Closed-form approximations in multi-asset market making

We fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a filtration $(\mathcal{F}_{t})_{t\in{\mathbb{R}}_{+}}$ satisfying the usual conditions. In what follows, we assume that all stochastic processes are defined on $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in{\mathbb{R}}_{+}},\mathbb{P})$. In all this paper, $\mathbb{R}_{+}$ denotes the set of nonnegative real numbers, and $\mathbb{R}_{+}^{*}$ denotes the set of positive real numbers.

For $i\in\{1,\ldots,d\}$, the reference price of asset $i$ is modeled by a process $(S_{t}^{i})_{t\in\mathbb{R}_{+}}$ with dynamics

$$dS^{i}_{t}=\sigma^{i}dW^{i}_{t},\quad\mbox{$S_{0}^{i}$ given,}$$

where $(W_{t}^{1},\ldots,W_{t}^{d})_{t\in\mathbb{R}_{+}}$ is a $d$-dimensional Brownian motion with correlation matrix $(\rho^{i,j})_{1\leqslant i,j\leqslant d}$ adapted to the filtration $(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}}$ – hereafter we denote by $\Sigma=(\rho^{i,j}\sigma^{i}\sigma^{j})_{1\leqslant i,j\leqslant d}$ the variance-covariance matrix associated with the process $(S_{t})_{t\in\mathbb{R}_{+}}=(S_{t}^{1},\ldots,S_{t}^{d})_{t\in\mathbb{R}_{+}}$.

The market maker chooses at each point in time the price at which she is ready to buy/sell each asset: for $i\in\{1,\ldots,d\}$, we let her bid and ask quotes for asset $i$ be modeled by two stochastic processes, respectively denoted by $(S_{t}^{i,b})_{t\in\mathbb{R}_{+}}$ and $(S_{t}^{i,a})_{t\in\mathbb{R}_{+}}$.

For $i\in\{1,\ldots,d\}$, we denote by $(N_{t}^{i,b})_{t\in\mathbb{R}_{+}}$ and $(N_{t}^{i,a})_{t\in\mathbb{R}_{+}}$ the two point processes modeling the number of transactions at the bid and at the ask, respectively, for asset $i$. We assume in this section that the transaction size for asset $i$ is constant and denoted by $z^{i}$. The inventory process of the market maker for asset $i$, denoted by $(q^{i}_{t})_{t\in\mathbb{R}_{+}}$, has therefore the dynamics

$$dq_{t}^{i}=z^{i}dN_{t}^{i,b}-z^{i}dN_{t}^{i,a},\quad\mbox{$q_{0}^{i}$ given,}$$

and we denote by $(q_{t})_{t\in\mathbb{R}_{+}}$ the (column) vector process $\left(q^{1}_{t},\ldots,q^{d}_{t}\right)^{\intercal}_{t\in\mathbb{R}_{+}}$.

For each $i\in\{1,\ldots,d\}$, we denote by $(\lambda_{t}^{i,b})_{t\in\mathbb{R}_{+}}$ and $(\lambda_{t}^{i,a})_{t\in\mathbb{R}_{+}}$ the intensity processes of $(N^{i,b}_{t})_{t\in\mathbb{R}_{+}}$ and $(N^{i,a}_{t})_{t\in\mathbb{R}_{+}}$, respectively. We assume that the market maker stops proposing a bid (respectively ask) price for asset $i$ when her position in asset $i$ following the transaction would exceed a given threshold $Q^{i}$ (respectively $-Q^{i}$).⁴⁴4$Q^{i}$ is assumed to be a multiple of $z^{i}$. It corresponds to the risk limit of the market maker for asset $i$.

Formally, we assume that the intensities verify

$$\lambda^{i,b}_{t}=\Lambda^{i,b}(\delta_{t}^{i,b})\mathds{1}_{\{q^{i}_{t-}+z^{i}\leq Q^{i}\}}\quad\mbox{and}\quad\lambda^{i,a}_{t}=\Lambda^{i,a}(\delta_{t}^{i,a})\mathds{1}_{\{q^{i}_{t-}-z^{i}\geq-Q^{i}\}},$$

where the processes $(\delta^{i,b}_{t})_{t\in\mathbb{R}_{+}}$ and $(\delta^{i,a}_{t})_{t\in\mathbb{R}_{+}}$ are defined by⁵⁵5It is often assumed in the literature that the point processes are independent of the Brownian motions. In that case, the quote processes $(\delta^{i,b}_{t})_{t\in\mathbb{R}_{+}}$ and $(\delta^{i,a}_{t})_{t\in\mathbb{R}_{+}}$ have to be independent of prices. In fact, the optimal control problem can be written in a weak form to show that this assumption is not necessary – see A for more details on the construction of the processes in that case.

$$\delta_{t}^{i,b}=S_{t}^{i}-S_{t}^{i,b}\quad\mbox{and}\quad\delta_{t}^{i,a}=S_{t}^{i,a}-S_{t}^{i},\quad\forall t\in\mathbb{R}_{+}.$$

Moreover, we assume that the functions $\Lambda^{i,b}$ and $\Lambda^{i,a}$ satisfy the following properties:

• 1.

  $\Lambda^{i,b}$ and $\Lambda^{i,a}$ are twice continuously differentiable,

• 2.

  $\Lambda^{i,b}$ and $\Lambda^{i,a}$ are decreasing, with $\forall\delta\in{\mathbb{R}}$, ${\Lambda^{i,b}}^{\prime}(\delta)<0$ and ${\Lambda^{i,a}}^{\prime}(\delta)<0$,

• 3.

  $\lim_{\delta\to+\infty}\Lambda^{i,b}(\delta)=\lim_{\delta\to+\infty}\Lambda^{i,a}(\delta)=0$,

• 4.

  $\sup_{\delta}\frac{\Lambda^{i,b}(\delta){\Lambda^{i,b}}^{\prime\prime}(\delta)}{\left({\Lambda^{i,b}}^{\prime}(\delta)\right)^{2}}<2\quad\text{and}\quad\sup_{\delta}\frac{\Lambda^{i,a}(\delta){\Lambda^{i,a}}^{\prime\prime}(\delta)}{\left({\Lambda^{i,a}}^{\prime}(\delta)\right)^{2}}<2.$
  

Finally, the process $(X_{t})_{t\in\mathbb{R}_{+}}$ modelling the amount of cash on the market maker’s cash account has the following dynamics:

	$\displaystyle dX_{t}$	$\displaystyle=\sum\limits_{i=1}^{d}S_{t}^{i,a}z^{i}dN_{t}^{i,a}-S_{t}^{i,b}z^{i}dN_{t}^{i,b}$
		$\displaystyle=\sum\limits_{i=1}^{d}(S_{t}^{i}+\delta_{t}^{i,a})z^{i}dN_{t}^{i,a}-(S_{t}^{i}-\delta_{t}^{i,b})z^{i}dN_{t}^{i,b}$
		$\displaystyle=\sum\limits_{i=1}^{d}\left(\delta_{t}^{i,b}z^{i}dN_{t}^{i,b}+\delta_{t}^{i,a}z^{i}dN_{t}^{i,a}\right)-\sum\limits_{i=1}^{d}S^{i}_{t}dq^{i}_{t}.$

We can consider two different optimization problems for the market maker. Following the initial model proposed by Avellaneda and Stoikov in [1], we can assume that she maximizes the expected value of a CARA utility function (with risk aversion parameter $\gamma>0$) applied to the mark-to-market value of her portfolio at a given time $T$. This mark-to-market value is the sum of the amount $X_{T}$ on the cash account and the mark-to-market value $\sum\limits_{i=1}^{d}q_{T}^{i}S_{T}^{i}$ of the assets remaining in the portfolio at date $T$.⁶⁶6In the literature there is sometimes a penalty function applied to the inventory at terminal time $T$ to “force” liquidation. Here, as we shall focus on the asymptotic regime of the optimal quotes, there is no point considering such a penalty. However, it is noteworthy that most of our non-asymptotic results could be generalized to the case of a quadratic terminal penalty. More precisely, her optimization problem writes

$\displaystyle\sup_{\begin{subarray}{c}(\delta_{t}^{1,b})_{t},\ldots,(\delta_{t}^{d,b})_{t}\in\mathcal{A}\\ (\delta_{t}^{1,a})_{t},\ldots,(\delta_{t}^{d,a})_{t}\in\mathcal{A}\end{subarray}}\mathbb{E}\left[-\exp\left(-\gamma\left(X_{T}+\sum\limits_{i=1}^{d}q_{T}^{i}S_{T}^{i}\right)\right)\right],$

where ${\mathcal{A}}$ is the set of predictable processes bounded from below. We call Model A our model with this first objective function.

Alternatively, as proposed by Cartea et al. in [10], we can consider a risk-adjusted expectation for the objective function of the market maker. In that case, the optimization problem writes

$\displaystyle\sup_{\begin{subarray}{c}(\delta_{t}^{1,b})_{t},\ldots,(\delta_{t}^{d,b})_{t}\in\mathcal{A}\\ (\delta_{t}^{1,a})_{t},\ldots,(\delta_{t}^{d,a})_{t}\in\mathcal{A}\end{subarray}}\mathbb{E}\left[X_{T}+\sum_{i=1}^{d}q_{T}^{i}S_{T}^{i}-\frac{1}{2}\gamma\int_{0}^{T}q_{t}^{\intercal}\Sigma q_{t}dt\right].$

We call Model B our model with this second objective function.

Let $\{e^{i}\}_{i=1}^{d}$ be the canonical basis of ${\mathbb{R}}^{d}$. The Hamilton-Jacobi-Bellman equation associated with Model A is

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle\partial_{t}u(t,x,q,S)+\frac{1}{2}\sum_{i,j=1}^{d}\rho^{i,j}\sigma^{i}\sigma^{j}\partial^{2}_{S^{i}S^{j}}u(t,x,q,S)$		(1)
			$\displaystyle+\sum_{i=1}^{d}\mathds{1}_{\{q^{i}+z^{i}\leq Q^{i}\}}\sup_{\delta^{i,b}}\Lambda^{i,b}(\delta^{i,b})\left(u(t,x-z^{i}S^{i}+z^{i}\delta^{i,b},q+z^{i}e^{i},S)-u(t,x,q,S)\right)$
			$\displaystyle+\sum_{i=1}^{d}\mathds{1}_{\{q^{i}-z^{i}\geq-Q^{i}\}}\sup_{\delta^{i,a}}\Lambda^{i,a}(\delta^{i,a})\left(u(t,x+z^{i}S^{i}+z^{i}\delta^{i,a},q-z^{i}e^{i},S)-u(t,x,q,S)\right),$

for all $(t,x,q,S)\in[0,T)\times\mathbb{R}\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\times\mathbb{R}^{d}$,⁷⁷7Given a positive number $z\in\mathbb{R}_{+}^{*}$, $z\mathbb{Z}$ denotes the set of multiples of $z$, i.e. $z\mathbb{Z}=\{\ldots,-2z,-z,0,z,2z,\ldots\}.$ with terminal condition

$$u(T,x,q,S)=-\exp\left(-\gamma\left(x+\sum_{i=1}^{d}q^{i}S^{i}\right)\right)\quad\forall(x,q,S)\in\mathbb{R}\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\times\mathbb{R}^{d}.$$

The Hamilton-Jacobi-Bellman equation associated with Model B is

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle\partial_{t}v(t,x,q,S)-\frac{1}{2}\gamma q^{\intercal}\Sigma q+\frac{1}{2}\sum_{i,j=1}^{d}\rho^{i,j}\sigma^{i}\sigma^{j}\partial^{2}_{S^{i}S^{j}}v(t,x,q,S)$		(2)
			$\displaystyle+\sum_{i=1}^{d}\mathds{1}_{\{q^{i}+z^{i}\leq Q^{i}\}}\sup_{\delta^{i,b}}\Lambda^{i,b}(\delta^{i,b})\left(v(t,x-z^{i}S^{i}+z^{i}\delta^{i,b},q+z^{i}e^{i},S)-v(t,x,q,S)\right)$
			$\displaystyle+\sum_{i=1}^{d}\mathds{1}_{\{q^{i}-z^{i}\geq-Q^{i}\}}\sup_{\delta^{i,a}}\Lambda^{i,a}(\delta^{i,a})\left(v(t,x+z^{i}S^{i}+z^{i}\delta^{i,a},q-z^{i}e^{i},S)-v(t,x,q,S)\right),$

for all $(t,x,q,S)\in[0,T)\times\mathbb{R}\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\times\mathbb{R}^{d}$ with terminal condition

$$v(T,x,q,S)=x+\sum_{i=1}^{d}q^{i}S^{i}\quad\forall(x,q,S)\in\mathbb{R}\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\times\mathbb{R}^{d}.$$

For each $i\in\{1,\ldots,d\}$ and $\xi\geq 0,$ let us define two Hamiltonian functions⁸⁸8It is noteworthy that our definition of $H_{\xi}^{i,b}$ and $H_{\xi}^{i,a}$ differs from that of [18] (by a factor $z^{i}$). The alternative definition we use in this paper is also that of [5] for $\xi=0$. $H_{\xi}^{i,b}$ and $H_{\xi}^{i,a}$ by

$$H^{i,b}_{\xi}(p)=\begin{cases}\underset{\delta}{\sup}\frac{\Lambda^{i,b}(\delta)}{\xi z^{i}}(1-\exp(-\xi z^{i}(\delta-p)))&\mbox{if }\xi>0,\\ \underset{\delta}{\sup}\Lambda^{i,b}(\delta)(\delta-p)&\mbox{if }\xi=0,\end{cases}$$

(3)

and

$$H^{i,a}_{\xi}(p)=\begin{cases}\underset{\delta}{\sup}\frac{\Lambda^{i,a}(\delta)}{\xi z^{i}}(1-\exp(-\xi z^{i}(\delta-p)))&\mbox{if }\xi>0,\\ \underset{\delta}{\sup}\Lambda^{i,a}(\delta)(\delta-p)&\mbox{if }\xi=0.\end{cases}$$

(4)

Using the ansatz introduced in [18] for the two functions $u:[0,T]\times\mathbb{R}\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ and $v:[0,T]\times\mathbb{R}\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\times\mathbb{R}^{d}\rightarrow\mathbb{R}$, i.e.

$$u(t,x,q,S)=-\exp\left(-\gamma\left(x+\sum_{i=1}^{d}q^{i}S^{i}+\theta(t,q)\right)\right)\text{ and }v(t,x,q,S)=x+\sum_{i=1}^{d}q^{i}S^{i}+\theta(t,q),$$

we see that solving the Hamilton-Jacobi-Bellman equations (1) and (2) boils down to finding the solution $\theta:[0,T]\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\to\mathbb{R}$ of the following Hamilton-Jacobi equation with $\xi=\gamma$ in the case of Model A and $\xi=0$ in the case of Model B:

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle\partial_{t}\theta(t,q)-\frac{1}{2}\gamma q^{\intercal}\Sigma q$
			$\displaystyle+\sum_{i=1}^{d}\mathds{1}_{\{q^{i}+z^{i}\leq Q^{i}\}}z^{i}H^{i,b}_{\xi}\left(\frac{\theta(t,q)-\theta(t,q+z^{i}e^{i})}{z^{i}}\right)+\sum_{i=1}^{d}\mathds{1}_{\{q^{i}-z^{i}\geq-Q^{i}\}}z^{i}H^{i,a}_{\xi}\left(\frac{\theta(t,q)-\theta(t,q-z^{i}e^{i})}{z^{i}}\right).$

In both cases, the terminal condition simply boils down to

$\displaystyle\theta(T,q)=0.$

(6)

From [18, Theorem 5.1], for a given $\xi\geq 0,$ there exists a unique $\theta:[0,T]\times\prod_{i=1}^{d}\left(z^{i}\mathbb{Z}\cap[-Q^{i},Q^{i}]\right)\to{\mathbb{R}}$, $C^{1}$ in time, solution of Eq. (2.3) with terminal condition (6). Moreover (see [18, Theorems 5.2 and 5.3]), a classical verification argument enables to go from $\theta$ to optimal controls for both Model A and Model B. The optimal quotes as functions of $\theta$ are recalled in the following theorems (for details, see [18]).

In the following two sections, we propose new methods to find approximations of the solution to the system of ordinary differential equations (ODEs) (2.3) with terminal condition (6). Eqs. (7) and (8) can then serve to go from approximations of $\theta$ (hereafter called – slightly abusively – the value function) to approximations of the optimal quotes. The resulting quotes correspond to what the reinforcement learning community calls the greedy quoting strategy associated with the proxy of the value function.⁹⁹9The true optimal quotes correspond to the greedy strategy with respect to the value function $u$ (in Model A) or $v$ (in Model B) deduced from the true $\theta$.

In the field of (stochastic) optimal control, finding value functions and optimal controls in closed form is the exception rather than the rule. One important exception goes with the class of Linear-Quadratic (LQ) and Linear-Quadratic-Gaussian (LQG) problems. Of course, the above market making problem does not belong to this class of control problems, for instance because the control of point processes is nonlinear by nature. Nevertheless, we see that price risk appears in both Model A and Model B through the quadratic term $\frac{1}{2}\gamma q^{\intercal}\Sigma q$ in the Hamilton-Jacobi equation (2.3). The main idea of this paper consists in replacing the Hamiltonian functions associated with our market making problem by quadratic functions that approximate them. The interest of quadratic Hamiltonian functions lies in that the resulting Hamilton-Jacobi equations can be solved in closed-form using the same tools as for LQ/LQG problems, i.e. Riccati equations.

At first sight, approximating the Hamiltonian functions involved in Eq. (2.3) by quadratic functions seems inappropriate. For all $i\in\{1,\ldots,d\}$, the functions $H_{\xi}^{i,b}$ and $H_{\xi}^{i,a}$ are indeed positive and decreasing and approximating them with U-shaped functions can only be valid locally. However, one has to bear in mind that our goal is to approximate the solution of the Hamilton-Jacobi equations and not the Hamiltonian functions. This remark is particularly important because the Hamiltonian terms involved in the Hamilton-Jacobi equations are (up to the indicator functions that we shall discard in what follows by considering the limit case where $\forall i\in\{1,\ldots,d\},Q^{i}=+\infty$) of the form

$$H^{i,b}_{\xi}\left(\frac{\theta(t,q)-\theta(t,q+z^{i}e^{i})}{z^{i}}\right)+H^{i,a}_{\xi}\left(\frac{\theta(t,q)-\theta(t,q-z^{i}e^{i})}{z^{i}}\right),$$

Assuming that $\frac{\theta(t,q)-\theta(t,q+z^{i}e^{i})}{z^{i}}\simeq-\frac{\theta(t,q)-\theta(t,q-z^{i}e^{i})}{z^{i}}$, we clearly see that, with respect to asset $i$, the function we need to approximate is $p\mapsto H^{i,b}_{\xi}(p)+H^{i,a}_{\xi}(-p)$ rather than $H^{i,b}_{\xi}$ and $H^{i,a}_{\xi}$ themselves, and it is natural to approximate the former function with a U-shaped one!

Let us replace for all $i\in\{1,\ldots,d\}$ the Hamiltonian functions $H_{\xi}^{i,b}$ and $H_{\xi}^{i,a}$ by the quadratic functions¹⁰¹⁰10We omit the subscript $\xi$ in the definition of $\check{H}^{i,b}$ and $\check{H}^{i,a}$. In particular, although the subscript $\xi$ is not written, the coefficients $\alpha^{i,b}_{0}$, $\alpha^{i,b}_{1}$, $\alpha^{i,b}_{2}$, $\alpha^{i,a}_{0}$, $\alpha^{i,a}_{1}$, and $\alpha^{i,a}_{2}$ do depend on $\xi$.

$$\check{H}^{i,b}:p\mapsto\alpha^{i,b}_{0}+\alpha^{i,b}_{1}p+\frac{1}{2}\alpha^{i,b}_{2}p^{2}\quad\textrm{and}\quad\check{H}^{i,a}:p\mapsto\alpha^{i,a}_{0}+\alpha^{i,a}_{1}p+\frac{1}{2}\alpha^{i,a}_{2}p^{2}.$$

We denote by $\check{\theta}$ the approximation of $\theta$ associated with the functions $(\check{H}^{i,b})_{i\in\{1,\ldots,d\}}$ and $(\check{H}^{i,a})_{i\in\{1,\ldots,d\}}$, i.e. if we consider the limit case where $\forall i\in\{1,\ldots,d\},Q^{i}=+\infty$, $\check{\theta}$ verifies

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle\partial_{t}\check{\theta}(t,q)-\frac{1}{2}\gamma q^{\intercal}\Sigma q+\sum_{i=1}^{d}z^{i}\left(\alpha^{i,b}_{0}+\alpha^{i,a}_{0}\right)$		(9)
			$\displaystyle+\sum_{i=1}^{d}\left(\alpha^{i,b}_{1}\left(\check{\theta}(t,q)-\check{\theta}(t,q+z^{i}e^{i})\right)+\alpha^{i,a}_{1}\left(\check{\theta}(t,q)-\check{\theta}(t,q-z^{i}e^{i})\right)\right)$
			$\displaystyle+\frac{1}{2}\sum_{i=1}^{d}\frac{1}{z^{i}}\left(\alpha^{i,b}_{2}\left(\check{\theta}(t,q)-\check{\theta}(t,q+z^{i}e^{i})\right)^{2}+\alpha^{i,a}_{2}\left(\check{\theta}(t,q)-\check{\theta}(t,q-z^{i}e^{i})\right)^{2}\right)$

and of course we consider the terminal condition

$$\check{\theta}(T,q)=0.$$

(10)

Eq. (9) with terminal condition (10) can be solved in closed form. To prove this point, we start with the following proposition: