High frequency trading, liquidity, and execution cost

Abstract

We build a model under the framework of discrete optimization to explain how high frequency trading (HFT) can be applied to supply liquidity and reduce execution cost. We derive the analytical properties of our model in finding the optimal solution to minimize the overall execution cost of HFT. We show that the execution cost can be reduced after increasing trading frequency (i.e., the higher the trading frequency, the lower the execution cost) with a simulation study. In addition, we conduct an empirical investigation with tick level data from US equity market through January 2008 to October 2010 to verify our conclusion drawn from the simulation study. Based on the simulation and empirical results we collected, we show that the HFT can reduce the execution cost when supplying liquidity.

Appendix

1.1 A.1 Proof of Proposition 1

For the induction basis at time t _N =T we have

For the induction step for some t _n ∈{t _m+1,…,t _N−1} we get

(14)

To obtain the minimum, we differentiate Eq. (14) with respect to $x_{t_{n}}$

(15)

Setting $\frac{\partial J}{\partial x_{t_{n}}}\stackrel{!}{=}0$ for Eq. (15) to obtain the optimal choice

$$ x_{t_n}=oD_{t_n}+wX_{t_n}+uF_{t_n}, $$

(16)

where

Putting Eq. (16) into Eq. (14) we obtain the optimal value function given by Eq. (2) and find the coefficients given by Eqs. (4)–(10). This completes the induction for t _n ∈{t _m+1,…,t _N }. We are unsure about market reaction to the event and the following change of the parameters that describe the market. At t _m we face the following problem

(17)

where q _u , κ _u , a _u and ρ _u should indicate that the current value of q and the future value of κ, ρ, and a _u are unknown.

Because the event is modeled as a discrete random variable, we obtain

(18)

We use $a_{i}=e^{\mu_{i} \tau}$ and $v_{i}=e^{(2\mu_{i}+\sigma_{i}^{2})\times\tau }$ and define

$$ \frac{1}{\overline{q}}=\sum_{i=1}^{r} p_i \times\frac{1}{q_i}, \qquad \overline{a}=\sum _{i=1}^{r} p_i \times \bigl(a_i^{N+1-(m+1)}\bigr), \qquad \overline{\lambda}=\sum _{i=1}^{r} p_i \times{ \lambda}_i. $$

Combining Eqs. (17) and (18) with this definitions, we find that

(19)

We then obtain the solution that minimizes Eq. (19) is

(20)

with

Inserting Eq. (20) into Eq. (19), we find the optimal value function given by Eq. (2) and the coefficients given by Eqs. (4)–(10). For the induction step for some t _n ∈{t ₀,…,t _m−1} we get

(21)

To obtain the minimum we differentiate Eq. (21) with respect to $x_{t_{n}}$

(22)

Setting $\frac{\partial J}{\partial x_{t_{n}}}\stackrel{!}{=}0$ for Eq. (22) to obtain the optimal choice

$$ x_{t_n}=oD_{t_n}+wX_{t_n}+uF_{t_n}, $$

(23)

where

Putting Eq. (23) into Eq. (21) we obtain the optimal value function given by Eq. (2) and the coefficients given by Eqs. (4)–(10). This concludes the induction.