We introduce a formal framework for analyzing trades in financial markets. An exchange is where multiple buyers and sellers participate to trade. These days, all big exchanges use computer algorithms that implement double sided auctions to match buy and sell requests and these algorithms must abide by certain regulatory guidelines. For example, market regulators enforce that a matching produced by exchanges should be , and . To verify these properties of trades, we first formally define these notions in a theorem prover and then give formal proofs of relevant results on matchings. Finally, we use this framework to verify properties of two important classes of double sided auctions. All the definitions and results presented in this paper are completely formalised in the Coq proof assistant without adding any additional axioms to it.
There is no prior work known to us which formalizes financial algorithms used by the exchanges. Passmore and Ignatovich in @cite_17 highlight the significance, opportunities and challenges involved in formalizing financial markets. Their work describes in detail the whole spectrum of financial algorithms that need to be verified for ensuring safe and fair markets. Matching algorithms used by the exchanges are at the core of this whole spectrum.
@cite_17
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Many deep issues plaguing today’s financial markets are symptoms of a fundamental problem: The complexity of algorithms underlying modern finance has significantly outpaced the power of traditional tools used to design and regulate them. At Aesthetic Integration, we have pioneered the use of formal verification for analysing the safety and fairness of financial algorithms. With a focus on financial infrastructure (e.g., the matching logics of exchanges and dark pools and FIX connectivity between trading systems), we describe the landscape, and illustrate our Imandra formal verification system on a number of real-world examples. We sketch many open problems and future directions along the way. In this article, we study the problem of online market clearing where there is one commodity in the market being bought and sold by multiple buyers and sellers whose bids arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to match without knowing what bids will arrive in the future. For maximizing profit, we present a (randomized) online algorithm with a competitive ratio of ln(pmax − pmin) p 1, when bids are in a range [pmin, pmax], which we show is the best possible. A simpler algorithm has a ratio twice this, and can be used even if expiration times are not known. For maximizing the number of trades, we present a simple greedy algorithm that achieves a factor of 2 competitive ratio if no money-losing trades are allowed. We also show that if the online algorithm is allowed to subsidize matches---match money-losing pairs if it has already collected enough money from previous pairs to pay for them---then it can actually be 1-competitive with respect to the optimal offline algorithm that is not allowed subsidy. That is, for maximizing the number of trades, the ability to subsidize is at least as valuable as knowing the future. We also consider objectives of maximizing buy or sell volume and social welfare. We present all of these results as corollaries of theorems on online matching in an incomplete interval graph.We also consider the issue of incentive compatibility, and develop a nearly optimal incentive-compatible algorithm for maximizing social welfare. For maximizing profit, we show that no incentive-compatible algorithm can achieve a sublinear competitive ratio, even if only one buy bid and one sell bid are alive at a time. However, we provide an algorithm that, under certain mild assumptions on the bids, performs nearly as well as the best fixed pair of buy and sell prices, a weaker but still natural performance measure. This latter result uses online learning methods, and we also show how such methods can be used to improve our “optimal” algorithms to a broader notion of optimality. Finally, we show how some of our results can be generalized to settings in which the buyers and sellers themselves have online bidding strategies, rather than just each having individual bids. We propose a survey of soft computing techniques applied to financial market.We surveyed several primary studies proposed in the literature.A framework for building an intelligent trading system was proposed.Future directions of this research field are discussed. Financial markets play an important role on the economical and social organization of modern society. In these kinds of markets, information is an invaluable asset. However, with the modernization of the financial transactions and the information systems, the large amount of information available for a trader can make prohibitive the analysis of a financial asset. In the last decades, many researchers have attempted to develop computational intelligent methods and algorithms to support the decision-making in different financial market segments. In the literature, there is a huge number of scientific papers that investigate the use of computational intelligence techniques to solve financial market problems. However, only few studies have focused on review the literature of this topic. Most of the existing review articles have a limited scope, either by focusing on a specific financial market application or by focusing on a family of machine learning algorithms. This paper presents a review of the application of several computational intelligent methods in several financial applications. This paper gives an overview of the most important primary studies published from 2009 to 2015, which cover techniques for preprocessing and clustering of financial data, for forecasting future market movements, for mining financial text information, among others. The main contributions of this paper are: (i) a comprehensive review of the literature of this field, (ii) the definition of a systematic procedure for guiding the task of building an intelligent trading system and (iii) a discussion about the main challenges and open problems in this scientific field. We study Fisher markets that admit equilibria wherein each good is integrally assigned to some agent. While strong existence and computational guarantees are known for equilibria of Fisher markets with additive valuations, such equilibria, in general, assign goods fractionally to agents. Hence, Fisher markets are not directly applicable in the context of indivisible goods. In this work we show that one can always bypass this hurdle and, up to a bounded change in agents' budgets, obtain markets that admit an integral equilibrium. We refer to such markets as pure markets and show that, for any given Fisher market (with additive valuations), one can efficiently compute a "near-by," pure market with an accompanying integral equilibrium. Our work on pure markets leads to novel algorithmic results for fair division of indivisible goods. Prior work in discrete fair division has shown that, under additive valuations, there always exist allocations that simultaneously achieve the seemingly incompatible properties of fairness and efficiency; here fairness refers to envy-freeness up to one good (EF1) and efficiency corresponds to Pareto efficiency. However, polynomial-time algorithms are not known for finding such allocations. Considering relaxations of proportionality and EF1, respectively, as our notions of fairness, we show that fair and Pareto efficient allocations can be computed in strongly polynomial time.