Insider trading with dynamic asset under market makers' partial observations

1.   Introduction

In recent years, the studying of financial micro-structures and characteristics for risky asset markets has become a hot topic. In a setting of semi-strong effective pricing rule, Kyle ^[1] first proposed an insider trading model of multi-stage auction with asymmetric information on a static risky asset and proved the existence and uniqueness of its linear sequential equilibrium, consisting of insider trading intensity and market liquidity. It showed that as the time step approaching zero, the equilibrium converges to a continuous-time limit version in which market liquidity is a time-independent constant and all information is incorporated in the market price. Later, Back ^[2] extended Kyle's model ^[1] to continuous-time version and also obtained a unique market equilibrium when the risky asset value follows more general distributions, where market liquidity is also independent of time. Collins-Dufresne and Fos ^[3] generalized Back's model by assuming that the liquidity trading volatility follows a general stochastic process and proved that market depth, market liquidity and price dynamics are characterized by a martingale, submartingale and stochastic Brown Bridge process, respectively. Yang, He and Huang ^[4] pointed out that even if noise traders have their own trading memories, the properties of equilibrium are similar to those of equilibrium in ^[3]. Real financial phenomena remind us that the value of risky asset often varies with time. Caldentey and Stacchetti ^[5] studied an insider trading model, where the risky asset value follows an arithmetic Brownian motion and the trade ends at a random time with life-time distributed. It indicates that in equilibrium both the market liquidity and the insider's value function are exponential functions with respect to time. In the market for defaultable claims, Campi, Çetin and Danilova ^[6] found that from the market's perspective, the information released by the insider while trading optimally makes the default time predictable in equilibrium. Furthermore, Ma, Sun and Zhou ^[7] considered the setting that the value of a risky is driven by a conditional mean-field Ornstein-Uhlenback-type dynamic and obtained a closed form of optimal trading intensity. There is much literature on continuous-time insider trading, see ^{[8,9,10,11,12,13,14,15,16]} and so on.

Note that all the above work on insider trading focuses on an insider with perfect information. However, insiders often acquire only partial information on the underlying risky asset. Back, Cao and Willard ^[17] took the lead in establishing a continuous-time insider trading model of imperfect competition among informed traders. Then, Back, Crotty and Li ^[18] investigated the case that an insider acquires partial information about a risky asset in a high or low probability way. Under the framework of Collins-Dufresne and Fos ^[3], Banerjee and Breon-Drish ^[19,20] assumed that the insider may need to pay a certain cost to acquire the dynamic information flow of asset and demonstrated that the market depth is a semimartingale. Recently, Han, Li, Ma and Kennedy ^[21] continued to explore insider trading behaviors that noise traders have their own memories of historical trading and found that in a transparent market, to prevent the private information from rapid information leakage, the insider should adopt a mixed insider trading strategy. Qiu and Zhou ^[22] solved the problem of insider trading that an insider possesses some memory about the on-going observation of the underlying asset. Moreover, Crane, Crotty and Umar ^[23] illustrated that hedge funds which acquire public information earn higher annualized abnormal returns than nonacquirers. In the study of a new economic benefit of common institutional ownership, Chen, Ma, Wu and Zhang ^[24] pointed out that there is a significant negative relation between common ownership and insider trading profitability. For a fuzzy model to consider the robust portfolio selection problem of an agent with limited attention, Ma and Li ^[25] established an explicit solution of robust optimal strategy, which shows that more attention leads to smaller variance in estimated return.

In fact, in real financial markets, market makers can also observe some information on the underlying risky asset. Nishide ^[26] added the correlation of liquidity trading and the public signal to the Back's model ^[2] and revealed that market liquidity in equilibrium is not necessarily monotonous with time and the public signal does not necessarily make the market more efficient. Zhou ^[27] showed that when market makers observe some partial observations on a risky asset at the very beginning, the market liquidity in equilibrium remains a constant and the partial observation makes more information on the risky asset to be incorporated into the market price, thus improving the informativeness of the market. Xiao and Zhou ^[28] verified this economic intuition further.

In this paper, we study an extension of Caldentey and Stacchetti's ^[5] continuous-time insider trading model, in which market makers can observe some on-going partial information on a dynamic risky asset. Since market makers observe not only the total market orders, but also some partial information about the dynamic asset, then when pricing they must consider impacts of the two kinds of information. Therefore, we introduce a linear Bayesian equilibrium, which consists of insider trading intensity, price pressure on market orders (market liquidity) and price pressure on asset observations. Then, we establish the existence and uniqueness of equilibrium for insider trading until a fixed time $T$ or a random time $\tau$ respectively. There are several findings in our study. First, when trading until at the fixed time $T$ price pressure on market orders is constant as in literature ^{[1,2,8,15,27]}, while it is an exponential function of time when trading until the random $\tau$ as in ^[5]. Second, at the end of the transaction in both cases, the insider's private information is completely released. Third, the more information observed by market makers, the smaller the weight that market makers give to price pressure on market orders, while the greater the weight to price pressure on asset observations and vice versa. Finally, the more information the market makers observe, the weaker the information advantage of the insider and the lower the expected profit earned by the insider as in ^[27]. These results show that the partial observations of market makers can prevent effectively the insider from profiteering by monopolizing information.

It is mentioned that our research is based on pure models, that is, only considering that all agents will process and respond to the information they have in the same way. However, empirically and theoretically it is not the case. For example, in a statistical method, Dang, Foerster, Li and Tang^[29] found that financial analysts with high ability can produce firmer specific information through more accurate forecasts, which can effectively reduce the critical information asymmetry between insiders and external investors. In the future research, we will check some famous insider trading events in history and link those to information and market conditions, especially when agents process their information in different ways.

This article consists of six sections. A model of continuous-time insider trading with partial observations is described in Section 2. In Section 3, some necessary conditions on market efficiency are given. The existence and uniqueness of linear Bayesian equilibria for insider trading until a fixed time or a random time is established in Section 4. Some comparative static analyses on properties of equilibria in some special markets are given in Section 5. Conclusions are drawn in Section 6.

2.   The model

We here consider an extension model of continuous-time insider trading in ^[5], which allows market makers to observe some partial information about a dynamic risky asset trading until either a deterministic time $T$ or a random time $\tau$. Since there could be many equilibria in a continuous-time insider trading market ^[2,15], then for simplification, only linear strategies for the insider and market makers are considered in our model. All of random variables or processes are assumed to be defined on a complete and filtered probability space $(\Omega, {\cal F}, \{{\cal F}_{t}\}_{t\geq0}, P)$ satisfying the usual condition ^[30].

In a financial market, there is a risky asset traded in continuous-time whose value $v_{t}$ evolves as

where $v_{0}$ is normally distributed as $N(0, \sigma_{v0}^{2})$, $\sigma_{vt}$ is a deterministic, differentiable and positive function, and $W_{t}^{v}$ is a standard Brownian motion ^[5,16]. There are three types of agents in the market:

(i) liquidity traders: who can not observe any information about the value of the risky asset and submit their trading volume $z_{t}$ randomly ^[8,10], which satisfies

where $\sigma_{zt}$ is a deterministic, differentiable and positive function and $W_{t}^{z}$ is a standard Brownian motion

(ii) an insider: who is risk-neutral and knows the realization of the risky asset value $v_{t}$ and the current price $p_t$ of the risky asset, then submits her/his trading volume $x_{t}$, evolving as in ^{[1,3,5,8,15,20]}

where $\beta$ is a trading strategy of the insider, called insider trading intensity, which is a deterministic, differentiable and positive function

(iii) market makers: who observe the aggregate trading volume

(but can not observe $x_{t}$ and $z_{t}$ separately, otherwise the market maker can infer the perfect information of risky asset through the trading volume submitted by the insider and there is no insider trading) and some partial information of the risky asset as

where $\varepsilon_{t} = \varepsilon_{0}+\int_0^t \sigma_{\varepsilon s}dW_{s}^{\varepsilon}$ with $\varepsilon_{0}$ normally distributed as $N(0, \sigma_{\varepsilon0}^{2})$, $\sigma_{\varepsilon t}$ is a deterministic, differentiable and positive function, $W_{t}^{\varepsilon}$ is a standard Brownian motion and set for the risky asset a market price $p_t$, whose dynamic is

where $p_0 = 0$, $\lambda_{1}$ and $\lambda_{2}$ are two deterministic, differentiable and positive functions, called price pressure on market orders and price pressure on asset observations respectively.

Here, $v_0$, $\varepsilon_{0}$, $\{W_{t}^{z}\}$, $\{W_{t}^{v}\}$ and $\{W_{t}^{\varepsilon}\}$ are supposed mutually independent. Denote the insider's information and market makers' information by ${\cal F}_{t}^{I} = \sigma\{p_{s}; \; 0\leq s\leq t\}\vee\sigma\{v_{s}; \; 0\leq s\leq t\}$ and ${\cal F}_{t}^{M} = \sigma\{y_{s}; \; 0\leq s\leq t\}\vee\sigma\{u_{s}; \; 0\leq s\leq t\}$, respectively. Then,

Case I: If the trading continues to a fixed time $T$, the insider's profit of self-financing ^[2] is

Case II: If the trading continues until a random time $\tau$, which is exponentially distributed with rate $\eta > 0$ and is independent of the history of transactions and prices, the insider's profit of self-financing ^[5] is

For well-postness in each of the two cases, given any insider trading intensity $\beta$ of the insider, any price pressure on market orders $\lambda_1$ and any price pressure on asset observations $\lambda_2$, there must be $E\int_{0}^{T}|\beta_{s}|(v_{s}-p_{s})^{2}ds < \infty$ in Case I and $E\int_{0}^{\infty}e^{-\eta s}|\beta_{s}|(v_{s}-p_{s})^{2}ds < \infty$ in Case II. $(\cal S, \cal P)$ denotes the choice space where $\cal S$ is the set of insider's strategies $\beta$ and $\cal P$ is the set of pricing rules $(\lambda_{1}, \lambda_{2})$.

Assume that market makers are all risk-neutral and have a Bertrand competition. Then, similar to those in ^[1,7,10], for any market trade volume $y_t$, the total profit of market makers should be $0$, that is,

or

which implies that the price $p_t$ satisfies semi-strong market efficiency.

Now a concept of linear Bayesian equilibrium in our model is given below.

Definition 2.1. A linear Bayesian equilibrium in the market is a triple $(\beta, (\lambda_{1}, \lambda_{2}))\in (\cal S, \cal P)$ such that for any time $t$,

(i) (maximization of profit) for the given $(\lambda_{1}, \lambda_{2})$, function $\beta$ in Case I (Case II) maximizes

(ii) (market efficiency) for the given $\beta$, $(\lambda_{1}, \lambda_{2})$ such that pricing $p$ satisfies

3.   Necessary conditions of market efficiency

Before to establish the existence of linear equilibrium, we first discuss some necessary conditions of market efficiency.

Proposition 3.1. Let a strategy profile $(\beta, (\lambda_{1}, \lambda_{2}))\in (\cal S, \cal P)$. If $(\lambda_{1}, \lambda_{2})$ such that the corresponding market price $p_t$ satisfies the market efficiency (2.11). Then,

where the residual information $\Sigma_{t} = E(v_{t}-p_{t})^{2}$ satisfies the following dynamic

with $\Sigma_0 = \frac{\sigma_{v0}^{2}\sigma_{\varepsilon0}^{2}}{\sigma_{v0}^{2}+\sigma_{\varepsilon0}^{2}}$.

Proof. According to Eqs (2.1)–(2.6), for market makers there is a signal-observation system of $(v_{t}$, $\xi_{t})$ satisfying

where

$v_{0}\sim N(0, \sigma_{v0}^{2})$, $\xi_{t} = \begin{pmatrix} y_{t} \\ u_{t} \end{pmatrix}$, $\xi_{0} = \begin{pmatrix} 0 \\ v_{0}+\varepsilon_{0} \end{pmatrix}$, $b_{1} = \begin{pmatrix} \sigma_{v} \end{pmatrix}$, $A_{0} = \begin{pmatrix} -p_t\beta_{t} \\ 0 \end{pmatrix}$, $A_{1} = \begin{pmatrix} \beta_{t} \\ 0 \end{pmatrix}$, $B_{1} = \begin{pmatrix} 0\\ \sigma_{vt} \end{pmatrix}$, $B_{2} = \begin{pmatrix} \sigma_{zt} & 0 \\ 0 & \sigma_{\varepsilon t} \end{pmatrix}$, $W_{1t} = \begin{pmatrix} W_{t}^{v} \end{pmatrix}$, and $W_{2t} = \begin{pmatrix} W_{t}^{z}\\ W_{t}^{\varepsilon} \end{pmatrix}$.

Denote $B\circ B = B_{1}B_{1}^{\ast}+B_{2}B_{2}^{\ast}$, $b \circ B = b_{1}B_{1}^{\ast}+b_{2}B_{2}^{\ast}$ and $b\circ b = b_{1}b_{1}^{\ast}+b_{2}b_{2}^{\ast}$. Then,

Since $p_t$ satisfies the market efficiency (2.9), that is,

Then, Theorem 12.7 in ^[31] (or see Lemma 3.3 in ^[27]) tells us that

where the residual information $\Sigma_{t}$ satisfies

with $\Sigma_0 = \frac{\sigma_{v0}^{2}\sigma_{\varepsilon0}^{2}}{\sigma_{v0}^{2}+\sigma_{\varepsilon0}^{2}}$ by Theorem 13.1 in ^[31].

Since $(\lambda_{1}, \lambda_{2})$ satisfies the market efficiency (2.11), that is,

the results follow from (3.4) and (3.5), and the proof is complete.

4.   Existence and uniqueness of linear Bayesian equilibrium

4.1.   Case I: When the asset value is released at the fixed time $T$

Let the pricing profile $(\lambda_1, \lambda_2)\in {\cal P}$ be given. Then, for any $(t, m)\in [0, T)\times R$, for any $\beta^{\prime}\in {\cal S}[t, T) = \{\beta^\prime : \beta^\prime(r) = \beta(r), t\leq r < T, \beta\in {\cal S}\}$, there is a gap process

which by Eqs (2.1)–(2.6), satisfies the stochastic differential equation

with $m_t = m$. Then, we have the conditional value function

Clearly this is a classical stochastic control problem, and by employing Proposition 3.5 of dynamic programming principle in ^[32], we can easily get Hamilton-Jacobi-Bellman equation below with its proof omitted.

Proposition 4.1. The Hamilton-Jacobi-Bellman equation of insider's value function (4.2) (if it exists in $C^{1, 2}([0, T)\times R)$) is driven by

Now the existence and uniqueness of linear Bayesian equilibrium can be given below.

Theorem 4.2. Let for any time $t\in [0, T)$

Then, if the following inequality holds for any time $t > 0$,

there is a unique linear Bayesian equilibrium $(\beta, (\lambda_1, \lambda_2))\in (\cal { P, S})$ satisfying

with

Also, the residual information $\Sigma_t$ satisfies

the value function is

and the expected total profit of insider is

Proof. The proof is broken down into three steps:

Step I: Find a solution $J_{1}$ to the Hamilton-Jacobi-Bellman equation (4.3).

According to the Hamilton-Jacobi-Bellman equation (4.3), the following system follows

By the first equation in the above system, we have

So, the second equation in (4.8) implies that

Then, by differentiating with respect to $m$,

So,

which reveals that $\lambda_{1t}$ is a constant, denoted by $\lambda^*$, that is, $\lambda_{1t}\equiv \lambda^*$.

Equation (4.9) shows that

for some determinate, continuous, differentiable function $g(t)$. Now plugging Eq (4.11) into Eq (4.10), we have

Since $\frac{d}{dt}(\frac{1}{\lambda_{1t}}) = 0$, then (4.12) degenerates to

Thus,

So,

which can be expressed as (4.7).

Step II: Find a necessary condition for the optimal insider trading intensity:

which means that

and

In fact, for any $\beta^{\prime}\in\cal S$, since stochastic process $m_{t}$ follows (4.1), then using Itô formula to $J_{1}(t, m_{t})$, we observe that

According to the properties of the value function $J_{1}$ in Step I,

and

Then, taking conditional expectation on both sides of the above equation, we see that

that is,

Note that $J_{1}(T, m_{T})\geq0$, which implies that

and the equality in (4.17) holds if and only if $E[J_{1}(T, m_{T})] = 0$.,

Indeed

which tells that $\lim_{t\rightarrow T}E[J(t, m_{t})] = 0$ if and only if $\lim_{t\rightarrow T}\Sigma_t = 0.$

On the other hand, by market efficiency (3.2) and the inequality condition (4.4), we get

So, using the requirement $\lim_{t\rightarrow T}\Sigma_t = 0$, we observe that

From this it follows that (4.19) satisfies the second equation in (4.6). Plugging (4.20) into (4.18) yields

Step III: By Proposition 3.1, taking $(\beta, (\lambda_1, \lambda_2))$ as the forms in (4.5), respectively. Thus, it is a market equilibrium:

(i) Given the pricing rule with $(\lambda_1, \lambda_2)$, by computing directly, the insider trading intensity $\beta$ is such that

that is, $\beta\in {\cal S}$ is optimal.

(ii) Given the insider trading intensity $\beta$, the local linear pricing rule $p_t$ with dynamics $dp_t = \lambda_1(t)dy_t+\lambda_2(t)du_t$ must satisfy the market efficiency $p_{t} = E[v_{t}|{\cal F}_{t}^{M}]$. In fact, there exists one signal-observation system in terms of $(v_{t}, \xi_{t})$ following

where

$v_{0}\sim N(0, \sigma_{v0}^{2})$, $\xi_{t} = \begin{pmatrix} y_{t} \\ u_{t} \end{pmatrix}$, $\xi_{0} = \begin{pmatrix} 0 \\ v_{0}+\varepsilon_{0} \end{pmatrix}$, $b_{1} = \begin{pmatrix} \sigma_{v} \end{pmatrix}$, $A_{0} = \begin{pmatrix} -p_t\beta_{t} \\ 0 \end{pmatrix}$, $A_{1} = \begin{pmatrix} \beta_{t} \\ 0 \end{pmatrix}$, $B_{1} = \begin{pmatrix} 0\\ \sigma_{vt} \end{pmatrix}$, $B_{2} = \begin{pmatrix} \sigma_{zt} & 0 \\ 0 & \sigma_{\varepsilon t} \end{pmatrix}$, $W_{1t} = \begin{pmatrix} W_{t}^{v} \end{pmatrix}$ and $W_{2t} = \begin{pmatrix} W_{t}^{z}\\ W_{t}^{\varepsilon} \end{pmatrix}$.

Let

with $\widetilde{p_0} = 0.$ Then, applying Theorem 12.7 in ^[31] (or see Lemma 3.3 in ^[27]) again, we know that

Now taking Eq (4.22) minus Eq (3.4) gives

with $\widetilde{p_0}-p_0 = 0$, which leads to $\widetilde{p_t}-p_t = 0$ a.s., that is $\widetilde{p_t} = p_t$ a.s. The proof is complete.

4.2.   Case II: When the asset value is released at a random time $\tau$

In this subsection, we further consider the case when the risky asset is traded until a random time $\tau$, life-time distributed with parameter $\eta > 0$.

Let the pricing profile $(\lambda_1, \lambda_2)\in {\cal P}$ be given. Then, for any $(t, m)\in [0, \infty)\times R$, for any $\beta^{\prime}\in {\cal S}[t, \infty) = \{\beta^\prime : \beta^\prime(r) = \beta(r), t\leq r < \infty, \beta\in {\cal S}\}$, the gap process $m_s = v_{s}-p_{s}, \ s\in[t, \infty)$ satisfies (4.1). Then, it follows from (2.8) that the conditional value function is

Again we can easily obtain the corresponding Hamilton-Jacobi-Bellman equation below.

Proposition 4.3. The Hamilton-Jacobi-Bellman equation of insider's value function (4.23) (if it exists in $C^{1, 2}([0, T)\times R)$) is driven by

Proof. By the value function (4.23), it can be written as

Let $\widetilde{J}(t, m) = e^{-\eta t}J_{2}(t, m)$. Then,

Hence, by employing Proposition 3.5 of dynamic programming principle in^[32], $\widetilde{J}(t, m)$ satisfies the following Hamilton-Jacobi-Bellman equation

It follows that

which directly leads to the Hamilton-Jacobi-Bellman equation (4.24). This proof is complete.

As in the previous subsection, the existence and uniqueness of linear Bayesian equilibrium can be given below.

Theorem 4.4. Let for any time $t\geq 0$,

Then, if the following inequalities hold

there is a unique linear Bayesian equilibrium $(\beta, (\lambda_1, \lambda_2))\in (\cal { P, S})$ satisfying

with

and

The residual information $\Sigma_t$ satisfies

the value function is

and the expected total profit of insider is

Proof. Similar to the proof of Theorem 4.2. The proof is divided into three steps:

Step I: Find a solution $J_{2}$ to the Hamilton-Jacobi-Bellman equation (4.24).

The Hamilton-Jacobi-Bellman equation (4.24) states that

By the first equation in (4.29), we have

where $\alpha_{t}$ and $\delta_{t}$ are two determinate, continuous, differentiable and positive functions with $\alpha_{t} = \frac{1}{2\lambda_{1t}}$.

Accordingly, it follows from the second equation in (4.29) that

So, similar to the proof as in ^[5], we can prove that

for some constant $\lambda_{1}^* > 0$ and that

that is,

Then, the value function $J_{2}(t, m_{t})$ can be written as (4.27).

Step II: Find a necessary condition for the optimal trading intensity:

which implies that

and

Let $\widetilde{J}(t, m) = e^{-\eta t}J_{2}(t, m)$. Then, from the value function (4.23), we see that

Indeed, for any $\beta^{\prime}\in\cal S$, by the Eq (4.1), applying Itô formula to $\widetilde{J}(t, m_{t})$ yields

that is,

Note that,

So,

which follows that

Multiplying both sides of the above equation by $e^{-\eta t}$ gives

Hence,

for any $\beta^{\prime}\in{\cal S}$. Since $\lim_{t\rightarrow \infty}e^{-\eta t}J_{2}(t, m_{t}(\beta^{\prime}))\geq0$, for any $t\in[0, \infty)$, then

and the above equality hold if and only if $\lim_{t\rightarrow \infty}e^{-\eta t}J_{2}(t, m_{t}(\beta^{\prime})) = 0$.

Since

then $\lim_{t\rightarrow \infty}e^{-\eta t}E[J_{2}(t, m_{t}(\beta))] = 0$ if and only if $\lim_{t\rightarrow \infty}\Sigma_{t} = 0.$

According to market efficiency (3.2), it shows that

Together with $\lim_{t\rightarrow \infty}\Sigma_{t} = 0$, we get

Thus, (4.2) holds. Taking $\lambda_{1}^*$ back into (4.33) deduces $E(J_2(0, m_0)) = \sqrt{{\Upsilon_{z0}}({\Sigma_0+\Upsilon_{v\varepsilon0}})}.$

Step III: From Proposition 3.1, taking $(\beta, (\lambda_1, \lambda_2)$ as the forms in (4.25) respectively, we now verify that it is a market equilibrium:

(i) For the given pricing rule with $(\lambda_1, \lambda_2)$, the insider trading intensity $\beta$ satisfies

which means that $\beta$ in ${\cal S}$ is optimal.

(ii) For the given insider trading intensity $\beta$, the local linear pricing $p_t$ with $(\lambda_1, \lambda_2)$ must satisfy the market efficiency by repeating the procedure of (ii) in Step 3 of the proof for Theorem 4.2. This proof is complete.

5.   Comparative statics

In this section we will investigate some influences of market makers' partial observation on insider trading in our model. Note that from Theorems 4.2 or 4.5, the corresponding price pressures and the insider's expected total profit depend on the volatility function $\sigma^{2}_{\varepsilon t}$ of market makers' observation intricately. To explain the economic implication better, we only consider some special cases, especially when the volatility function $\sigma^{2}_{\varepsilon t}$ keeps constant.

5.1.   In Case I.

Now recalling from Theorem 4.2, we can obtain the following proposition.

Proposition 5.1. Let $(\beta, (\lambda_{1}, \lambda_{2}))\in (\cal S, \cal P)$ be the linear Bayesian equilibrium in Case I

Then

In particular,

(i) if $\sigma^{2}_{\varepsilon0}\rightarrow \infty$, $\sigma^{2}_{\varepsilon}\rightarrow \infty$, then

(ii) if $\sigma^{2}_{\varepsilon0}\rightarrow0$, $\sigma^{2}_{\varepsilon}\rightarrow0$, then

(iii) $E(J_{1}(\sigma_{\varepsilon0}^{2}, \sigma_{\varepsilon}^{2})) < E(J_{1}(\infty, \infty))$.

Proof. Let $\sigma_{vt}$, $\sigma_{zt}$ and $\sigma_{\varepsilon t}$ be the corresponding constants in the assumption. Then, by Theorem 4.2, it follows that

where the residual information $\Sigma_{t}$ and the expected total profit $E(J_{1}(\beta^{*}))$ satisfy respectively

Finally, it is easy to obtain our results above from Eqs (5.5) and (5.6). This proof is complete.

Beyond that, some propositions for several special cases when the value of risky asset is static, that is, $\sigma_{vt}\equiv 0$, will be listed below one by one:

1) In the case when market makers only observe the total market order $y_{t} = x_{t}+z_{t}$, which means $\sigma_{\varepsilon0}^{2}\rightarrow \infty$, it follows from Theorem 4.2 that

which are the same as those results in^[1,2,8,15].

2) In the case when market makers can observe two signals $y_{t} = x_{t}+z_{t}$ and $u_{t} = v_{0}+\varepsilon_{0}+\int_{0}^{t}\sigma_{\varepsilon t}dW_{t}^{\varepsilon}$ in the market, it is easy to see from Theorem 4.2 that

which are consistent with those results in ^[27].

5.2.   In Case II

Similarly, some properties about linear Bayesian equilibrium for special markets in Case II will be presented below with their proofs omitted.

Proposition 5.2. Let $(\beta, (\lambda_{1}, \lambda_{2}))\in (\cal S, \cal P)$ be the linear Bayesian equilibrium in Case II when

Then,

In particular,

(i) if $\sigma^{2}_{\varepsilon0}\rightarrow \infty$, $\sigma^{2}_{\varepsilon}\rightarrow \infty$, then

and

(ii) if $\sigma^{2}_{\varepsilon0}\rightarrow0$, $\sigma^{2}_{\varepsilon}\rightarrow0$, then

(iii) $E(J_{2}(\sigma_{\varepsilon0}^{2}, \sigma_{\varepsilon}^{2})) < E(J_{2}(\infty, \infty)).$

We remark that our results (5.8), (5.9) and (5.10) in Proposition 5.2 show that the model in our setting degenerates into the case of Caldentey and Stacchetti ^[5].

As in the previous subsection, from Theorem 4.5 some propositions for two more special cases if the asset value is static, that is, $\sigma_{vt}\equiv 0$, are stated as follows:

1) When $\sigma_{\varepsilon t} = 0$ and $\sigma_{z t} = \sigma_z > 0$,

2) In the above case when market makers only observe the total market order $y_{t} = x_{t}+z_{t}$, which means $\sigma_{\varepsilon0}^{2}\rightarrow \infty$,

6.   Summary

Based on the Caldentey-Stacchetti's model ^[5], this article investigates a new insider trading model, in which all information about a dynamic risky asset is known to an insider, while some partial information are observed by market makers. By applying filtering theory and dynamic programming principle, we establish the existence and uniqueness of linear Bayesian equilibrium trading either until a fixed maturity time $T$ or a random time $\tau$, respectively, which consists of insider trading intensity, price pressure on market orders and price pressure on asset observations. It shows that in equilibrium, $\lim_{t\rightarrow T}\Sigma_t = 0$ which means that all information on the risky asset is incorporated in the market price (see Theorems 4.2 and 4.5). Our results cover some classical results in literature ^{[1,2,5,8,15,27]}.

To explain the economic implication better, we further study our insider trading model for some special settings especially when the volatility function $\sigma^{2}_{\varepsilon t}$ keeps constant. According to Proposition 5.1 for Case I or Proposition 5.2 for Case II, it shows that the larger the noise $\sigma_{\varepsilon}^{2}$, the greater the weight that market makers give to price pressure on market orders, but the smaller the weight to price pressure on asset observations such that the insider earns more profit. Particularly, when $\sigma_{\varepsilon0}^{2}\rightarrow \infty$ and $\sigma_{\varepsilon}^{2}\rightarrow \infty$ (which means market makers observe fewer information on the asset), the price pressure on asset observations tends to 0, which reveals that the weight that market makers give to asset observations in pricing tends to 0. In addition, the expected aggregate profit of the insider reaches the maximum. When $\sigma_{\varepsilon0}^{2}\rightarrow0$ and $\sigma_{\varepsilon}^{2}\rightarrow0$ (which means market makers observe almost of information on the asset), the price pressure on market orders tends to 0 and the price pressure on asset observations tends to 1, which reveals that the weight that market makers give to market orders in pricing tends to 0, the weight to asset observations tends to 1 and the insider can make no money. These results suggest that market makers can acquire information about the risky asset through a variety of channels to prevent the insider from monopolizing the market to seek excessive profit.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are supported partially by Guizhou QKZYD[2022]4055, Natural Science Research Project of Guizhou Provincial Department of Education (QJJ[2023]011), Chinese NSF Grant (11861025).

We would like to express our sincere gratitude to three anonymous referees for their careful reading of the manuscript and very insightful suggestions, which helped us to improve the quality of the paper significantly, and three funds for their funding.

Conflict of interest

All authors declare no conflict of interest that could affect the publication of this paper.