Optimal portfolio liquidation with additional information

Abstract

We consider the problem of how to optimally close a large asset position in a market with a linear temporary price impact. We take the perspective of an agent who obtains a signal about the future price evolvement. By means of classical stochastic control we derive explicit formulas for the closing strategy that minimizes the expected execution costs. We compare agents observing the signal with agents who do not see it. We compute explicitly the expected additional gain due to the signal, and perform a comparative statics analysis.

Appendix

The coefficients of the value function in Theorem 1 are given by

$$\begin{aligned} b(t)= & {} -\eta \frac{1}{T-t}, \\ c(t)= & {} \frac{T}{T-t}h^{-1}(0,t) \int _t^T \beta (u)\frac{T-u}{T}h(0,u) du, \\ d^{\text {hom}}(t)= & {} \exp \left( -\int _0^t 2 \beta (u)du \right) \\ d(t)= & {} d^\text {hom}(t) \int _t^T \frac{c^2(u)}{4 \eta d^\text {hom}(u)} du \\ e(t)= & {} \frac{T}{T-t} \int _t^T (c(u) + 1) \alpha (u)\frac{T-u}{T} du, \\ f^{\text {hom}}(t)= & {} \exp \left( -\int _0^t \beta (u)du\right) \\ f(t)= & {} f^\text {hom}(t) \int _t^T \left( \frac{1}{2\eta } c(u) e(u) + 2 \alpha (u) d(u)\right) \frac{1}{f^\text {hom}(u)} du \\ g(t)= & {} \int _t^T \left( \frac{e^2(u)}{4 \eta } + \alpha (u) f + \sigma ^2 d(u) \right) du, \end{aligned}$$

with $t \in [0,T]$. We remark that Theorem 1 can be derived from Theorem 2 in [15]. The proof given below is completely different though, using classical verification arguments.

Proof of Theorem 1

Let $w(t,x,s) = b(t) x^2 + c(t) xs + d(t)s^2 + e(t) x + f(t) s + g(t)$. We first show that the value function satisfies $V \le w$. Notice that w is a solution of the HJB Eq. (4) and satisfies the terminal condition (5). This follows from the fact that the coefficients satisfy the following ODEs

$$\begin{aligned}&\displaystyle -b_t - \frac{1}{\eta } b^2 = 0 \\&\displaystyle -c_t - \frac{1}{\eta } bc -\beta c -\beta = 0 \\&\displaystyle -d_t - \frac{1}{4 \eta } c^2 - 2\beta d = 0 \\&\displaystyle -e_t - \frac{1}{\eta } be - \alpha c - \alpha = 0 \\&\displaystyle -f_t - \frac{1}{2\eta } ce - 2\alpha d - \beta f = 0 \\&\displaystyle -g_t - \frac{1}{4\eta } e^2 - \alpha f - \sigma ^2 d = 0. \end{aligned}$$

Since the functions $\alpha$ and $\beta$ are bounded, there exists a constant $C\in \mathbb {R}_+$ such that

$$\begin{aligned} |c(t)| + |d(t)| + |e(t)|+ |f(t)|+|g(t)| \le C (T-t) \end{aligned}$$

(20)

for all $t \in [0,T]$. Moreover, we have $|b(t)| \le C\frac{1}{T-t}$.

Let $\xi \in \mathcal {A}(t,x)$ be an arbitrary admissible control and let X be its associated position process. Let $\tau < T$. Itô’s formula implies

$$\begin{aligned} w(\tau ,X_\tau ,S_\tau )= & {} w(t,x,s) + \int _t^\tau \frac{1}{2} \sigma ^2 w_{ss}(u,X_u,S_u) du + M_\tau \\&+ \int _t^\tau [w_t(u,X_u,S_u) - w_x(u,X_u,S_u) \xi _u + a(u,S_u) w_s(u,X_u,S_u)] du, \end{aligned}$$

where $M_s = \int _t^s w_s(u,X_u,S_u) \sigma dW_u$. As $(X_t)_{t \in [0,\tau ]}$ is $L^2$-bounded and all functions b, c, d, e, f, g and their derivatives are bounded on $[t,\tau ], M$ is a strict martingale on $[t,\tau ]$. Taking expectations, therefore, leads to

$$\begin{aligned} E(w(\tau ,X_{\tau },S_{\tau }))= & {} w(t,x,s)+E\left( \int _t^{\tau } \left( w_t-w_x \xi +aw_s+\frac{1}{2} \sigma ^2w_{ss}\right) (u,X_u,S_u)du\right) {\nonumber }\\\le & {} w(t,x,s)+E\left( \int _t^{\tau }(-a(u,S_u) X_u+ \eta \xi ^2_u)du\right) . \end{aligned}$$

(21)

As $\xi$ is square integrable (Condition (A1)). This further implies that we have

$$\begin{aligned} \lim _{\tau \rightarrow T} E\left( \int _t^{\tau }(-a(u,S_u) X_u+\eta \xi ^2_u)du\right) =J(t,x,s,\xi ). \end{aligned}$$

Moreover, since also $\left( \frac{X_t^2}{T-t}\right) _{t \in [0,T)}$ is uniformly integrable and $\lim _{t \rightarrow T}\frac{X_t^2}{T-t}=0$, we have $\lim _{\tau \rightarrow T} E[w(\tau ,X_{\tau },S_{\tau })]=0$. Inequality (21), therefore, implies $w(t,x,s) \ge J(t,x,s,\xi )$. Taking the supremum over all admissible controls, one has $V(t,x,s)\le w(t,x,s)$.

Secondly, we show that the control $(\xi ^*_t)_{t\in [0,T]}$ is admissible. Using the majoration (20) on the coefficients c, b and e, one can show that there exists a constant C such that

$$\begin{aligned} \left| [c(u)S_u+e(u)]\right| \frac{ T}{(T-u)}\le C(\left| S_u\right| +1) \end{aligned}$$

for all $u \in [0,T]$. With (8) we obtain that $|X_t^*|\le C(T-t)(1 +\int _0^t \left| S_u\right| du)$

and hence Condition (A2) is satisfied.

Condition (A1) is a consequence of $\xi ^2_t\le C(b(t)^2X_t^2+ b(t)X_t)\le C$.

Equality holds in Inequality (21) by choosing $\xi =\xi ^*$. This proves that $J(t,x,s,\xi ^*) =w(t,s,x)$. Thus the proof is complete. $\square$