Asymptotic synthesis of contingent claims with controlled risk in a sequence of discrete‐time markets
研究了当离散时间模型概率律收敛到Black-Scholes-Merton模型且最大步长趋于零时，有界连续未定权益可被渐近合成并控制风险，使期望效用最大化消费者在离散时间经济中获得与连续时间极限相同的效用。
We examine the connection between discrete‐time models of financial markets and the celebrated Black–Scholes–Merton (BSM) continuous‐time model in which “markets are complete.” Suppose that (a) the probability law of a sequence of discrete‐time models converges to the law of the BSM model and (b) the largest possible one‐period step in the discrete‐time models converges to zero. We prove that, under these assumptions, every bounded and continuous contingent claim can be asymptotically synthesized, controlling for the risks taken in a manner that implies, for instance, that an expected‐utility‐maximizing consumer can asymptotically obtain as much utility in the (possibly incomplete) discrete‐time economies as she can at the continuous‐time limit. Hence, in economically significant ways, many discrete‐time models with frequent trading resemble the complete‐markets model of BSM.