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.

Appendix

1.1 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(t, s) 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(t, s; T), 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.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 $g(t,s) \ge \xi (t,s)$.