套利与记账单位的关系

The Relation of Arbitrage and Numeraire

通常说一策略是套利策略,是指其初始资产为0,而最后的资产以正概率严格大于0,也就是无中生有。如果只是看净收益是否严格大于0还不能判断该策略是否是套利策略。当策略的初始资产不是0时,如何判断该策略是否是套利策略?将就套利给出另一种定义,并证明一策略是否为套利策略只要判断其折准净收益是否以正概率严格大于0。给出了原市场和不同折准市场下无套利数学表达式的等价形式,并给出了在选择不同的记账单位作折准及对策略的不同要求(如:可取,折准可取,折准可取等)时无套利的数学表达式的等价形式。

Generally speaking,a strategy Ψ is said to be an arbitrage if it makes something out of nothing without risk,i.e V_T(Ψ) is strictly positive with positive probability and with its initial wealth 0.If the net capital gains are strictly positive with positive probability,we can't tell whether it is an arbitrage.How do you tell whether a strategy is an arbitrage with its initial wealth unequal to 0? We give another definition of arbitrage and have shown that judging whether a strategy is an arbitrage is to judge whether its discounted gains are strictly positive with positive probability. We give the equivalent expression forms in mathematics of the original market and different discounted markets with respect to no arbitrage ,and correspondingly we get equivalent expression forms in mathematics with respect to no arbitrage with different strategy requirements.