A Comparison Of Two No-arbitrage Conditions Under Nonlinear Trading Strategies

A comparison of two essential no-arbitrage conditions for the fundamental theorem of asset pricing was established in previous article,which trading strategy depends only linearly on the time variable .The two essential no-arbitrage conditions are so called the no free lunch with vanishing risk condition and the no good deal condition,respectively.However,the real market is complex and changeable,and the linear simulation cannot accurately reflect the relationship between these two basic no-arbitrage conditions.In this paper,under the index model closer to the real market,we will promote previous articles,namely when the time variable of trading strategy satisfies the exponential function models,the relationship between the two basic no-arbitrage conditions will be established from the perspective of random analysis.The research method in this paper is also applicable to the case where the trading strategy is the logarithm function of time variable t,that is,the logarithm function can be used to fit the market changes.In fact,the logarithm function can be viewed as the inverse of the exponential function.The paper is arranged as follows:In chapter one,we review the background of the regularization framework and briefly describe the significance.In chapter two,we build up a general continuous market based on the fundamental theorems of asset pricing.For this,we first introduce some basic concepts of the Fundamental Theorems of asset pricing and measure theory.Then,the definitions of these two basic no-arbitrage conditions are given respectively.In chapter three,we give a comparison of these two essential no-arbitrage conditions and study under what conditions they can be derived from each other.Firstly,the specific expressions of these two no-arbitrage conditions are given,and the no free lunch with vanishing risk condition is classified and scaled so as to be used in the subsequent mutual derivation.Next,we will use (?) to fit trading strategy changes.At the same time,if (?) ? 0,we can say the index trading strategy increases exponentially over time add.On the contrary,if (?)< 0,we say that the index trading strategy decreases over time.It^o stochastic calculus will be used many times.Meanwhile,Girsanov transformation and equivalent martingale measure will also be used.