Optimal portfolio liquidation with additional information


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.

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  1. 1.

    Almgren, R., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3, 5–39 (2000)

    Google Scholar 

  2. 2.

    Amendinger, J., Becherer, D., Schweizer, M.: A monetary value for initial information in portfolio optimization. Finance Stoch. 7(1), 29–46 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Amendinger, J., Imkeller, P., Schweizer, M.: Additional logarithmic utility of an insider. Stoch. Process. Appl. 75(2), 263–286 (1998)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Ankirchner, S., Dereich, S., Imkeller, P.: The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34(2), 743–778 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Ankirchner, S., Imkeller, P.: Financial markets with asymmetric information: information drift, additional utility and entropy. In Stochastic processes and applications to mathematical finance, pp. 1–21. World Sci. Publ., Hackensack (2007)

  6. 6.

    Baudoin, F.: Modeling Anticipations on Financial Markets. In Paris-Princeton Lectures on Mathematical Finance, 2002, Volume 1814 of Lecture Notes in Math, pp. 43–94. Springer, Berlin (2003)

    Google Scholar 

  7. 7.

    Bertsimas, D., Lo, A.W.: Optimal control of execution costs. J. Financ. Markets 1, 1–50 (1998)

    Article  Google Scholar 

  8. 8.

    Corcuera, J.M., Imkeller, P., Kohatsu-Higa, A., Nualart, D.: Additional utility of insiders with imperfect dynamical information. Fin. Stoch. 8, 437–450 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Eyraud-Loisel, A.: Backward stochastic differential equations with enlarged filtration. Option hedging of an insider trader in a financial market with jumps. Stoch. Process. Their Appl. 115(11), 1745–1763 (2005)

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Grorud, A., Pontier, M.: Insider trading in a continuous time market model. Int. J. Theor. Appl. Fin. 1(3), 331–347 (1998)

    MATH  Article  Google Scholar 

  11. 11.

    Hillairet, C.: Comparison of insiders’ optimal strategies depending on the type of side-information. Stoch. Process. Appl. 115(10), 1603–1627 (2005)

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Ihara, S.: Information Theory for Continuous Systems. World Scientific Publishing Co., Inc., River Edge, NJ (1993)

    MATH  Book  Google Scholar 

  13. 13.

    Karatzas, I., Pikovsky, I.: Anticipative portfolio optimization. Adv. Appl. Probab. 28(4), 1095–1122 (1996)

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Mansuy, R., Yor, M.: Random Times and Enlargements of Filtrations in a Brownian Setting, Volume 1873 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2006)

    Google Scholar 

  15. 15.

    Schied, A.: A control problem with fuel constraint and Dawson-Watanabe superprocesses. Ann. Appl. Probab. 23(6), 2472–2499 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  16. 16.

    Schied, A., Schöneborn, T., Tehranchi, M.: Optimal basket liquidation for CARA investors is deterministic. Appl. Math. Finance 17(6), 471–489 (2010)

    MATH  MathSciNet  Article  Google Scholar 

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The authors thank anonymous referees for very useful comments. The first author gratefully acknowledges the financial support by the Ecole Centrale de Lyon during his visit in March 2012.

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Correspondence to Christophette Blanchet-Scalliet.



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

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

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 bcdefg and their derivatives are bounded on \([t,\tau ], M\) is a strict martingale on \([t,\tau ]\). Taking expectations, therefore, leads to

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

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 cb 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. \(\square \)

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Ankirchner, S., Blanchet-Scalliet, C. & Eyraud-Loisel, A. Optimal portfolio liquidation with additional information. Math Finan Econ 10, 1–14 (2016). https://doi.org/10.1007/s11579-015-0147-3

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  • Optimal liquidation
  • Price impact
  • Additional information
  • Enlargement of filtration
  • HJB equation