Building good deals with arbitrage-free discrete time pricing models

Recent literature has proved that many classical very important pricing models of Financial Economics (Black and Scholes, Heston, etc.) and risk measures (VaR, CVaR, etc.) may lead to “pathological meaningless situations”, since there exist sequences of portfolios whose negative risk and positive expected return are unbounded. Such a sequence of strategies will be called “good deal”.This paper focuses on a discrete time arbitrage-free and complete pricing model and goes beyond existence properties. It deals with the effective construction of good deals, i.e., sequences (ym)m=1∞ of portfolios such that(VaR(ym),CVaR(ym),Expected_return(ym))(VaR(ym),CVaR(ym),Expected_return(ym))tends to (− ∞ , − ∞ , + ∞). Under quite general conditions the explicit expression of a good deal is given, and practical algorithms are provided. The sensitivity of our results with respect to measurement errors or dynamic changes of the parameters is analyzed, and numerical experiments are presented with the binomial model.