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 de ned as supE POPT E PX E where E ranges over exchange rate sequences POPT E is the number of yen obtained by an optimal o line algo rithm and PX E is the number of yen obtained by the on line algorithm X We also consider a contin uous version of the problem in which the exchange rate varies over a continuous time interval The on line player s a priori information about the uctua tion of exchange rates distinguishes di erent variants of the problem For three variants we show that a simple threat based strategy is optimal for the on line player and determine its competitive ratio We 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 we consider a bidirectional conversion prob lem in which the player may trade dollars for yen or yen for dollars