AHEAD: Ad hoc electronic auction design

Joffrey Derchu; Philippe Guillot; Thibaut Mastrolia; Mathieu Rosenbaum

doi:10.3934/fmf.2024004

Article Contents

2024, Volume 3, Issue 2: 163-213. Doi: 10.3934/fmf.2024004 open access

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AHEAD: Ad hoc electronic auction design

1.
École Polytechnique, CMAP, France
2.
Autorité des Marchés Financiers, France
3.
UC Berkeley, Industrial Engineering & Operations Research Department, USA

^*Corresponding author: Thibaut Mastrolia
^*Corresponding author: Thibaut Mastrolia

Received: February 2023

Revised: February 2024

Early access: March 13, 2024

Published online: March 13, 2024

This work benefits from the financial support of the ERC Grant 679836 Staqamof, the Chaires Analytics and Models for Regulation, Financial Risk, Deep finance and statistics, Machine learning and systematic methods, the ANR ReLISCoP ANR-21-CE40-0001. The authors are also thankful to Alexandra Givry, Iris Lucas and Eric Va for insightful discussions.

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Abstract

We introduce a new matching design for financial transactions in an electronic market. In this mechanism, called ad hoc electronic auction design (AHEAD), market participants can trade between themselves at a fixed price and trigger an auction when they are no longer satisfied with this fixed price. In this context, we prove that a Nash equilibrium is obtained between market participants. Furthermore, we are able to assess quantitatively the relevance of ad hoc auctions and to compare them with periodic auctions and continuous limit order books. We show that from the investors' viewpoint, the microstructure of the asset is usually significantly improved when using AHEAD.

Keywords:

Mathematics Subject Classification: Primary: 91G15, 91A15; Secondary: 49N70.

Citation:

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Figure 1. AHEAD model sketch.

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Figure 2. Simplified setting when $ P^*_0 = P+\frac{\delta}{2} $, for $ 0\leq t\leq 2 $.

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Figure 3. $ V^a $, with $ 8 $ time steps, for $ v^a = 1>v^b = 0.5 $. The x-axis gives $ \hat n $. There are 3 plots for each set of parameters: blue (no continuous trading), orange (continuous trading allowed, $ P^*_0 = P-\frac{\delta}{2} $) and green (continuous trading allowed, $ P^*_0 = P+\frac{\delta}{2} $).

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Figure 4. $ V^a $, with $ 8 $ time steps, for $ v^a = 0.5<v^b = 1 $. The x-axis gives $ \hat n $. There are 3 plots for each set of parameters: blue (no continuous trading), orange (continuous trading allowed, $ P^*_0 = P-\frac{\delta}{2} $) and green (continuous trading allowed, $ P^*_0 = P+\frac{\delta}{2} $).

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Figure 5. $ V^a $, with $ 8 $ time steps, for $ v^a = v^b = 1 $. The x-axis gives $ \hat n $. There are 3 plots for each set of parameters: blue (no continuous trading), orange (continuous trading allowed, $ P^*_0 = P-\frac{\delta}{2} $) and green (continuous trading allowed, $ P^*_0 = P+\frac{\delta}{2} $).

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Figure 6. Values of the game and average duration of the continuous phase, at time $ t = 0 $, as functions of $ P-P^*_0 $, with $ v^a = v^b = 0.1 $.

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Figure 7. Values of the game and average duration of the continuous phase, at time $ t = 0 $, as functions of $ P-P^*_0 $, with $ v^a = 0.1 $ and $ v^b = 0.05 $.

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Figure 8. Values of the game and average duration of the continuous phase, at time $ t = 0 $, as functions of $ P-P^*_0 $, with $ v^a = 0.1 $ and $ v^b = 0.15 $.

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Figure 9. On the left, $ \xi^a_\tau $ as a function of $ N^a_\tau-v^a\tau $, for different values of $ N^a_+ $, with $ N^b_+ = 0 $, $ N^b_\tau-v^b\tau = 0 $ and $ h = 30s $. On the right, $ \xi^a_\tau $ as a function of $ N^b_\tau-v^b\tau $, for different values of $ N^a_+ $, with $ N^b_+ = 0 $, $ N^a_\tau-v^a\tau = 0 $ and $ h = 30s $, $ q = 0.1 $.

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Figure 10. $ \xi^a_\tau $ as a function of $ h $, with $ N^a_\tau-v^a\tau = 0 $, $ N^b_\tau-v^b\tau = 0 $. On the left, we fix $ N^b_+ = 0 $ and consider multiple values of $ N^a_+ $. On the right, we fix $ N^a_+ = 0 $ and consider multiple values of $ N^b_+ $.

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Table 1. Cost/gain for Player $ a $/$ b $ depending on whether Player $ a $/$ b $ stops or not the discrete game played at time $ k\delta $.

a$ \backslash $ b	stops	continues
stops	$ (\frac{L^a_{k\delta}+g^{\text{sim}}_a(k\delta,N^a_{k\delta},N^b_{k\delta})}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{sim}}_b(k\delta,N^a_{k\delta},N^b_{k\delta})}{k\delta+h}) $	$ (\frac{L^a_{k\delta}+g^{\text{first}}_a(k\delta,N^a_{k\delta},N^b_{k\delta})}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{second}}_b(k\delta,N^a_{k\delta},N^b_{k\delta})}{k\delta+h}) $
continues	$ (\frac{L^a_{k\delta}+g^{\text{second}}_a(k\delta,N^a_{k\delta},N^b_{k\delta})}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{first}}_b(k\delta,N^a_{k\delta},N^b_{k\delta})}{k\delta+h}) $	$ (\mathbb{E}^{\lambda^{a,}_k,\lambda^{b,}_k}_{k\delta}\big[U^a_{k+1}\big],\mathbb{E}^{\lambda^{a,}_k,\lambda^{b,}_k}_{k\delta}\big[U^b_{k+1}\big]) $

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Table 2. Cost/gain for Player $ a $/$ b $ depending on whether Player $ a $/$ b $ stops or not the simplified game played at time $ k\Delta T $.

a$ \backslash $ b	stops	continues
stops	$ (\frac{L^a_{k\Delta T}+g^{\text{sim}}_a}{k\Delta T+h},\frac{-L^b_{k\Delta T}+g^{\text{sim}}_b}{k\Delta T+h}) $	$ (\frac{L^a_{k\Delta T}+g^{\text{first}}_a}{k\Delta T+h},\frac{-L^b_{k\Delta T}+g^{\text{second}}_b}{k\Delta T+h}) $
continues	$ (\frac{L^a_{k\Delta T}+g^{\text{second}}_a}{k\Delta T+h},\frac{-L^b_{k\Delta T}+g^{\text{first}}_b(k\Delta T,N^a_{k\Delta T},N^b_{k\Delta T})}{k\Delta T+h}) $	$ \hat V^a,\hat V^b $

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Table 3. Mean of squared imbalance in the auction $ \mathbb E^{\lambda^{a,*},\lambda^{b,*}}[(P^*_{\tau+h}-P^{auc})^2] $, with $ 8 $ time steps, for $ v^a = 1>v^b = 0.5 $, in three cases: no continuous trading, continuous trading allowed with $ P^*_0 = P-\frac{\delta}{2} $, continuous trading allowed with $ P^*_0 = P+\frac{\delta}{2} $. We take $ q = 1 $ and $ \delta = 0.5 $. Those results are stable with $ \delta $ and $ q>0.5 $.

$\hat n$	0	1	2	3	4
no continuous trading	1	1	1	1	4
continuous trading allowed, $P^*_0=P-\frac{\delta}{2}$	1	1	0.63	0.71	1.16
continuous trading allowed, $P^*_0=P+\frac{\delta}{2}$	1	1	0.72	1.74	2.49

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Table 4. $ V^a $ and average duration of the continuous trading phase for different values of $ v^a $ and $ v^b $ with $ q = 0.01 $.

	$V^a$ (1e-6)				Average duration
Market design	$h = 20$, $\hat{n}=3$		$h = 20$, $\hat{n}=1$	CLOB	$h = 20$, $\hat{n}=3$		$h = 20$, $\hat{n}=1$	CLOB
continuous trading allowed	Yes	No	No	-	Yes	No	No	-
$v^a=0.1$, $v^b=0.1$	3685.6	384.6	0.0	10000.0	21.3s	6.0s	0.0s	10.0s
$v^a=0.05$, $v^b=0.1$	392.8	-6666.7	-5000.0	5000.0	33.3s	10.0s	0.0s	20.0s
$v^a=0.1$, $v^b=0.05$	7800.0	11606.7	10000.0	10000.0	33.3s	10.0s	0.0s	10.0s
$v^a=0.15$, $v^b=0.1$	9397.9	8680.0	10000.0	15000.0	9.0s	5.0s	0.0s	6.7s
$v^a=0.1$, $v^b=0.15$	-2841.8	0.0	0.0	10000.0	9.0s	5.0s	0.0s	10s

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DownLoad: CSV

Table 5. Cost/gain for Player $ a $/$ b $ depending on whether Player $ a $/$ b $ stops or not the discrete game played at time $ k\delta $.

a$ \backslash $b	stops	continues
stops	$ (\frac{L^a_{k\delta}+g^{\text{first}}_a}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{first}}_b}{k\delta+h}) $	$ (\frac{L^a_{k\delta}+g^{\text{first}}_a}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{first}}_b}{k\delta+h}) $
continues	$ (\frac{L^a_{k\delta}+g^{\text{first}}_a}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{first}}_b}{k\delta+h}) $	$ (\mathbb{E}^{\lambda^{a,}_k,\lambda^{b,}_k}_{k\delta}\big[U^a_{k+1}\big],\mathbb{E}^{\lambda^{a,}_k,\lambda^{b,}_k}_{k\delta}\big[U^b_{k+1}\big]) $

| Show Table

DownLoad: CSV

Table 6. Cost/gain for Player $ a $/$ b $ depending on whether Player $ a $/$ b $ stops or not the discrete game played at time $ k\delta $.

a$ \backslash $b	stops	continues
stops	$ (\frac{L^a_{k\delta}+g^{\text{sim}}_a}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{sim}}_b}{k\delta+h}) $	$ (\frac{L^a_{k\delta}+g^{\text{first}}_a}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{second}}_b}{k\delta+h}) $
continues	$ (\frac{L^a_{k\delta}+g^{\text{second}}_a}{k\delta+h},\frac{-L^b_{k\delta}+g^{\text{first}}_b}{k\delta+h}) $	$ (\mathbb{E}^{\lambda^{a,}_k,\lambda^{b,}_k}_{k\delta}\big[U^a_{k+1}\big],\mathbb{E}^{\lambda^{a,}_k,\lambda^{b,}_k}_{k\delta}\big[U^b_{k+1}\big]) $

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Algorithm 1: Computation of the value functions in the discretised game

Result: Value functions and probability of triggering an auction

Set

$ \begin{equation*} (U^a_{T/\delta},U^b_{T/\delta}) = (\frac{L^a_T+g^{\text{T}}_a(T,P^*_T,N^a_T,N^b_T)}{T+h}, \frac{-L^b_T+g^{\text{T}}_b(T,P^*_T,N^a_T,N^b_T)}{T+h}) \end{equation*} $

for $ k\in\{T/\delta-1,...,0\} $ do

end

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Algorithm 2: Computation of the value functions in the discretised game

Result: Value functions and probability of triggering an auction

Set

$ \begin{equation*} (U^a_{T/\delta},U^b_{T/\delta}) = (\frac{L^a_T+g^{\text{T}}_a(T,P^*_T,N^a_T,N^b_T)}{T+h}, \frac{-L^b_T+g^{\text{T}}_b(T,P^*_T,N^a_T,N^b_T)}{T+h}) \end{equation*}$

for $ k\in\{T/\delta-1,...,0\} $ do

end

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Algorithm 3: Computation of the average duration of the continuous phase in the discretised game

Result: Average duration of the continuous phase

Set

$ \begin{equation*} E_{T/\delta} = 0 \end{equation*} $

for $ k\in\{T/\delta-1,...,0\} $ do

end

| Show Table

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Table 7. $ V^a $ and average duration of the continuous trading phase for different values of $ v^a $ and $ v^b $ with $ q = 0.005 $.

	$V^a$ (1e-6)				Average duration
Market design	$h = 20$, $\hat{n}=3$		$h = 20$, $\hat{n}=1$	CLOB	$h = 20$, $\hat{n}=3$		$h = 20$, $\hat{n}=1$	CLOB
continuous trading allowed	Yes	No	No	-	Yes	No	No	-
$v^a=0.1$, $v^b=0.1$	16458.6	27424.3	12000.5	10000.0	40.9s	5.6s	0.0s	10.0s
$v^a=0.05$, $v^b=0.1$	6533.5	11581.1	2000.5	5000.0	56.9s	14.9s	0.0s	10.0s
$v^a=0.1$, $v^b=0.05$	14196.4	22874.1	12000.5	10000.0	56.9s	14.9s	0.0s	20.0s
$v^a=0.15$, $v^b=0.1$	27287.7	30051.5	31501.0	15000.0	27.1s	3.9s	0.0s	6.7s
$v^a=0.1$, $v^b=0.15$	16450.0	16309.9	9510.7	10000.0	27.1s	3.9s	0.0s	10.0s

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