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Speculative Futures Trading under Mean Reversion

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Tim Leung
Tim Leung
Tim Leung
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Jiao Li at Columbia University
Jiao Li
Xin Li at Columbia University
Xin Li
Zheng Wang at Columbia University
Zheng Wang
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Abstract and Figures

This paper studies the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross (CIR), or exponential Ornstein-Uhlenbeck (XOU) model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor's timing option to enter or exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short.
Left VIX futures historical prices on Nov 20, 2008 with the current VIX value at 80.86. The days to expiration range from 26 to 243 days (Dec–Jul contracts). Calibrated parameters: μ~=4.59,θ~=40.36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 4.59, \tilde{\theta } = 40.36$$\end{document} under the CIR/OU model, or μ~=3.25,θ~=3.65,σ=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 3.25, \tilde{\theta } = 3.65, \sigma = 0.15$$\end{document} under the XOU model. Right VIX futures historical prices on Jul 22, 2015 with the current VIX value at 12.12. The days to expiration ranges from 27 to 237 days (Aug–Mar contracts). Calibrated parameters: μ~=4.55,θ~=18.16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 4.55, \tilde{\theta } = 18.16$$\end{document} under the CIR/OU model, or μ~=4.08,θ~=3.06,σ=1.63\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 4.08, \tilde{\theta } = 3.06, \sigma = 1.63$$\end{document} under the XOU model.
Left VIX futures historical prices on Nov 20, 2008 with the current VIX value at 80.86. The days to expiration range from 26 to 243 days (Dec–Jul contracts). Calibrated parameters: μ~=4.59,θ~=40.36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 4.59, \tilde{\theta } = 40.36$$\end{document} under the CIR/OU model, or μ~=3.25,θ~=3.65,σ=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 3.25, \tilde{\theta } = 3.65, \sigma = 0.15$$\end{document} under the XOU model. Right VIX futures historical prices on Jul 22, 2015 with the current VIX value at 12.12. The days to expiration ranges from 27 to 237 days (Aug–Mar contracts). Calibrated parameters: μ~=4.55,θ~=18.16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 4.55, \tilde{\theta } = 18.16$$\end{document} under the CIR/OU model, or μ~=4.08,θ~=3.06,σ=1.63\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu } = 4.08, \tilde{\theta } = 3.06, \sigma = 1.63$$\end{document} under the XOU model.
… 
Optimal long-short boundaries with/without transaction costs for futures trading under the CIR model in a and b respectively, optimal short-long boundaries with/without transaction costs in c and d respectively, and optimal boundaries with/without transaction costs in e and f respectively. Parameters: T^=22252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}=\frac{22}{252}$$\end{document} , T=66252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=\frac{66}{252}$$\end{document}, r=0.05,σ=5.33,θ=17.58,θ~=18.16,μ=8.57,μ~=4.55,c=c^=0.005.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r=0.05, \sigma =5.33, \theta = 17.58,\tilde{\theta }=18.16,\mu =8.57,\tilde{\mu }=4.55, c=\hat{c}=0.005.$$\end{document}
Optimal long-short boundaries with/without transaction costs for futures trading under the CIR model in a and b respectively, optimal short-long boundaries with/without transaction costs in c and d respectively, and optimal boundaries with/without transaction costs in e and f respectively. Parameters: T^=22252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}=\frac{22}{252}$$\end{document} , T=66252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=\frac{66}{252}$$\end{document}, r=0.05,σ=5.33,θ=17.58,θ~=18.16,μ=8.57,μ~=4.55,c=c^=0.005.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r=0.05, \sigma =5.33, \theta = 17.58,\tilde{\theta }=18.16,\mu =8.57,\tilde{\mu }=4.55, c=\hat{c}=0.005.$$\end{document}
… 
Optimal boundaries with and without transaction costs for futures trading under the CIR model in a and b respectively. Parameters: T^=22252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}=\frac{22}{252}$$\end{document}, T=66252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=\frac{66}{252}$$\end{document}, r=0.05,σ=5.33,θ=17.58,θ~=18.16,μ=8.57,μ~=4.55,c=c^=0.005.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r=0.05, \sigma =5.33, \theta = 17.58,\tilde{\theta }=18.16,\mu =8.57,\tilde{\mu }=4.55, c=\hat{c}=0.005. $$\end{document}
Optimal boundaries with and without transaction costs for futures trading under the CIR model in a and b respectively. Parameters: T^=22252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}=\frac{22}{252}$$\end{document}, T=66252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=\frac{66}{252}$$\end{document}, r=0.05,σ=5.33,θ=17.58,θ~=18.16,μ=8.57,μ~=4.55,c=c^=0.005.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r=0.05, \sigma =5.33, \theta = 17.58,\tilde{\theta }=18.16,\mu =8.57,\tilde{\mu }=4.55, c=\hat{c}=0.005. $$\end{document}
… 
Optimal boundaries with transaction costs for futures trading. a OU spot model with σ=18.7,θ=17.58,θ~=18.16,μ=8.57,μ~=4.55\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =18.7, \theta = 17.58,\tilde{\theta }=18.16,\mu =8.57,\tilde{\mu }=4.55$$\end{document}. b XOU spot model with σ=1.63,θ=3.03,θ~=3.06,μ=8.57,μ~=4.08\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =1.63, \theta = 3.03,\tilde{\theta }=3.06, \mu =8.57, \tilde{\mu }=4.08$$\end{document}. Common parameters: T^=22252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}=\frac{22}{252}$$\end{document}, T=66252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=\frac{66}{252}$$\end{document}, c=c^=0.005.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ c=\hat{c}=0.005.$$\end{document}
Optimal boundaries with transaction costs for futures trading. a OU spot model with σ=18.7,θ=17.58,θ~=18.16,μ=8.57,μ~=4.55\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =18.7, \theta = 17.58,\tilde{\theta }=18.16,\mu =8.57,\tilde{\mu }=4.55$$\end{document}. b XOU spot model with σ=1.63,θ=3.03,θ~=3.06,μ=8.57,μ~=4.08\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =1.63, \theta = 3.03,\tilde{\theta }=3.06, \mu =8.57, \tilde{\mu }=4.08$$\end{document}. Common parameters: T^=22252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}=\frac{22}{252}$$\end{document}, T=66252\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=\frac{66}{252}$$\end{document}, c=c^=0.005.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ c=\hat{c}=0.005.$$\end{document}
… 
The value fuctions P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}$$\end{document}, A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}, and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} plotted against the spot price at time 0. The parameters are the same as those in Fig. 4
The value fuctions P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}$$\end{document}, A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}, and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} plotted against the spot price at time 0. The parameters are the same as those in Fig. 4
… 
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Asia-Pacific Finan Markets (2016) 23:281–304
DOI 10.1007/s10690-016-9215-9
Speculative Futures Trading under Mean Reversion
Tim Leung1·Jiao Li2·Xin Li1·Zheng Wang1
Published online: 18 April 2016
© Springer Japan 2016
Abstract This paper studies the problem of trading futures with transaction costs
when the underlying spot price is mean-reverting. Specifically, we model the spot
dynamics by the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, or exponential Ornstein–
Uhlenbeck model. The futures term structure is derived and its connection to futures
price dynamics is examined. For each futures contract, we describe the evolution of
the roll yield, and compute explicitly the expected roll yield. For the futures trading
problem, we incorporate the investor’s timing option to enter or exit the market, as well
as a chooser option to long or short a futures upon entry. This leads us to formulate and
solve the corresponding optimal double stopping problems to determine the optimal
trading strategies. Numerical results are presented to illustrate the optimal entry and
exit boundaries under different models. We find that the option to choose between a
long or short position induces the investor to delay market entry, as compared to the
case where the investor pre-commits to go either long or short.
The authors would like to thank Sebastian Jaimungal and Peng Liu for their helpful remarks, as well as
the participants of the Columbia-JAFEE Conference 2015, especially Jiro Akahori, Junichi Imai, Yuri
Imamura, Hiroshi Ishijima, Keita Owari, Yuji Yamada, Ciamac Moallemi, Marcel Nutz, and Philip Protter.
BTim Leung
tl2497@columbia.edu
Jiao Li
jl4170@columbia.edu
Xin Li
xl2206@columbia.edu
Zheng Wang
zw2192@columbia.edu
1IEOR Department, Columbia University, New York, NY 10027, USA
2APAM Department, Columbia University, New York, NY 10027, USA
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Numerical analysis of SGDCT for two common financial models is included in Sections 5.1, 5.2, and 5.5. The first is the well-known Ornstein-Uhlenbeck (OU) process (for examples in finance, see [19], [20], [29], and [18]). The second is the multidimensional CIR process which is a common model for interest rates (for examples in finance, see [4], [22], [11], [5], and [12]). ...
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This paper analyzes the problem of starting and stopping a Cox-Ingersoll-Ross (CIR) process with fixed costs. In addition, we also study a related optimal switching problem that involves an infinite sequence of starts and stops. We establish the conditions under which the starting-stopping and switching problems admit the same optimal starting and/or stopping strategies. We rigorously prove that the optimal starting and stopping strategies are of threshold type, and give the analytical expressions for the value functions in terms of confluent hypergeometric functions. Numerical examples are provided to illustrate the dependence of timing strategies on model parameters and transaction costs.
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Mean Reversion in Equilibrium Asset Prices: Evidence from the Futures Term Structure
Article
  • Aug 1993
We use price data from an array of futures markets to test whether investors expect spot asset prices to revert, and we identify two sources of equilibrium mean reversion: negative covariation between prices and interest rates, and positive covariation between prices and benefits to holding the asset. We find evidence of mean reversion in every market we examine, although magnitudes and sources differ across markets. For agricultural commodities and crude oil, the mean reversion is strong and arises solely from positive co- movement between prices and benefits to holders of the spot asset. For metals, the mean reversion arises from both sources, but is weaker. For financial assets, mean reversion is weak and is attributable entirely to interest rate sensitivity.
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