The underlying trading mechanisms of electronic securities exchanges have mostly stayed the same over the years with some additions and improvements. However, over the recent decade, high-frequency traders using algorithmic trading have shifted the field using practices that many consider unfair or unethical. In addition, insider trading continues to cause trust issues on certain trading platforms. In this paper, we present PET-Exchange, a privacy-preserving framework for trading securities on an electronic stock exchange. By using homomorphic encryption, PET-Exchange prevents information disclosures and unfair advantages in the trading processes. By matching and trading encrypted orders, we study the performance under various volumes and timing constraints, and compare this to the unencrypted counterparts. Our analysis of PET-Exchange using market trade data shows the privacy and cryptographic tradeoffs, demonstrating it to be suitable for small-scale trading and privacy-preserving auctions. Finally, we discuss the potential impact on transparency, fairness, and opportunities for financial crime in an electronic securities exchange. The insights we provide take us one step closer to a privacy-aware and fair public securities exchange.