Analytic value function for a pairs trading strategy with a Lévy-driven Ornstein–Uhlenbeck process

Abstract

This paper studies the performance of pairs trading strategy under a specific spread model. Based on the empirical evidence of mean reversion and jumps in the spread between pairs of stocks, we assume that the spread follows a Lévy-driven Ornstein–Uhlenbeck process with two-sided jumps. To evaluate the performance of a pairs trading strategy, we propose the expected return per unit time as the value function of the strategy. Significantly different from the current related works, we incorporate an excess jump component into the calculation of return and time cost. Further, we obtain the analytic expression of strategy value function, where we solve out the probabilities of crossing thresholds via the Laplace transform of first passage time of the Lévy-driven Ornstein–Uhlenbeck process in one-sided and two-sided exit problems. Through numerical illustrations, we calculate the value function and optimal thresholds for a spread model with symmetric jumps, reveal the non-negligible contribution of incorporating the excess jumps into the value function, and analyze the impact of model parameters on the strategy performance.

Keywords:

• Pairs trading
• Lévy-driven Ornstein–Uhlenbeck process
• Optimal thresholds
• Two-sided exit problem

JEL Classification:

• C61
• C63

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It should be noted that the initial value x can take any values in theory and practice. Since pairs trading is an consecutive process, the value of $X_{\nu }$ or $X_{\eta }$ is exactly the initial value of next trading. Due to the double exponential jumps in the OUDEJ process, the overshoot/undershoot allows $X_{\nu }$ and $X_{\eta }$ to exceed the predetermined thresholds by any distance, so the initial value of next trade can take any value theoretically. In practice, the investor should adjust investment ratio β dynamically, so the initial value of spread can change with strategy adjustment.

2 The existence of $G_2^{\ast }\left(x\right)$ and $H_3^{\ast }\left(x\right)$ is verified in Appendix A.5.

3 In the following, when x = 0, we drop the subscript of ${\mathbb{E}}_x[\cdot ]$, ${\mathbb{P}}_x(\cdot )$ and $v_x(\cdot ).$

4 By calculating the first-order and second-order moments of $X_t$, we can show that the benchmark parameter setting agrees with the common sense in market. Specifically, from ABC-ICBC spread minute-by-minute data in 2011 analyzed in Section 2.1, we calculate the daily mean and standard derivation as 0 and 0.3. In the other hand, from the results (2.3) in Zhou et al. (2017), we can calculate the expectation and variance of $X_t$. $\begin{array}{rl}{\mathbb{E}}_x\left(X_t\right)& =\underset{q\downarrow 0}{lim}\frac{\mathrm{\partial }}{\mathrm{\partial }q}\ln \left[{\mathbb{E}}_x\left({\mathrm{e}}^{q{X}_t}\right)\right]\\ & ={\mathrm{e}}^{-\kappa t}x+\left(1-{\mathrm{e}}^{-\kappa t}\right)\left[\alpha +\frac{1}{\kappa }\left(\mu +\frac{\lambda p}{{\gamma }_1}-\frac{\lambda \left(1-p\right)}{{\gamma }_2}\right)\right],\\ {\mathrm{V}\mathrm{a}\mathrm{r}}_x\left(X_t\right)& =\underset{q\downarrow 0}{lim}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{q}^2}\ln \left[{\mathbb{E}}_x\left({\mathrm{e}}^{q{X}_t}\right)\right]-{\mathbb{E}}_x^2\left(X_t\right)\\ & =\left(1-{\mathrm{e}}^{-2\kappa t}\right)\left[\frac{{\sigma }^2}{2\kappa }+\frac{\lambda }{\kappa }\left(\frac{p}{{\gamma }_1^2}+\frac{1-p}{{\gamma }_2^2}\right)\right].\end{array}$ When t is large enough, the asymptotic expectation and standard derivation are $\begin{array}{rl}\underset{t\to \mathrm{\infty }}{lim}{\mathbb{E}}_x\left(X_t\right)& =\alpha +\frac{1}{\kappa }\left(\mu +\frac{\lambda p}{{\gamma }_1}-\frac{\lambda \left(1-p\right)}{{\gamma }_2}\right),\mathrm{a}\mathrm{n}\mathrm{d}\\ \underset{t\to \mathrm{\infty }}{lim}S{D}_x\left(X_t\right)& ={\left[\frac{{\sigma }^2}{2\kappa }+\frac{\lambda }{\kappa }\left(\frac{p}{{\gamma }_1^2}+\frac{1-p}{{\gamma }_2^2}\right)\right]}^{1/2}.\end{array}$ Using empirical results, we set the benchmark parameters subject to that the asymptotic mean and standard derivation equal to 0 and 0.3, respectively.

5 This point will be explained in detail in next section.

Additional information

Funding

The research of Wu and Zang is supported by the Key Laboratory of Mathematical Economics and Quantitative Finance (Ministry of Education) at Peking University.