SUMMARY ORDER

Plaintiff Andrew Walzer, pro se and as custodian for his daughter Arielle Walzer, sued to recover the approximately $140,000 he lost by investing in the stock of defendant UAL Corporation (“UAL”). He now appeals from the district court’s dismissal of his complaint and denial of leave to amend. We review the dismissal de novo, liberally construing the pro se complaint, accepting its factual allegations as true, and drawing all reasonable inferences in the plaintiffs favor. See Boykin v. KeyCorp, 521 F.3d 202, 214 (2d Cir. 2008); Vietnam Ass’n for Victims of Agent Orange v. Dow Chem. Co., 517 F.3d 104, 115 (2d Cir. 2008). We review the denial of leave to amend for abuse of discretion. See McCarthy v. Dun & Bradstreet Corp., 482 F.3d 184, 200 (2d Cir. 2007). In doing so, we assume the parties’ familiarity with the facts and the record of prior proceedings, which we reference only as necessary to explain our decision to affirm.

1. Dismissal of Complaint

Our de novo review confirms that, even if liberally construed, the complaint fails to state a claim under any of the myriad legal theories Walzer identifies. Accordingly, we affirm the judgment of dismissal for *553substantially the same reasons stated by the district court in its thorough and well-reasoned opinion. See Walzer v. UAL Corp., No. 05 Civ. 0581, 2008 WL 87944 (S.D.N.Y. Jan. 2, 2008).

2. Denial of Leave To Amend

Although leave to amend is generally granted at least once, especially to pro se litigants, the district court need not permit amendment when, as in this case, a liberal reading of the complaint gives no indication that a valid claim might be stated. See Davis v. Goord, 320 F.3d 346, 352 (2d Cir. 2003); Branum v. Clark, 927 F.2d 698, 705 (2d Cir. 1991). Here, Walzer never filed a proposed amended complaint and only vaguely alluded to the claims he intended to raise. Moreover, Walzer’s original complaint unequivocally acknowledged that his financial losses were caused by UAL’s filing for bankruptcy and defendants’ general mismanagement of UAL, not by any of defendants’ alleged misrepresentations or omissions. Accordingly, the district court did not abuse its discretion in concluding that, regardless of any future claims Walzer might allege, he could not state a claim that any fraud “caused the loss for which the plaintiff seeks to recover.” Dura Pham., Inc. v. Broudo, 544 U.S. 336, 341-42, 125 S.Ct. 1627, 161 L.Ed.2d 577 (2005).

On appeal, Walzer presents two new theories of loss causation. His first theory attributes his loss to the UAL directors’ failure to disclose that they had retained bankruptcy attorneys. Silence, however, does not give rise to securities fraud unless defendants had a duty to disclose information, and Walzer has not alleged any such duty here. See Chiarella v. United States, 445 U.S. 222, 232, 100 S.Ct. 1108, 63 L.Ed.2d 348 (1980) (“[T]he element required to make silence fraudulent — a duty to disclose — is absent in this case.”).1 Walzer’s second theory attributes his loss to UAL’s inconsistent or inaccurate disclosure of the number of shares outstanding at the end of 2000. In fact, UAL’s Form 10-K states that 16,295,475 of the 68,834,167 shares then issued were held in treasury. Given that outstanding shares comprise “stock that is held by investors and has not been redeemed by the issuing corporation,” Black’s Law Dictionary 1552 (9th ed. 2009); see also Douglas K. Moll, Shareholder Oppression and “Fair Value”: Of Discounts, Dates, and Dastardly Deeds in the Close Corporation, 54 Duke L.J. 293, 328 n. 129 (2004) (“ ‘Treasury shares are carried on the books as authorized and still issued but not outstanding.’ ” (quoting James D. Cox & Thomas Lee Hazen, Cox & Hazen on Corporations, § 21.11, at 1287 (2d ed. 2003))), Walzer cannot claim that the market was deceived by the statement that 52,538,692 shares were outstanding at the end of 2000.

Accordingly, we identify no abuse of discretion in the district court’s denial of leave to amend to plead these theories.

*554We have considered Walzer’s other arguments on appeal and conclude that they lack merit. Accordingly, we AFFIRM the judgment of the district court.