How do your solve for trader's optimal demand in market similar to Kyle's model?

Question

Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a standard probability space and $Z_t=(Z_t^1,Z_t^2)$ is a two dimensional Brownian motion with the filtration $\mathcal{F}^Z_{t}$ and $Z_t^1$, $Z_t^2$ are correlated with $\rho\in(-1,1)$. Let $\mathcal{E}_t$ the endowment of a trader with the following dynamics

$$d\mathcal{E}_t=\mu_1 dt + \sigma_1 dZ^1_t$$

The utility payoff of the investor is concave and increasing with respect to her demand $\alpha_t$ at any $t\in [0, T]$ and to keep simple the problem we assume that $\forall t\in [0,T]$ the trader does not have any budget constraint and can reallocate her demand with respect to a risky asset $V$, that is used to hedge her endowment $\mathcal{E}_t$, with the following dynamics

$$dV_t = \mu_2 dt + \sigma_2 dZ^2_t $$

the position of the trader is

$$W_t = \mathcal{E}_t+\alpha_t (P_t - V_t)$$

where $P_t$ stands for the price of the risky asset $V$ on date $t$ when the demand of the trader will be $\alpha_t$, and $P_t$ is $F_t^Z$-adapted. The expected utility payoff will be

$$\mathbb{E}(U(\mathcal{E}_{t}+\alpha_{t} (P_{t} - V_{t}))|F_0^Z)$$

At any $t\in [0,T]$ there exist also noise traders who who have random, price-inelastic demands and we denote by $B_t$ their cumulative orders at any $t$ and let's assume that $B$ is also an $\mathcal{F}^Z_{t}$-adapted Brownian motion that is independent of $Z=(Z^1,Z^2)$. Hence the total order is

$$Y_t = B_t + \alpha_t$$

where $\alpha$ is the demand of the trader and the price dynamics are

$$P_{t} = \mathbb{E}(V_{t}|Y_{t})$$

where (I believe that) the price dynamics resemble the market efficiency hypothesis as in Kyle 1985. Note here that the trader knows $P_t$ when she trades and that the market maker cannot designate the order $\alpha$ from $B$ and hence she can not know the exactly know the order of the trader.

How could someone solve for the optimal demand $\alpha_t^*$ such that

$$\alpha^*=\operatorname{argmax}\{\mathbb{E}(U(\mathcal{E}_{T}+\alpha_{T} (P_{T} - V_{T}))|F_0^Z)\}$$ when $(t,\alpha_t,V_t) \in \mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}$?

$\mathbb{Remark:}$ In contrast to the classic model of Kyle, there is an endowment $\mathcal{E}$ that refers to the hedging needs of the trader as well. More precisely, the trader will give an order $a^*$ not only because of her informational advantage (which she wants to exploit) on the dynamics of $V$, but also based on her hedging needs, which would make more difficult for the market maker to elicit the (private) information that the trader holds about $V$ at any $t\in[0,T]$ even in the case where no noise traders exist.

Answer 1

Generic knowledge about this kind of models

Let me try to get your model close to elements that are known:

1. Time continuous Kyle's model is something that is solved in Çetin, Umut, and Albina Danilova. "Markovian Nash equilibrium in financial markets with asymmetric information and related forward–backward systems." (2016): 1996-2029.
   Such a model shares a lot of your features:

   • the informed trader observes the value of an asset $\cal E$
   • she can trade it to optimise her PnL at $T$
   • in front of a Market Maker (MM) who does not see the true value of $\cal E$ but observes the flow $\alpha dt$ (same notation as yours!) of the informed trader, the is mixed with noise trades' demand
   • hence the MM "see" a diffusion with a controlled drift: $dY = \alpha dt + \sigma dB$.
   • the MM has a CARA utility function (that is more general than your conditional expectation)
   • the difference is that the informed trader does not have another instrument $V$ to trade, but this is already complicated
   • something that you fail to properly define in your model is how the market maker and the informed trader share (or not) information; in their paper, Cetin and Dsnilova have to use a Brownian Bridge.
     If you want. to do something like them, they have a great book on this topic: Çetin, Umut, and Albina Danilova. Dynamic Markov Bridges and Market Microstructure: Theory and Applications. Vol. 90. Springer, 2018.

2. Optimal execution with hedging is another story and focuses on simultaneously trading $\cal E$ and $V$ to hedge some market risk on the most liquid instrument ($V$ for you). There is this paper that shown how to do it on a portfolio: Lehalle, Charles-Albert. "Rigorous strategic trading: Balanced portfolio and mean-reversion." The Journal of Trading 4, no. 3 (2009): 40-46.

   • it deals with a full portfolio: the investor trades all thehe components of the portfolio that are correlated and do not have all the same liquidity
   • but it is done inside an "Almgren-Chriss" framework, that is discrete and not really well adapted to yours (or Cetin-Danilova's one)

My advice would be to use a Cartea-Jaimungal framework and write your optimal trading with an instrument properly. It is not very difficult (I gave it at one of my exams, hence I prefer not to write the full solution there...), but mixing it with Cetin and Danilova may be tricky....

Start of analysis: the one period model

Let me help you in the context of a one period model.

The mechanism of the proof should be this one

1. you need to choose a pricing function for your Market Maker (MM), let me use the notation $f_\theta(\alpha)$, where $\theta$ are the parameters of the pricing function.
   If you want to get inspiration you can have a look at Lehalle, Charles-Albert, Eyal Neuman, and Segev Shlomov. "Phase Transitions in Kyle's Model with Market Maker Profit Incentives." arXiv preprint arXiv:2103.04481 (2021) where a neural network is used (and theoretical results provided).
2. let me replace the generic CARA function by the cash account of the informed trader, she want to maximise $$\mathbb{E}((Q_0-\alpha){\cal E} + \alpha (f_\theta(\alpha)-V)).$$ I let you check: she can liquidate her position (of size $Q_0-\alpha$) in the initial security and the value of the remaining position $\alpha$ is clear.
3. To maximise this it is enough to find $\alpha^*$ such that $$f_\theta(\alpha^*)+\alpha^*f_\theta'(\alpha^*)=V-{\cal E}.$$ This is nothing more that the derivative of the upper expression with respect to $\alpha$.
4. Now you have a relation between $\alpha^*$ and $(\theta,{\cal E})$ that is of primary importance in this kind of game (this is a kind of Stackelberg game, see Vasal, Deepanshu, and Randall Berry. "Master Equation for Discrete-Time Stackelberg Mean Field Games with a Single Leader." In 2022 IEEE 61st Conference on Decision and Control (CDC), pp. 5529-5535. IEEE, 2022.)
5. This relation has to be reinjected in the pricing model $P=\mathbb{E}(V|B+\alpha)$, giving birth to something like $$P=f_\theta(\alpha^*(\theta,{\cal E}))=\mathbb{E}(V|B+\alpha^*(\theta,{\cal E})).$$ This is the formula in $\theta$ of a regression of $V$ on $\alpha$.
   
6. usually you try to get there a regression of $\cal E$ on $\alpha$ and not of $V$ on it.

Conclusion and advice

I hope you understand that there is a lot of knowledge and literature on this topic. You should read more about it before attacking your problem: I am sure the reading will tell you how to modify your problem so that it reflects what you really want. I am not sure that at this stage it is really the case.