Competitive analysis of financial games

In the unidirectional conversion problem an on-line player is given the task of converting dollars to yen over some period of time. Each day, a new exchange rate is announced and the player must decide how many dollars to convert. His goal is to minimize the competitive ratio. defined as sup_E (P_OPT(E)/P_XE) where E ranges over exchange rate sequences. P_OPT(E) is the number of yen obtained by an optimal off-line algorithm, and Px(E) is the number of yen obtained by the on-line algorithm X. The authors also consider a continuous version of the problem. in which the exchange rate varies over a continuous time interval. The on-line line players a priori information about the fluctuation of exchange rates distinguishes different variants of the problem. For three variants they show that a simple threat-based strategy is optimal for the on-line player and determine its competitive ratio. They also derive and analyze an optimal policy for the on-line player when he knows the probability distribution of the maximum value that the exchange rate will reach. Finally, they consider a bidirectional conversion problem, which the player may trade dollars for yen or yen for dollars