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![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.](publication/285393064/figure/fig1/AS:961460394024962@1606241429410/Left-VIX-futures-historical-prices-on-Nov-20-2008-with-the-current-VIX-value-at-8086.gif)
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.
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![The value fuctions P\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/285393064/figure/fig5/AS:961460398215168@1606241430063/The-value-fuctions-Pdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
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 dynamic...
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Citations
... Perceived knowledge about derivatives was found low in Europe (Abreu & Mendes, 2010). The complexity of the Derivative instruments like Futures and Options (F&O) are regularly used as hedging instruments, but these are also used as speculative instruments among investors (Leung et al., 2016). Some past studies proposed that the main use of the derivatives market is to provide hedging against price risk (Pan et al., 2003) while a wider perspective shows that it is also used as speculation, especially during earning results, and high volatility periods. ...
The objective of the study is to explore the motives of investors to invest in derivative markets. It is a quantitative study where a survey method was used to collect data from the investors using a probability sampling method. The data was analyzed using PLS-SEM to test the conceptual model. The results of the study show that speculation, hedging, and financial literacy are strong predictors of investors’ motives to invest in the derivatives market. The R2 was 0.447 implying speculation, hedging, and financial literacy explain 44.7% of the variance of the dependent variable, that is, the motives of investors to invest in equity derivatives, and the adjusted R-square is 0.432 (43.2%) which validates the model. Few studies explore the reasons to invest in derivatives using secondary data. However, to the best of the author’s knowledge studies exploring the motives of investors are rare, and there have been none using primary data from an Indian perspective. The study provides empirical evidence that could be useful to companies, investors, brokers, and policymakers to understand the motives of investors to invest in derivatives.
... There are a few alternative approaches and applications of dynamic futures portfolios. In [10] and [11], an optimal stopping approach for futures trading is studied. In practice, dynamic futures portfolios are also commonly used to track a commodity index [12]. ...
We study the problem of dynamically trading multiple futures whose underlying asset price follows a multiscale central tendency Ornstein-Uhlenbeck (MCTOU) model. Under this model, we derive the closed-form no-arbitrage prices for the futures contracts. Applying a utility maximization approach, we solve for the optimal trading strategies under different portfolio configurations by examining the associated system of Hamilton-Jacobi-Bellman (HJB) equations. The optimal strategies depend on not only the parameters of the underlying asset price process but also the risk premia embedded in the futures prices. Numerical examples are provided to illustrate the investor's optimal positions and optimal wealth over time.
... Duffie and Jackson (1990) is among the earliest to study the optimal hedging problem with standard futures. For further development along this line, please see Duffie and Richardson (1991), Dai et al. (2011), Leung et al. (2016), Angoshtari and Leung (2019) and the references therein. For Merton's problem with derivatives (not necessarily futures), we refer to Carr et al. (2001) for an exponential Lévy price model and Liu and Pan (2003) for a jump-diffusion model with stochastic volatility. ...
We consider an optimal trading problem for an investor who trades Bitcoin spot and Bitcoin inverse futures, plus a risk-free asset. The investor seeks an optimal strategy to maximize her expected utility of terminal wealth. We obtain explicit solutions to the investor's optimal strategies under both exponential and power utility functions. Empirical studies confirm that optimal strategies perform well in terms of Sharpe ratio and Sortino ratio and beat the long-only strategy in Bitcoin spot.
... All these studies propose a stochastic control approach to futures trading. In contrast, Leung et al. (2016) introduce an optimal stopping approach to determine the optimal timing to open or close a futures position under three single-factor mean-reverting spot models. Futures portfolios are also often used to track the spot price movements, and we refer to Ward (2015, 2018) for examples using gold and VIX futures. ...
We study a stochastic control approach to managed futures portfolios. Building on the (Schwartz, 1997) stochastic convenience yield model for commodity prices, we formulate a utility maximization problem for dynamically trading a single-maturity futures or multiple futures contracts over a finite horizon. By analyzing the associated Hamilton–Jacobi–Bellman (HJB) equation, we solve the investor’s utility maximization problem explicitly and derive the optimal dynamic trading strategies in closed form. We provide numerical examples and illustrate the optimal trading strategies using WTI crude oil futures data.
... All these studies propose a stochastic control approach to futures trading. In contrast, Leung et al. (2016) introduce an optimal stopping approach to determine the optimal timing to open or close a futures position under three single-factor mean-reverting spot models. Futures portfolios are also often used to track the spot price movements, and we refer to Ward (2015, 2018) for examples using gold and VIX futures. ...
We study a stochastic control approach to managed futures portfolios. Building on the Schwartz 97 stochastic convenience yield model for commodity prices, we formulate a utility maximization problem for dynamically trading a single-maturity futures or multiple futures contracts over a finite horizon. By analyzing the associated Hamilton-Jacobi-Bellman (HJB) equation, we solve the investor's utility maximization problem explicitly and derive the optimal dynamic trading strategies in closed form. We provide numerical examples and illustrate the optimal trading strategies using WTI crude oil futures data.
... All these studies propose a stochastic control approach to futures trading. In contrast, Leung et al. (2016) introduce an optimal stopping approach to determine the optimal timing to open or close a futures position under three single-factor mean-reverting spot models. Futures portfolios are also often used to track the spot price movements, and we refer to Ward (2015, 2018) for examples using gold and VIX futures. ...
... 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]). ...
Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. SGDCT performs an online parameter update in continuous time, with the parameter updates satisfying a stochastic differential equation. We prove that where is a natural objective function for the estimation of the continuous-time dynamics. The convergence proof leverages ergodicity by using an appropriate Poisson equation to help describe the evolution of the parameters for large times. SGDCT can also be used to solve continuous-time optimization problems, such as American options. For certain continuous-time problems, SGDCT has some promising advantages compared to a traditional stochastic gradient descent algorithm. As an example application, SGDCT is combined with a deep neural network to price high-dimensional American options (up to 100 dimensions).
... In addition, taking into account the mean-reverse property of the VIX index (Dueker [1997]; Whaley [2008]; Leung, Li, Li, and Wang [2016]), it is likely that investors get more incentives to trade VIX ETPs during time periods when equity market volatility is very extreme (high or low). More specifically, investors are motivated to take long positions in VIX ETPs in normal (inverse) groups when the VIX index is extremely low (high). ...
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... Analogously, when the equity market is highly volatile, investors tend to be long in VIX ETP groups that apply inverse tracking strategies. These findings provide supportive evidence that market participants expect the VIX index to move in a mean-reverse way (Whaley [2008]; Leung, Li, Li, and Wang [2016]). ...
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... which can be found in (Leung et al., 2016, Sect. 2.2). ...
This paper studies the market phenomenon of non-convergence between futures and spot prices in the grains market. We postulate that the positive basis observed at maturity stems from the futures holder's timing options to exercise the shipping certificate delivery item and subsequently liquidate the physical grain. In our proposed approach, we incorporate stochastic spot price and storage cost, and solve an optimal double stopping problem to give the optimal strategies to exercise and liquidate the grain. Our new models for stochastic storage rates lead to explicit no-arbitrage prices for the shipping certificate and associated futures contract. We calibrate our models to empirical futures data during the periods of observed non-convergence, and illustrate the premium generated by the shipping certificate.
We study the optimal timing strategies for trading a mean-reverting price process with afinite deadline to enter and a separate finite deadline to exit the market. The price process is modeled by a diffusion with an affine drift that encapsulates a number of well-known models,including the Ornstein-Uhlenbeck (OU) model, Cox-Ingersoll-Ross (CIR) model, Jacobi model,and inhomogeneous geometric Brownian motion (IGBM) model.We analyze three types of trading strategies: (i) the long-short (long to open, short to close) strategy; (ii) the short-long(short to open, long to close) strategy, and (iii) the chooser strategy whereby the trader has the added flexibility to enter the market by taking either a long or short position, and subsequently close the position. For each strategy, we solve an optimal double stopping problem with sequential deadlines, and determine the optimal timing of trades. Our solution methodology utilizes the local time-space calculus of Peskir (2005) to derive nonlinear integral equations of Volterra-type that uniquely characterize the trading boundaries. Numerical implementation ofthe integral equations provides examples of the optimal trading boundaries.
