<p>Not really an answer, but too long for comment. </p>

<p>The <span class="math-container">$P$</span> in your 
<span class="math-container">$$y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$$</span> 
expression from the insider's problem and the <span class="math-container">$P$</span> in the expression 
<span class="math-container">$$
\min_{P(\cdot)} \cdots
$$</span> 
from the market maker's problem should not be the same. That is <em>not</em> the definition of rational expectation equilibrium in this context.</p>

<p>The insider takes the pricing rule as given, per definition of rational expectation equilibrium. This means insider has optimal demand 
<span class="math-container">$$
y(u,v, P_1(\cdot))^*= \frac{v-P_1(y^*)}{P_1'(y^*)} +u
$$</span> for any pricing rule <span class="math-container">$P_1(\cdot)$</span>. Then market maker minimizes 
<span class="math-container">$$
E[ ( y(u,v, P_1(\cdot))^* - u) \cdot ( v - P_2(y(u,v, P_1(\cdot))^*) )  ]
$$</span> over <span class="math-container">$P_2$</span>---taking <span class="math-container">$P_1$</span> as fixed. It's a simultaneous game.
If the minimizing <span class="math-container">$P_2$</span> coincides with <span class="math-container">$P_1$</span>, then <span class="math-container">$P= P_1 = P_2$</span> is an equilibrium pricing rule. Doing this in the one-period Kyle setting recovers Kyle's lambda, whereas your calculation would not.</p>

<p>The optimization problem you're solving is therefore not the correct one.</p>

<p>The insider optimizes with respect to a <em>conjectured</em> pricing rule <span class="math-container">$P_1$</span>. In equilibrium, the market maker confirms the insider's conjecture, making the latter's expectation rational.</p>

<p>Your calculation assumes that the market maker acts after the insider. This is not the case in Kyle and I don't believe it's the case in Rochet and Vila either. </p>

CC BY-SA 4.0
2019-08-08T12:03:19.480Z
Also, the Euler-Lagrange equation should appear somewhere in a calculus of variations calculation.
Michael
CC BY-SA 4.0
2020-08-23T09:36:19.927Z
Do you know where can I find any details about defining some problem like this, using the calculus of variations? Do we need to introduce the time dimension in order to define an Euler-Lagrange differentiaal equation for our problem or we can do it for a static model also?
Nav89
CC BY-SA 4.0
2019-08-08T12:00:17.463Z
2019-08-08T23:51:35.207Z
2019-08-08T23:51:35.207Z
Michael
<p>The problem, I am trying to solve is based on the paper by Rochet and Vila 1994 (see literature below). In fact, it is a variant of the seminal paper of Kyle 1985 in the finance/economics literature. Despite the problem being framed in the finance literature, the mathematical problem associated with it is far more general and is based on calculus of variations. Let me briefly introduce the problem:</p>

<ul>
<li>Let <span class="math-container">$V \sim N(0, \sigma^2_V)$</span> denote the random asset value, <span class="math-container">$U\sim N(0, \sigma^2_U)$</span> the random liquidity trader's demand, <span class="math-container">$X$</span> the order by the strategic trader, and <span class="math-container">$Y=X+U$</span> the total order flow</li>
<li>The strategic trader observes the realization of the asset value <span class="math-container">$V=v$</span>, the exogenous liquidity trades <span class="math-container">$U=u$</span>, and submits an order <span class="math-container">$x(u,v)$</span> that maximizes his trading profits: <span class="math-container">$\pi(x,u,v)=x(v-P(y))$</span></li>
<li>The market maker observes total order flow <span class="math-container">$Y=y$</span> and sets the trading price equal to the conditional expected value of the asset, i.e. <span class="math-container">$P(y)=E[V|y]$</span>.</li>
<li>A market equilibrium is characterized by a trading strategy <span class="math-container">$x(u,v)$</span> and a pricing rule <span class="math-container">$P(y)$</span> such that: i) Given the pricing rule, for every <span class="math-container">$(u,v)$</span>, the trading strategy <span class="math-container">$x(u,v)$</span> maximizes the informed trader's profit <span class="math-container">$\pi$</span> and ii) given the trading strategy, the pricing rule <span class="math-container">$P(y)$</span> sets the price equal to the expected value of <span class="math-container">$V$</span> conditional on <span class="math-container">$Y=y$</span>.</li>
</ul>

<p>Now the authors show that this problem is equivalent to the following optimization problem <span class="math-container">$\min_{P(\cdot)} E [ \max_x (v-P(x+u))x]$</span>, that is, the problem can be solved by the calculus of variations. They state the answer to be <span class="math-container">$P(y)=\frac{\sigma_V}{\sigma_U}y$</span> which is true, once we "guess and verify" the solution. However, I want to find the solution from scratch, that is, without "guess and verify".</p>

<p>My problems arise as the function over which the functional of <span class="math-container">$P$</span> is defined is itself an optimal value. We know that the insider maximizes <span class="math-container">$x(v-P(x+u))$</span> or equivalently <span class="math-container">$(y-u)(v-P(y))$</span>. From the first-order condition, we can calculate the optimal demand <span class="math-container">$y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$</span> which results in an optimal profit of <span class="math-container">$\pi(y^*,u,v)=\frac{(v-P(y^*))^2}{P'(y^*)}$</span>. </p>

<p>Therefore, the market maker's optimization problem can be rewritten as <span class="math-container">$$\min_{P(\cdot)} E [ \max_x (v-P(x+u))x]= \int_{ \infty}^{\infty} \int_{-\infty}^{\infty} \pi(y^*,u,v) f_V(v) f_U(u) du dv = \int_{ \infty}^{\infty} \int_{-\infty}^{\infty} \frac{(v-P(y^*))^2}{P'(y^*)} f_V(v) f_U(u) du dv  $$</span> with <span class="math-container">$f_V$</span> and <span class="math-container">$f_U$</span> being the density functions of <span class="math-container">$V$</span> and <span class="math-container">$U$</span> respectively.</p>

<p>From the fact that <span class="math-container">$\pi^*$</span> is the maximum of the informed trader's objective function, we can use the envelope theorem to simplify: i) <span class="math-container">$\frac{d \pi(y^*)}{d u}=\frac{\partial \pi(y,u,v)}{\partial u}|_{y=y^*}=-(v-P(y^*))$</span> and  ii) <span class="math-container">$\frac{d \pi(y^*)}{d v}=(y^*-u)$</span>.</p>

<p>This is where I am stuck. My problem is the fact that the argument <span class="math-container">$y^*(u,v)$</span> of the function we are searching for <span class="math-container">$P(\cdot)$</span> is itself the solution of a maximization problem. Especially, I do not know how to calculate the integral with respect to <span class="math-container">$v$</span> and <span class="math-container">$u$</span>.</p>

<p>Any comments are highly appreciated. Thank you!</p>

<p>Literature:</p>

<ul>
<li>Rochet and Vila 1994, "Insider Trading wihtout Normality", The Review of Economic Studites, Vol. 61, No. 1, pp. 131-152</li>
<li>Kyle 1985, "Continuous Auctions and Insider Trading", Econometrica, Vol. 52, No. 6, pp 1315-1335</li>
</ul>

<finance><academic-graduate><optimal-control><trading>
Optimal price function: application of calculus of variations
Matthias 
# Optimal price function: application of calculus of variations

(Asked by: username_0 on 2019-08-07)

The problem, I am trying to solve is based on the paper by Rochet and Vila 1994 (see literature below). In fact, it is a variant of the seminal paper of Kyle 1985 in the finance/economics literature. Despite the problem being framed in the finance literature, the mathematical problem associated with it is far more general and is based on calculus of variations. Let me briefly introduce the problem:

* Let $V \sim N(0, \sigma^2\_V)$ denote the random asset value, $U\sim N(0, \sigma^2\_U)$ the random liquidity trader's demand, $X$ the order by the strategic trader, and $Y=X+U$ the total order flow
* The strategic trader observes the realization of the asset value $V=v$, the exogenous liquidity trades $U=u$, and submits an order $x(u,v)$ that maximizes his trading profits: $\pi(x,u,v)=x(v-P(y))$
* The market maker observes total order flow $Y=y$ and sets the trading price equal to the conditional expected value of the asset, i.e. $P(y)=E[V|y]$.
* A market equilibrium is characterized by a trading strategy $x(u,v)$ and a pricing rule $P(y)$ such that: i) Given the pricing rule, for every $(u,v)$, the trading strategy $x(u,v)$ maximizes the informed trader's profit $\pi$ and ii) given the trading strategy, the pricing rule $P(y)$ sets the price equal to the expected value of $V$ conditional on $Y=y$.

Now the authors show that this problem is equivalent to the following optimization problem $\min\_{P(\cdot)} E [ \max\_x (v-P(x+u))x]$, that is, the problem can be solved by the calculus of variations. They state the answer to be $P(y)=\frac{\sigma\_V}{\sigma\_U}y$ which is true, once we "guess and verify" the solution. However, I want to find the solution from scratch, that is, without "guess and verify".

My problems arise as the function over which the functional of $P$ is defined is itself an optimal value. We know that the insider maximizes $x(v-P(x+u))$ or equivalently $(y-u)(v-P(y))$. From the first-order condition, we can calculate the optimal demand $y(u,v)^\*= \frac{v-P(y^\*)}{P'(y^\*)} +u$ which results in an optimal profit of $\pi(y^\*,u,v)=\frac{(v-P(y^\*))^2}{P'(y^\*)}$.

Therefore, the market maker's optimization problem can be rewritten as $$\min\_{P(\cdot)} E [ \max\_x (v-P(x+u))x]= \int\_{ \infty}^{\infty} \int\_{-\infty}^{\infty} \pi(y^\*,u,v) f\_V(v) f\_U(u) du dv = \int\_{ \infty}^{\infty} \int\_{-\infty}^{\infty} \frac{(v-P(y^\*))^2}{P'(y^\*)} f\_V(v) f\_U(u) du dv $$ with $f\_V$ and $f\_U$ being the density functions of $V$ and $U$ respectively.

From the fact that $\pi^\*$ is the maximum of the informed trader's objective function, we can use the envelope theorem to simplify: i) $\frac{d \pi(y^\*)}{d u}=\frac{\partial \pi(y,u,v)}{\partial u}|\_{y=y^\*}=-(v-P(y^\*))$ and ii) $\frac{d \pi(y^\*)}{d v}=(y^\*-u)$.

This is where I am stuck. My problem is the fact that the argument $y^\*(u,v)$ of the function we are searching for $P(\cdot)$ is itself the solution of a maximization problem. Especially, I do not know how to calculate the integral with respect to $v$ and $u$.

Any comments are highly appreciated. Thank you!

Literature:

* Rochet and Vila 1994, "Insider Trading wihtout Normality", The Review of Economic Studites, Vol. 61, No. 1, pp. 131-152
* Kyle 1985, "Continuous Auctions and Insider Trading", Econometrica, Vol. 52, No. 6, pp 1315-1335

**Tags:** finance, academic-graduate, optimal-control, trading


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## 1 answers

### Answer by username_1 on 2019-08-08. Score: 4

Not really an answer, but too long for comment.

The $P$ in your
$$y(u,v)^\*= \frac{v-P(y^\*)}{P'(y^\*)} +u$$
expression from the insider's problem and the $P$ in the expression
$$
\min\_{P(\cdot)} \cdots
$$
from the market maker's problem should not be the same. That is *not* the definition of rational expectation equilibrium in this context.

The insider takes the pricing rule as given, per definition of rational expectation equilibrium. This means insider has optimal demand
$$
y(u,v, P\_1(\cdot))^\*= \frac{v-P\_1(y^\*)}{P\_1'(y^\*)} +u
$$ for any pricing rule $P\_1(\cdot)$. Then market maker minimizes
$$
E[ ( y(u,v, P\_1(\cdot))^\* - u) \cdot ( v - P\_2(y(u,v, P\_1(\cdot))^\*) ) ]
$$ over $P\_2$---taking $P\_1$ as fixed. It's a simultaneous game.
If the minimizing $P\_2$ coincides with $P\_1$, then $P= P\_1 = P\_2$ is an equilibrium pricing rule. Doing this in the one-period Kyle setting recovers Kyle's lambda, whereas your calculation would not.

The optimization problem you're solving is therefore not the correct one.

The insider optimizes with respect to a *conjectured* pricing rule $P\_1$. In equilibrium, the market maker confirms the insider's conjecture, making the latter's expectation rational.

Your calculation assumes that the market maker acts after the insider. This is not the case in Kyle and I don't believe it's the case in Rochet and Vila either.

> **Comments:**
> **username_1**: Also, the Euler-Lagrange equation should appear somewhere in a calculus of variations calculation.  
> **username_2**: Do you know where can I find any details about defining some problem like this, using the calculus of variations? Do we need to introduce the time dimension in order to define an Euler-Lagrange differentiaal equation for our problem or we can do it for a static model also?  

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