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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.

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  • $\begingroup$ not sure I understand the question: shouldn't you specify the dynamics of $P_t$? at least this process has to be ${\cal F}^Z$-adapted, but can you tell more? $\endgroup$
    – lehalle
    Apr 1, 2023 at 17:04
  • $\begingroup$ @lehalle does this extra information help? $\endgroup$ Apr 1, 2023 at 19:29
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    $\begingroup$ (1) it starts to be more clear. First remark: I guess that your $a$ in $Y$ is $\alpha$. Another remark is that to be able to write $\mathbb{E}(V|Y)$ you need $V$ and $Y$ to share the same filtration: please add $B$ in $\cal F$. $\endgroup$
    – lehalle
    Apr 2, 2023 at 5:56
  • $\begingroup$ (2) Your model seems to be close to Kyle's ones: you need a way to bind $V$ to $Y$ so that your $P:=\mathbb{E}(V|Y)$ makes sense. In Kyle, the "informed investor" is informed: she or he knows $P$. Is it your case too? And the market makers does not observe $P$ and sets the market price (that would be for you $V$). $\endgroup$
    – lehalle
    Apr 2, 2023 at 6:07
  • $\begingroup$ (3) even if you answer to my questions (1) and (2) I am not sure to be able to help. What is exactly the situation you want to model? could it be simply be an optimal liquidation (by your trader) with market impact on the main asset $P$, and no market impact on $V$ (because it would de "infinitely more liquid") such that $V$ and $P$ are correlated? your trader would be able to use $V$ to hedge the risk on $P$, decreasing the market risk on $P$ and hence allowing to not trade too fast (and hence pay too much) on $P$? $\endgroup$
    – lehalle
    Apr 2, 2023 at 6:08

1 Answer 1

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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.

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  • $\begingroup$ Well, what I want to do is to have a market maker who is strategic, but to do so I need n first place to define and solve this problem in the right direction, and to avoid the noise traders because it is not a convenient assumption and the literature has not done so many things to drop this assumption. I know some papers in the static time who avoid using the assumption of noise traders, but nothing has done in continuous or dynamic time. The endowmend will help exaclty to avoid this assumption, because the market maker may elicit a noisy signal by the informed trader which will $\endgroup$ Apr 3, 2023 at 8:02
  • $\begingroup$ which will be like a blend of both the hedging needs and the private information of her, and this means that though the informed trader would know exactly the diffusion process of $V$, the market maker who will take the order she will not elicit $V$ BUT a noisy estimate from both $V$ and $\mathcal{E}$ even if she know exactly the stochastic order of the noise traders or even the latter did not exist. $\endgroup$ Apr 3, 2023 at 8:06
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    $\begingroup$ I did my best: my answer is not the max I can do with the information I have $\endgroup$
    – lehalle
    Apr 4, 2023 at 7:12
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    $\begingroup$ Thanks for all the help! It is very important to me! $\endgroup$ Apr 4, 2023 at 19:22
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    $\begingroup$ send me an email (my email is easy to find) if you want to follow up. $\endgroup$
    – lehalle
    Apr 5, 2023 at 10:25

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