Trading a Fraction of Total Volume (part 2)

How to trade optimally with a volume constraint

(The following is adapted from chapter 9.2 in A. Cartea, S. Jaimungal, J. Penalva’s book¹ )

Last week, we examined the question of optimally selling a stock position, with a target speed expressed as a fraction ρ of the market’s overall speed of selling μ_t, assumed to be stochastic. We carried out the computation in the case where there is only temporary market impact, of strength kappa>0. This week, let’s add a permanent market impact term, with intensity c > 0, and, to make sure we can still get an explicit solution, let’s also assume μ_t is a constant = μ > 0.

Our usual stochastic processes (see last week’s post for notations) now satisfy

\((1)\ \ \ \ \ \ \begin{equation*} \left\{ \begin{split} &ds_t = -c\lambda_t dt + \sigma dW_t\\ \\ &dq_t = - \lambda_t dt,\ \ \ \ \lambda_t>0,\ \ \ \ q_0 = Q\\ \\ &dx_t = -\left(s_t - k\lambda_t\right)dq_t, \end{split} \right. \end{equation*}\)

and the value function u, defined by

\((2)\ \ \ \ \ \ u(t,x,s,q) = \sup_{\lambda_t}\mathbb{E}_{t,x,s,q}\left( x_T + q_Ts_T - \kappa\int_t^T(\lambda_s - \rho\mu)^2 ds\right)\)

is shown to satisfy the following HJB equation

\((3)\ \ \ \ \ \ \begin{equation*} \left\{ \begin{split} &u_t + \frac{\sigma^2}{2}u_{ss} + \sup_\lambda\left\{ (s - k\lambda)\lambda u_x - \lambda u_q - c\lambda u_s -\kappa(\lambda-\rho\mu)^2 \right\} = 0\\ \\ &u(t=T,x,s,q,\mu) = x + qs \end{split} \right. \end{equation*}\)

This leads us to the Ansatz

\((4)\ \ \ \ \ \ \begin{equation*} \begin{split} &u(t,x,s,q) = x + qs + h(t,q)\ \ \ \ (\mu = \hbox{const})\\ \\ &h(t,q) = h^0(t) + h^1(t) q + h^2(t) q^2 \end{split} \end{equation*}\)

and the optimal policy (speed of selling) λ*

\((5)\ \ \ \ \ \ \lambda^* = \frac{1}{2(k+\kappa)}\left(2\rho\mu\kappa - h_q - cq\right)\)

From (3) and (4) we immediately see that h is the unique solution to the ODE

\((6)\ \ \ \ \ \ \begin{equation*} \left\{ \begin{split} &h_t + \frac{1}{4(k+\kappa)} \left(h_q + cq - 2\rho\mu\kappa\right)^2 - \kappa\rho^2\mu^2 = 0\\ \\ &h(t=T) = 0 \end{split} \right. \end{equation*}\)

from which we deduce the following two ODE for h^1 and h^2

\((7)\ \ \ \ \ \ \begin{equation*} \begin{split} &h^1_t + \frac{1}{k+\kappa}\left(h^2 + \frac{c}{2}\right)\left(h^1 - 2\rho\mu\kappa\right) = 0\\ \\ &h^2_t + \frac{1}{k + \kappa}\left(h^2 + \frac{c}{2}\right)^2 = 0,\\ \end{split} \end{equation*}\)

along with a zero boundary condition at t = T. Here, we disregard the equation for h^0, since this term which doesn’t appear in (5). (Our goal being to express the optimal speed of selling λ*.)

Solving (7) for h^1 and h^2 and replacing into (5), we get the optimal speed of selling under our assumptions:

\((8)\ \ \ \ \ \ \ \boxed{\lambda^* = \frac{\rho\mu\kappa - \frac{c}{2}q}{k+\kappa - \frac{c}{2}(T-t)}}\)

Let’s examine two asymptotic regimes. First, if the permanent market impact’s strength c is set to 0, we recover last week’s formula (6) — our speed of selling is the target speed ρμ, corrected for the ration between temporary market impact intensity k and penalizing constant kappa. Second, if c is sent to infinity, we find a constant speed of selling (a linearly decreasing inventory down to 0 at time t=T), which is intuitive since in that case the constraint no longer matters.

Quantitatively Yours,

1. Algorithmic and High-Frequency Trading, A. Cartea, S. Jaimungal, J. Penalva, Cambridge U.P. (2015)