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Optimal Pairs Trading with Time-Varying Volatility
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Abstract
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.
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We analyze the classical Merton's portfolio optimization problem when the risky asset follows an exponential Ornstein-Uhlenbeck process, also known as the Schwartz mean-reversion dynamics. The corresponding Hamilton-Jacobi-Bellman equation is a two-dimensional nonlinear parabolic partial differential equation. We produce an explicit solution to this equation by reducing it to a simpler one-dimensional linear parabolic equation. This reduction is achieved through a Cole-Hopf type transformation, recently introduced in portfolio optimization theory by Zariphopoulou [9]. A verification argument is then used to prove that this solution coincides with the value function of the control problem. The optimal investment strategy is also given explicitly. On the technical side, the main problem we are facing here is the necessity to identify conditions on the parameters of the control problem ensuring uniform integrability of a family of random variables that, roughly speaking, are the exponentials of squared Wiener integrals.
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.
We present a simple, purely analytic method for proving the convergence of a wide class of approximation schemes to the solution of fully non linear second-order elliptic or parabolic PDE. Roughly speaking, we prove that any monotone, stable and consistent scheme converges to the correct solution provided that there exists a comparison principle for the limiting equation. This method is based on the notion of viscosity solution of Crandall and Lions and it gives completely new results concerning the convergence of numerical schemes for stochastic differential games.
We assume that the drift in the returns of asset prices consists of an idiosyncratic component and a common component given by a co-integration factor. We analyze the optimal investment strategy for an agent who maximizes expected utility of wealth by dynamically trading in these assets. The optimal solution is constructed explicitly in closed-form and is shown to be affine in the co-integration factor. We calibrate the model to three assets traded on the Nasdaq exchange (Google, Facebook, and Amazon) and employ simulations to showcase the strategy’s performance.
Under the assumption that two asset prices follow an uncertain volatility model, the maximal and minimal solution values of
an option contract are given by a two-dimensional Hamilton–Jacobi–Bellman Partial Differential Equation (PDE). A fully implicit,
unconditionally monotone finite difference numerical scheme is developed in this article. Consequently, there are no timestep
restrictions due to stability considerations. The discretized algebraic equations are solved using policy iteration. Our discretization
method results in a local objective function, which is a discontinuous function of the control. Hence, some care must be taken
when applying policy iteration. The main difficulty in designing a discretization scheme is development of a monotonicity
preserving approximation of the cross derivative term in the PDE. We derive a hybrid numerical scheme, which combines use
of a fixed point stencil and a wide stencil based on a local coordinate rotation. The algorithm uses the fixed point stencil
as much as possible to take advantage of its accuracy and computational efficiency. The analysis shows that our numerical
scheme is l ∞ l∞ stable, consistent in the viscosity sense and monotone. Thus, our numerical scheme guarantees convergence to the viscosity
solution.
Motivated by the industry practice of pairs trading, we study the optimal timing strategies for trading a mean-reverting price spread. An optimal double stopping problem is formulated to analyze the timing to start and subsequently liquidate the position subject to transaction costs. Modeling the price spread by an Ornstein-Uhlenbeck process, we apply a probabilistic methodology and rigorously derive the optimal price intervals for market entry and exit. As an extension, we incorporate a stop-loss constraint to limit the maximum loss. We show that the entry region is characterized by a bounded price interval that lies strictly above the stop-loss level. As for the exit timing, a higher stop-loss level always implies a lower optimal take-profit level. Both analytical and numerical results are provided to illustrate the dependence of timing strategies on model parameters such as transaction costs and stop-loss level.
We study the optimal trading policy of an arbitrageur who can exploit temporary mispricing in a market with two convergent assets. We build on the model of Liu and Timmermann (2013) and include transaction costs, which impose additional limits to the implementation of such convergence trade strategy. We show that the presence of transaction costs could reveal an endogenous stop-loss concern in a certain economy, which affects the optimal policy of the arbitrageur in significant ways. Using pairs of Chinese stock shares that are dual-listed in Shanghai and in Hong Kong, we show that our co-integration strategy is generally superior to the relative-value pairs trading strategy studied in Gatev et al. (2006). Several extensions of our model are also discussed.
This paper investigates the impact of stochastic volatility on the Dickey–Fuller unit root test. Monte Carlo simulations show that the test size is seriously distorted if nonstationary stochastic volatility is ignored. To improve the performance of the test, we propose a Bayesian test for unit root that is robust in the presence of stationary and nonstationary stochastic volatility. The finite sample property of the proposed test statistic is evaluated using Monte Carlo studies. Applying the developed method, we test the policy effect of the Split Share Structure Reform, which is an important milestone for the development of the emerging stock market in China.
We propose a model for analyzing dynamic pairs trading strategies using the stochastic control approach. The model is explored in an optimal portfolio setting, where the portfolio consists of a bank account and two co-integrated stocks and the objective is to maximize for a fixed time horizon, the expected terminal utility of wealth. For the exponential utility function, we reduce the problem to a linear parabolic Partial Differential Equation which can be solved in closed form. In particular, we exhibit the optimal positions in the two stocks.
This paper studies Merton's portfolio problem for an investor who can trade a risk-free asset and a stock. We consider the case where the stock price follows a constant elasticity of variance (CEV) dynamics. By applying stochastic optimal control theory and variable change technique, we derive an explicit solution for the CRRA utility function. The solution consists of a moving Merton strategy and a modified factor. The moving Merton strategy is equivalent to the original Merton strategy but it has an updated volatility at the current time. The modified factor denotes a supplement part resulted from the change of the volatility.
Deregulation of energy commodity markets in the last decade, together with the growth of world consumption and the attractive returns on commodities over the period 2000-2007, has generated a dramatic increase in trading volumes and, in turn, spikes and random changes in the volatility of commodity spot prices.This article introduces the constant elasticity of variance (CEV) model for commodity prices and examines its calibration to four strategic commodity trajectory prices over the period 1990-2007 by using a Generalized Method of Moments. Six other models are compared to the CEV model by performing a test of goodness-of-fit. Estimating the model for crude oil, coal, copper, and gold and comparing the results during the sub-periods 1990-1999 and 2000-2007 shows that the constant elasticity of variance exponent can efficiently account for the stochastic volatility observed after 2000 in commodity prices. Moreover, the article exhibits that although mean-reverting processes well captured the pattern of commodity prices prevailing before 2000, they do not apply to the recent past.
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.
We study the arbitrage free optionpricing problem for the constant elasticity of variance (CEV) model. To treatthestochastic aspect of the CEV model, we direct attention to the relationship between the CEV modeland squared Bessel processes. Then we show the existence of a unique equivalentmartingale measure and derive the Cox's arbitrage free option pricing formulathrough the properties of squared Bessel processes. Finally we show that the CEVmodel admits arbitrage opportunities when it is conditioned to be strictlypositive.
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.
This paper examines the structure of option valuation problems and develops a new technique for their solution. It also introduces several jump and diffusion processes which have not been used in previous models. The technique is applied to these processes to find explicit option valuation formulas, and solutions to some previously unsolved problems involving the pricing of securities with payouts and potential bankruptcy.
This paper investigates theoretical and practical aspects of options that are based upon two or more assets which are co-integrated. For this purpose, a new, discrete-time model of asset prices is developed, a model featuring both the co-integration property as well as stochastic volatilities. Using a GARCH, equilibrium-based option pricing approach, it is shown that when volatilities are deterministic the option prices do not depend on the co-integration parameters, except for the mis-specification effect as to the manner in which the volatilities are estimated. However, with stochastic volatilities the option prices explicitly depend upon the co-integration parameters. In order to understand these results better, this paper also examines a continuous-time, diffusion limit of the asset price system and empirically studies the co-integration effect using spread options based upon the S&P500 and the NASDAQ100. These numerical results suggest that consideration of co-integration can substantially alter the value, delta and vega of a spread option.
This paper describes a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction. It also establishes consistency of the estimated covariance matrix under fairly general conditions.
This paper studies estimators that make sample analogues of population orthogonality conditions close to zero. Strong consistency and asymptotic normality of such estimators is established under the assumption that the observable variables are stationary and ergodic. Since many linear and nonlinear econometric estimators reside within the class of estimators studied in this paper, a convenient summary of the large sample properties of these estimators, including some whose large sample properties have not heretofore been discussed, is provided.
Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.
Notes on Option pricing I: constant elasticity of diffusions. A revised version of this paper was
- J C Cox
Cox, J. C. (1975). Notes on Option pricing I: constant elasticity of diffusions. A revised version of
this paper was published in the Journal of Portfolio management in 1996.
Optimal Trading of Cointegrated Portfolios in Arbitrary Dimensions with an Application to Arbitrage in Bitcoin Markets
- P Lintilhac
Lintilhac, P. (2014). Optimal Trading of Cointegrated Portfolios in Arbitrary Dimensions with
an Application to Arbitrage in Bitcoin Markets. Master's thesis, Courant Institute, New York
University, 251 Mercer Street, New York, NY 10012.