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Infinite-Horizon Optimal Switching Regions for a Pair-Trading Strategy with Quadratic Risk Aversion Considering Simultaneous Multiple Switchings: A Viscosity Solution Approach
Abstract
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.
... 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.
Continuous-time pairs trading rules are often developed based on the diffusion limit of the first-order vector autoregressive (VAR(1)) cointegration models. Empirical identification of cointegration effects is generally made according to discrete-time error correction representation of vector autoregressive (VAR(p)) processes, allowing for delayed adjustment of the price deviation. Motivated by this, we investigate the continuous-time dynamic pairs trading problem under a class of path-dependent models. 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 the means of functional Ito’s calculus. We conduct a numerical study to analyze the sensitivities of the pairs trading strategy with respect to the initial market conditions and the memory length.
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 one short the other and short one long the
other. A fixed commission cost is charged with each transaction. We use a
viscosity solutions approach to prove the existence and the explicit
characterization of cut-off points via the resolution of quasi-algebraic
equations. We illustrate our results by numerical simulations.
This work provides an optimal trading rule that allows buying and selling an asset sequentially over time. The asset price follows a switchable mean-reversion model with a Markovian jump. Such a model can be applied to assets with a “staircase” price behavior and yet is simple enough to allow an analytic solution. The objective is to determine a sequence of trading times to maximize an overall return. The corresponding value functions are characterized by a set of quasi-variational inequalities. A closed-form solution is obtained under suitable conditions. The sequence of trading times can be given in terms of a set of threshold levels. Finally, numerical examples are given to demonstrate the results.
This paper is concerned with a pairs trading rule. The idea is to monitor two
historically correlated securities. When divergence is underway, i.e., one
stock moves up while the other moves down, a pairs trade is entered which
consists of a pair to short the outperforming stock and to long the
underperforming one. Such a strategy bets the "spread" between the two would
eventually converge. In this paper, a difference of the pair is governed by a
mean-reverting model. The objective is to trade the pair so as to maximize an
overall return. A fixed commission cost is charged with each transaction. In
addition, a stop-loss limit is imposed as a state constraint. The associated
HJB equations (quasi-variational inequalities) are used to characterize the
value functions. It is shown that the solution to the optimal stopping problem
can be obtained by solving a number of quasi-algebraic equations. We provide a
set of sufficient conditions in terms of a verification theorem. Numerical
examples are reported to demonstrate the results.
'Pairs Trading' is an investment strategy used by many Hedge Funds. Consider two similar stocks which trade at some spread. If the spread widens short the high stock and buy the low stock. As the spread narrows again to some equilibrium value, a profit results. This paper provides an analytical framework for such an investment strategy. We propose a mean-reverting Gaussian Markov chain model for the spread which is observed in Gaussian noise. Predictions from the calibrated model are then compared with subsequent observations of the spread to determine appropriate investment decisions. The methodology has potential applications to generating wealth from any quantities in financial markets which are observed to be out of equilibrium.
In this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process. We use backward stochastic dierential equations and Snell envelopes. Finally we give some numerical results. AMS Classification subjects: 60G40 ; 93E20 ; 62P20 ; 91B99.
We consider a stochastic control problem that has emerged in the economics literature as an investment model under uncertainty. This problem combines features of both stochastic impulse control and optimal stopping. The aim is to discover the form of the optimal strategy. It turns out that this has a priori rather unexpected features. The results that we establish are of an explicit nature. We also construct an example whose value function does not possess C1 regularity.
This paper studies the optimal switching problem for a general one-dimensional diffusion with multiple (more than two) regimes. This is motivated in the real options literature by the investment problem of a firm managing several production modes while facing uncertainties. A viscosity solutions approach is employed to carry out a fine analysis on the associated system of variational inequalities, leading to sharp qualitative characterizations of the switching regions. These characterizations, in turn, reduce the switching problem into one of finding a finite number of threshold values in state that would trigger switchings. The results of our analysis take several qualitatively different forms depending on model parameters, and the issue of when and where it is optimal to switch is addressed. The general results are then demonstrated by the three-regime case, where a quasi-explicit solution is obtained, and a numerical procedure to find these critical values is devised in terms of the expectation functionals of hitting times for one-dimensional diffusions.
We address the problem of determining in an optimal way the sequence of times at which a firm can enter or exit an economic activity. In particular, we consider an investment model which involves production scheduling as well as a sequence of entry and exit decisions. The pricing of an investment conforming with this model gives rise to a stochastic impulse control problem that we explicitly solve. Our solution takes qualitatively different forms, depending on the problem's data.
This paper studies the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solutions approach is employed to carry out a finne analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. This problem is in relation with the valuation of firms in a financial market.
In this paper we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is in relation with the valuation of firms in a financial market.
We consider the problem of optimal multiple switching in finite horizon, when the state of the system, including the switching costs, is a general adapted stochastic process. The problem is formulated as an extended impulse control problem and completely solved using probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. Finally, when the state of the system is a Markov diffusion process, we show that the vector of value functions of the optimal problem is a viscosity solution to a system of variational inequalities with inter-connected obstacles.
We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions. Comment: Keywords: Optimal switching problem, optimal stopping problem, It\^{o} diffusions
In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.
Suzuki [Automatica, 2016, 67, 33–45] solves the optimal, finitely iterative, three-regime switching problem for investing in a mean-reverting asset that follows an Ornstein–Uhlenbeck price process and find explicit solutions. The remarkable feature of this model is that the investor can explicitly take either a long, short or square position and can switch the position, with transaction costs, during the investment period. We run empirical simulations of such multiple-regime switching models. There are very few such attempts in the existing literature because it is difficult to find, first, an explicit solution to the problem and second, appropriate financial assets that follow the artificial stochastic process required by the mathematical model. According to the Monte Carlo simulations of the optimal pair-trading strategy, the mean daily Sharp ratio is more than 2.3, whereas the mean Sharp ratio for the historical simulation of the ‘stub’ pairs (combinations of parent/subsidiary companies) is 0.6886. We believe that the results obtained from performing the empirical simulations are remarkable and consider that the optimal switching strategy of the rigorous mathematical model is applicable to businesses in the real world. For the reference many pseudo-program codes are added, which can help to replicate the optimal trading strategies.
We solve optimal iterative three-regime switching problems with transaction costs, with investment in a mean-reverting asset that follows an Ornstein–Uhlenbeck process and find the explicit solutions. The investor can take either a long, short or square position and can switch positions during the period. Modeling the short sales position is necessary to study optimal trading strategies such as the pair trading. Few studies provide explicit solutions to problems with multiple (more than two) regimes (states). The value function is proved to be a unique viscosity solution of a Hamilton–Jacobi–Bellman variational inequality (HJB-VI). Multiple-regime switching problems are more difficult to solve than conventional two-regime switching problems, because they need to identify not only when to switch, but also where to switch. Therefore, multiple-regime switching problems need to identify the structure of the continuation/switching regions in the free boundary problem for each regime. If the number of the states is two, only two regions have to be identified, but if , regions have to be detected. We identify the structure of the switching regions for each regime using the theories related to the viscosity solution approach.
A stochastic optimal switching and impulse control problem in a finite horizon is studied. The continuity of the value function, which is by no means trivial, is proved. The Bellman dynamic programming principle is shown to be valid for such a problem. Moroever, the value function is characterized as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation.
We consider the problem of the optimal trading of an asset in the presence of xed transaction costs where the asset price satises an SDE of the formdSt = dBt+h(Xt)dt where Bt is a Brownian motion, h is a known function and Xt is a Markov Chain. We look at two versions of the problem, maximising the long term gain per unit time and maximising a form of discounted gain. It is well known that the optimal trading strategy for such a problem is the solution of a free-boundary problem; we present an intuitive derivation by viewing the optimal trading problem as a pair of simultaneous optimal stopping problems. We also give explicit solutions for a range of examples, and give bounds on the transaction cost above which it is optimal never to buy the asset at all. We show that in the case where Markov Chain Xt is independent of the Brownian motion and has a nite statespace, this critical transaction cost has a simple form.
This paper considers the problem of finding the optimal sequence of opening (starting) and closing (stopping) times of a multi- activity production process, given the costs of opening, running, and closing the activities and assuming that the state of the economic system is a stochastic process. The problem is formulated as an extended impulse control problem and solved using stochastic calculus. As an application, the optimal starting and stopping strategy are explicitly found for a resource extraction when the price of the resource is following a geometric Brownian motion.
Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.
This paper studies the asset allocation problem of optimally tracking a target mix of asset categories when there are transaction costs. We consider the trading strategy for an investor who is trying to minimize both fixed and proportional transaction costs while simultaneously minimizing the tracking error with respect to a specified, target asset mix. We use imupulse control theory in a continuous-time, dynamic setting to deal with this problem in a general and analytical way, showing that the optimal trading strategy can be characterized in terms of a quasi-variational inequality. We derive an explicit solution for the two-asset case, and we use this to provide a sensitivity analysis, showing how the optimal strategy depends upon individual input parameters. We also use some theory for one-dimensional diffusion processes to derive analytical expressions for various measures of performance such as the average time between transactions.
In this paper, we propose a stochastic control approach to the problem of pairs trading. We model the log-relationship between a pair of stock prices as an Ornstein-Uhlenbeck process and use this to formulate a portfolio optimization based stochastic control problem. We are able to obtain the optimal solution to this control problem in closed form via the corresponding Hamilton-Jacobi-Bellman equation. We also provide closed form maximum-likelihood estimation values for the parameters in the model. The approach is illustrated with a numerical example involving simulated data for a pair of stocks.
Many recent theoretical papers have come under attack for modeling prices as Geometric Brownian Motion. This process can diverge over time, implying that firms facing this price process can earn infinite profits. We explore the significance of this attack and contrast investment under Geometric Brownian Motion with investment assuming mean reversion. While analytically more complex, mean reversion in many cases is a more plausible assumption, allowing for supply responses to increasing prices. We show that cumulative investment is generally unaffected by the use of a mean reversion process rather than Geometric Brownian Motion and provide an explanation for this result.
This paper is concerned with an optimal trading (buy and sell) rule. The underlying asset price is governed by a mean-reverting model. The objective is to buy and sell the asset so as to maximize the overall return. Slippage cost is imposed on each transaction. The associated HJB equations (quasi-variational inequalities) are used to characterize the value functions. It is shown that the solution to the original optimal stopping problem can be obtained by solving two quasi-algebraic equations. Sufficient conditions are given in the form of a verification theorem. A numerical example is reported to demonstrate the results.
Many economic variables of interest exhibit a tendency to revert to predictable long-run levels. However, mean reverting processes are rarely used in investment models in the literature. In most models, geometric Brownian motion processes are used for tractability. In this paper, a firm's entry and exit decisions when the output equilibrium price follows an exogenous mean reverting process are examined, and then compared to the decisions of the firm under the usually employed assumption of lognormally distributed output price, presented in Dixit (1989a). By extending previous work by Sarkar (2003) to account for costly reversibility, we show that mean reversion has a significant effect, not only on firm-specific entry and exit decisions, but also on the balance of entering and exiting firms in an industry/market. Thus it would be erroneous to use the more tractable geometric Brownian motion process as an approximation for a mean-reverting process in models and investigations of aggregate industry investment.
Mean-Reverting processes are appropriate for most “real option” investment models, yet Geometric Brownian Motion (GBM) processes are generally used for tractability. Hassett and Metcalf (J. Econom. Dyn. Control 19 (1995) 1471–1488) argue that using a GBM process is justified because mean-reversion has two opposing effects—it brings the investment trigger closer, and also reduces the conditional volatility (thereby lowering the likelihood of reaching the trigger)—and the overall effect on investment should be negligible for most reasonable parameter values. This paper extends the Hassett and Metcalf model by incorporating a third factor, the effect of mean-reversion on systematic risk. The main result is that mean reversion, in general, does have a significant impact on investment. Moreover, this effect could be either positive or negative, depending on various factors such as project duration, cost of investing, interest rate, etc. Thus it is generally inappropriate to use the GBM process to approximate a mean-reverting process.
This paper considers the problem of determining the optimal sequence of stopping times for a diusion process subject to regime switching decisions. This is motivated in the economics literature, by the investment problem under uncertainty for a multi- activity firm involving opening and closing decisions. We use a viscosity solutions approach combined with the smooth-fit property, and explicitly solve the problem in the two regime case when the state process is of geometric Brownian nature. The results of our analysis take several qualitatively dierent forms, depending on model parameter values.
We consider the problem of optimal multi-modes switching in finite horizon,
when the state of the system, including the switching cost functions are
arbitrary (). We show existence of the optimal strategy, and
give when the optimal strategy is finite via a verification theorem. Finally,
when the state of the system is a markov process, we show that the vector of
value functions of the optimal problem is the unique viscosity solution to the
system of m variational partial differential inequalities with
inter-connected obstacles.
A firm's entry and exit decisions when the output price follows a random walk are examined. An idle firm and an active firm are viewed as assets that are call options on each other. The solution is a pair of trigger prices for entry and exit. The entry trigger exceeds the variable cost plus the interest on the entry cost, and the exit trigger is less than the variable cost minus the interest on the exit cost. These gaps produce "hysteresis." Numerical solutions are obtained for several parameter values; hysteresis is found to be significant even with small sunk costs. Copyright 1989 by University of Chicago Press.
In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one‐factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.
A finite horizon optimal multiple switching problem
- B Djehiche
- S Hamadne
- A Popier
Djehiche B, Hamadne S, Popier A (2009) A finite horizon optimal multiple switching problem. SIAM J. Control Optim. 48(4):2751-2770.