Speculative Futures Trading under Mean Reversion

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Notes

  1. 1.

    Statistics taken from Acworth (2015).

  2. 2.

    See p. 615 of Elton et al. (2009) for a discussion.

  3. 3.

    See Deconstructing Futures Returns: The Role of Roll Yield, Campbell White Paper Series, February 2014.

  4. 4.

    By taking a short futures position, the investor is required to sell the underlying spot at maturity at a pre-specified price. In contrast to the short sale of a stock, a short futures does not involve share borrowing or re-purchasing.

  5. 5.

    The spot price is positive, thus \(s\in \mathbb {R}_+\), under the CIR and XOU models.

  6. 6.

    For a detailed discussion on the projected SOR method, we refer to Wilmott et al. (1995).

References

  1. Acworth, W. (2015). 2014 FIA annual global futures and options volume: Gains in North America and Europe offset declines in Asia-Pacific. [Online; posted 09-March-2015].

  2. Bali, T. G., & Demirtas, K. O. (2008). Testing mean reversion in financial market volatility: Evidence from S&P 500 index futures. Journal of Futures Markets, 28(1), 1–33.

    Article  Google Scholar 

  3. Bessembinder, H., Coughenour, J. F., Seguin, P. J., & Smoller, M. M. (1995). Mean reversion in equilibrium asset prices: Evidence from the futures term structure. The Journal of Finance, 50(1), 361–375.

    Article  Google Scholar 

  4. Brennan, M. J., & Schwartz, E. S. (1990). Arbitrage in stock index futures. Journal of Business, 63(1), S7–S31.

    Article  Google Scholar 

  5. Cartea, A., Jaimungal, S., & Penalva, J. (2015). Algorithmic and high-frequency trading. Cambridge: Cambridge University Press.

    Google Scholar 

  6. Casassus, J., & Collin-Dufresne, P. (2005). Stochastic convenience yield implied from commodity futures and interest rates. The Journal of Finance, 60(5), 2283–2331.

    Article  Google Scholar 

  7. Cox, J. C., Ingersoll, J., & Ross, S. A. (1981). The relation between forward prices and futures prices. Journal of Financial Economics, 9(4), 321–346.

    Article  Google Scholar 

  8. Dai, M., Zhong, Y., & Kwok, Y. K. (2011). Optimal arbitrage strategies on stock index futures under position limits. Journal of Futures Markets, 31(4), 394–406.

    Article  Google Scholar 

  9. Detemple, J., & Osakwe, C. (2000). The valuation of volatility options. European Finance Review, 4(1), 21–50.

    Article  Google Scholar 

  10. Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009). Modern portfolio theory and investment analysis (8th ed.). New York: Wiley.

    Google Scholar 

  11. Geman, H. (2007). Mean reversion versus random walk in oil and natural gas prices. In M. C. Fu, R. A. Jarrow, J.-Y. J. Yen, & R. J. Elliot (Eds.), Advances in mathematical finance, applied and numerical harmonic analysis (pp. 219–228). Boston: Birkhuser Boston.

    Google Scholar 

  12. Gorton, G. B., Hayashi, F., & Rouwenhorst, K. G. (2013). The fundamentals of commodity futures returns. Review of Finance, 17(1), 35–105.

    Article  Google Scholar 

  13. Grübichler, A., & Longstaff, F. (1996). Valuing futures and options on volatility. Journal of Banking and Finance, 20(6), 985–1001.

    Article  Google Scholar 

  14. Irwin, S. H., Zulauf, C. R., & Jackson, T. E. (1996). Monte Carlo analysis of mean reversion in commodity futures prices. American Journal of Agricultural Economics, 78(2), 387–399.

    Article  Google Scholar 

  15. Leung, T., & Li, X. (2015). Optimal mean reversion trading with transaction costs and stop-loss exit. International Journal of Theoretical & Applied Finance, 18(3), 15500.

    Article  Google Scholar 

  16. Leung, T., Li, X., & Wang, Z. (2014). Optimal starting–stopping and switching of a CIR process with fixed costs. Risk and Decision Analysis, 5(2), 149–161.

    Google Scholar 

  17. Leung, T., Li, X., & Wang, Z. (2015). Optimal multiple trading times under the exponential OU model with transaction costs. Stochastic Models, 31(4), 554–587.

    Article  Google Scholar 

  18. Leung, T., & Liu, P. (2012). Risk premia and optimal liquidation of credit derivatives. International Journal of Theoretical & Applied Finance, 15(8), 1250059.

    Article  Google Scholar 

  19. Leung, T., & Shirai, Y. (2015). Optimal derivative liquidation timing under path-dependent risk penalties. Journal of Financial Engineering, 2(1), 1550004.

    Article  Google Scholar 

  20. Lu, Z., & Zhu, Y. (2009). Volatility components: The term structure dynamics of VIX futures. Journal of Futures Markets, 30(3), 230–256.

    Google Scholar 

  21. Mencía, J., & Sentana, E. (2013). Valuation of VIX derivatives. Journal of Financial Economics, 108(2), 367–391.

    Article  Google Scholar 

  22. Monoyios, M., & Sarno, L. (2002). Mean reversion in stock index futures markets: A nonlinear analysis. The Journal of Futures Markets, 22(4), 285–314.

    Article  Google Scholar 

  23. Moskowitz, T. J., Ooi, Y. H., & Pedersen, L. H. (2012). Time series momentum. Journal of Financial Economics, 104(2), 228–250.

    Article  Google Scholar 

  24. Ribeiro, D. R. & Hodges, S. D. (2004). A two-factor model for commodity prices and futures valuation. EFMA 2004 Basel Meetings Paper.

  25. Schwartz, E. (1997). The stochastic behavior of commodity prices: Implications for valuation and hedging. The Journal of Finance, 52(3), 923–973.

    Article  Google Scholar 

  26. Wang, Z., & Daigler, R. T. (2011). The performance of VIX option pricing models: Empirical evidence beyond simulation. Journal of Futures Markets, 31(3), 251–281.

    Article  Google Scholar 

  27. Wilmott, P., Howison, S., & Dewynne, J. (1995). The mathematics of financial derivatives: A student introduction (1st ed.). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  28. Zhang, J. E., & Zhu, Y. (2006). VIX futures. Journal of Futures Markets, 26(6), 521–531.

    Article  Google Scholar 

  29. Zhu, S.-P., & Lian, G.-H. (2012). An analytical formula for VIX futures and its applications. Journal of Futures Markets, 32(2), 166–190.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tim Leung.

Additional information

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.

Appendix

Appendix

Numerical Implementation

We apply a finite difference method to compute the optimal boundaries in Figs. 2, 3 and 4. The operators \({\mathcal {L}} \,^{(i)}\), \(i\in \{1,2,3\}\), defined in (4.2)–(4.4) correspond to the OU, CIR, and XOU models, respectively. To capture these models, we define the generic differential operator

$$\begin{aligned} {\mathcal {L}} \,\{\cdot \}:= -r \cdot + \frac{\partial \cdot }{\partial t} + \varphi (s) \frac{\partial \cdot }{\partial s} + \frac{\sigma ^2(s)}{2}\frac{\partial ^2 \cdot }{\partial s^2}, \end{aligned}$$

then the variational inequalities (4.5), (4.6), (4.7), (4.8) and (4.9) admit the same form as the following variational inequality problem:

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {L}} \,g(t,s) \le 0, g(t,s) \ge \xi (t,s), &{}\quad (t,s) \in [0,\hat{T}) \times {\mathbb {R}}_+, \\ \\ ({\mathcal {L}} \,g(t,s)) (\xi (t,s) - g(t,s))= 0, &{}\quad (t,s) \in [0,\hat{T}) \times {\mathbb {R}}_+,\\ \\ g (\hat{T},s) = \xi (\hat{T},s), &{}\quad s \in {\mathbb {R}}_+. \end{array}\right. \end{aligned}$$
(6.1)

Here, g(ts) represents the value functions \({\mathcal {V}}(t,s)\), \({\mathcal {J}}(t,s)\), \(-{\mathcal {U}}(t,s)\), \({\mathcal {K}}(t,s)\), or \({\mathcal {P}}(t,s)\). The function \(\xi (t,s)\) represents \(f(t,s;T) - c\), \(({\mathcal {V}}(t,s) - (f(t,s;T) + \hat{c}))^+\), \(-(f(t,s;T) + \hat{c})\), \((f(t,s;T) - c) - {\mathcal {U}}(t,s))^+\), or \(\max \{ {\mathcal {A}}(t,s), {\mathcal {B}}(t,s) \}\). The futures price f(tsT), with \(\hat{T} \le T\), is given by (2.1), (2.4), and (2.10) under the OU, CIR, and XOU models, respectively.

We now consider the discretization of the partial differential equation \( {\mathcal {L}} \,g(t,s) =0\), over an uniform grid with discretizations in time (\(\delta t = \frac{\hat{T}}{N}\)), and space (\(\delta s = \frac{S{\max }}{M}\)). We apply the Crank–Nicolson method, which involves the finite difference equation:

$$\begin{aligned} -\alpha _i g_{i-1,j-1} + (1-\beta _i) g_{i,j-1} - \gamma _i g_{i+1,j-1}=\alpha _i g_{i-1,j} + (1+\beta _i) g_{i,j} + \gamma _i g_{i+1,j} , \end{aligned}$$

where

$$\begin{aligned} g_{i,j}&= g(j \delta t, i \delta s ), \quad \xi _{i,j} = \xi (j \delta t, i \delta s ), \quad \varphi _i = \varphi (i \delta s), \quad \sigma _i = \sigma (i \delta s). \\ \alpha _i&= \frac{\delta t}{4 \delta s}\big ( \frac{\sigma ^2 _i }{\delta s} - \varphi _i \big ), \quad \beta _i = -\frac{\delta t}{2} \big (r + \frac{\sigma ^2 _i}{(\delta s)^2}\big ), \quad \gamma _i = \frac{\delta t}{4 \delta s}\big ( \frac{\sigma ^2 _i }{\delta s} + \varphi _i \big ), \end{aligned}$$

for \(i=1,2,\ldots ,M-1\) and \(j=1,2,\ldots ,N-1\). The system to be solved backward in time is

$$\begin{aligned} \mathbf {M_1 g_{j-1}=r_j}, \end{aligned}$$

where the right-hand side is

$$\begin{aligned} \mathbf {r_j=M_2 g_{j}}+\alpha _1 \begin{bmatrix} g_{0,j-1}+g_{0,j} \\ 0 \\ \vdots \\ 0 \end{bmatrix} + \gamma _{M-1} \begin{bmatrix} 0 \\ \vdots \\ 0 \\ g_{M,j-1}+g_{M,j}, \end{bmatrix}, \end{aligned}$$

and

$$\begin{aligned} \mathbf {M_1}&= \left[ \begin{array}{cccccc} 1- \beta _1 &{}\quad -\gamma _1 &{}\quad &{}\quad &{}\quad \\ -\alpha _2 &{}\quad 1- \beta _2 &{}\quad -\gamma _2 &{}\quad &{}\quad \\ &{}\quad -\alpha _3 &{}\quad 1- \beta _3 &{}\quad -\gamma _3 &{}\quad \\ &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \\ &{}\quad &{}\quad &{}\quad - \alpha _{M-2} &{}\quad 1- \beta _{M-2} &{}\quad -\gamma _{M-2} \\ &{}\quad &{}\quad &{}\quad &{}\quad - \alpha _{M-1} &{}\quad 1- \beta _{M-1} \end{array} \right] ,\\ \mathbf {M_2}&\quad = \left[ \begin{array}{cccccc} 1+ \beta _1 &{}\quad \gamma _1 &{}\quad &{}\quad &{}\quad \\ \alpha _2 &{}\quad 1+ \beta _2 &{}\quad \gamma _2 &{}\quad &{}\quad \\ &{}\quad \alpha _3 &{}\quad 1+ \beta _3 &{}\quad \gamma _3 &{}\quad \\ &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \\ &{}\quad &{}\quad &{}\quad \alpha _{M-2} &{}\quad 1+ \beta _{M-2} &{}\quad \gamma _{M-2} \\ &{}\quad &{}\quad &{}\quad &{}\quad \alpha _{M-1} &{}\quad 1+ \beta _{M-1} \end{array} \right] ,\\ \mathbf {g_j}&\quad =\begin{bmatrix} g_{1,j}, g_{2,j}, \ldots , g_{M-1,j} \end{bmatrix} ^T. \end{aligned}$$

This leads to a sequence of stationary complementarity problems. Hence, at each time step \(j \in \left\{ 1, 2, \ldots , N-1\right\} \), we need to solve

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{}\mathbf {M_1 g_{j-1}} \ge \mathbf {r_j}, \\ \\ &{}\mathbf {g_{j-1}} \ge \varvec{\xi _{j-1}}, \\ \\ &{}(\mathbf {M_1 g_{j-1}} -\mathbf {r_j})^T (\varvec{\xi _{j-1}} - \mathbf {g_{j-1}}) = 0. \end{aligned} \end{array}\right. } \end{aligned}$$

To solve the optimal problem, our algorithm enforces the constraint explicitly as follows

$$\begin{aligned} g_{i,j-1}^{new}=\max \big \{g_{i,j-1}^{old},\xi _{i,j-1}\big \}. \end{aligned}$$
(6.2)

The projected SOR method is used to solve the linear system.Footnote 6 At each time j, we iteratively solve

$$\begin{aligned} \begin{aligned} g_{1,j-1}^{(k+1)}&= \max \left\{ \xi _{1,j-1} \,,\, g_{1,j-1}^{(k)} + \frac{\omega }{1-\beta _1} \left[ r_{1,j}-(1-\beta _1) g_{1,j-1}^{(k)}+\gamma _1 g_{2,j-1}^{(k)}\right] \right\} ,\\ g_{2,j-1}^{(k+1)}&= \max \left\{ \xi _{2,j-1} \,,\, g_{2,j-1}^{(k)} + \frac{\omega }{1-\beta _2} \left[ r_{2,j}+\alpha _2 g_{1,j-1}^{(k+1)}-(1-\beta _2) g_{2,j-1}^{(k)}\right. \right. \\&\quad \left. \left. +\gamma _2 g_{3,j-1}^{(k)}\right] \right\} ,\\ \vdots&\; \\ g_{M-1,j-1}^{(k+1)}&= \max \left\{ \xi _{M-1,j-1} \,,\, g_{M-1,j-1}^{(k)} \right. \\&\left. + \frac{\omega }{1-\beta _{M-1}} \left[ r_{M-1,j}+\alpha _{M-1} g_{M-2,j-1}^{(k+1)}-(1-\beta _{M-1}) g_{M-1,j-1}^{(k)}\right] \right\} , \end{aligned} \end{aligned}$$
(6.3)

where k is the iteration counter and \(\omega \) is the overrelaxation parameter. The iterative scheme starts from an initial point \(\mathbf {g}_j ^{(0)}\) and proceeds until a convergence criterion is met, such as \(|| \mathbf {g}_{j-1} ^{(k+1)} - \mathbf {g}_{j-1} ^{(k)} || < \epsilon ,\) where \(\epsilon \) is a tolerance parameter. The optimal boundary \(S_f(t)\) can be identified by locating the boundary that separates the regions where \(g(t,s)=\xi (t,s)\), or \(g(t,s) \ge \xi (t,s)\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Leung, T., Li, J., Li, X. et al. Speculative Futures Trading under Mean Reversion. Asia-Pac Financ Markets 23, 281–304 (2016). https://doi.org/10.1007/s10690-016-9215-9

Download citation

Keywords

  • Optimal stopping
  • Mean reversion
  • Futures trading
  • Roll yield
  • Variational inequality

Mathematics Subject Classification

  • 60G40
  • 62L15
  • 91G20
  • 91G80

JEL Classification  

  • C41
  • G11
  • G13