Optimal Algorithms for k-Search with Application in Option Pricing

• Published in: Algorithmica Vol. 55; no. 2; pp. 311 - 328
• Main Authors: Lorenz, Julian, Panagiotou, Konstantinos, Steger, Angelika
• Format: Journal Article Conference Proceeding
• Language: English
• Published: New York Springer-Verlag 01.10.2009; Springer
• Subjects: Algorithm Analysis and Problem Complexity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Inference from stochastic processes; time series analysis; Mathematics; Mathematics of Computing; Operational research and scientific management; Operational research. Management science; Portfolio theory; Probability and statistics; Sciences and techniques of general use; Statistics; Theoretical computing; Theory of Computation; Online algorithms; One-way trading; Competitive analysis; Option pricing; Time series search; Performance evaluation; Search problem; Arbitration; On line; Optimal algorithm; Worst case method; Competitiveness; Time series; Minimization; Approximation algorithm; Randomized algorithm; Combinatorial optimization; Search algorithm; Time series seach; One-way treading; Randomized design; Stock exchange; Derivative; Tariffication; Deterministic algorithms
• ISSN: 0178-4617; 1432-0541; 1432-0541
• DOI: 10.1007/s00453-008-9217-8

In the k -search problem, a player is searching for the k highest (respectively, lowest) prices in a sequence, which is revealed to her sequentially. At each quotation, the player has to decide immediately whether to accept the price or not. Using the competitive ratio as a performance measure, we give optimal deterministic and randomized algorithms for both the maximization and minimization problems, and discover that the problems behave substantially different in the worst-case. As an application of our results, we use these algorithms to price “lookback options”, a particular class of financial derivatives. We derive bounds for the price of these securities under a no-arbitrage assumption, and compare this to classical option pricing.