [27,30]. Chiu and Wong [12] prove mathematically that a time-consistent mean-variance (TCMV) strategy of cointegrated assets ensures the existence of a statistical arbitrage in the sense of [23]. ...
