Differentiation of nonnegative measurable function Choquet integral over real fuzzy measure space and its application to financial option trading model

Fuzzy measure shift differentiation of the Choquet integral for a nonnegative measurable function taken with respect to a fuzzy measure over a real fuzzy measure space is proposed. It is applied to financial engineering. First, a real interval limited Choquet integral for a nonnegative measurable function taken with respect to a fuzzy measure over a real fuzzy measure space is given, then a fuzzy measure left shift differential coefficient, a fuzzy measure right shift differential coefficient, a fuzzy measure shift differential coefficient, and a fuzzy measure shift derived function of the real interval limited Choquet integral for a nonnegative measurable function over a real fuzzy measure space along the domain are defined by the limitation process of a fuzzy measure shift. Two examples of a fuzzy measure shift differentiation are given, where fuzzy measure distributions are either a continuous distribution or a discrete distribution, to understand the notion of the fuzzy measure shift differentiation. Moreover, they are applied to financial option trading. The pricing models of a European call option premium and a European put option premium are defined using the real interval limited Choquet integral for a nonnegative measurable function over a real fuzzy measure space. Then, the distribution of underlying securities of an option trading at the expiration date is given as a λ-fuzzy measure, where the total fuzzy measure is equal to one. An important risk index, the delta, which is the rate of change of the premium with respect to underlying security price is defined using the fuzzy measure shift differentiation of the real interval limited Choquet integral for a nonnegative measurable function over a real fuzzy measure space. Finally, these option trading models based on the real interval limited Choquet integral over a real fuzzy measure space is tested with the real market data and is compared with the popular option trading model based on the probability measure and logarithmic normal distribution defined by Black and Sholes (1973)