Notion of Neutrosophic Risk and Financial Markets Prediction

The efficient market hypothesis based primarily on the statistical principle of Bayesian inference has been proved to be only a special-case scenario. The generalized financial market, modeled as a binary, stochastic system capable of attaining one of two possible states (High  1, Low  0) with finite probabilities, is shown to reach efficient equilibrium with p . M = p if and only if the transition probability matrix M2x2 obeys the additionally imposed condition {m11 = m22, m12 = m21}, where mij is an element of M (Bhattacharya, 2001). [1] Efficient equilibrium is defined as the stationery condition p = [0.50, 0.50] i.e. the state in t + 1 is equiprobable between the two possible states given the market vector in time t. However, if this restriction {m11 = m22, m12 = m21} is removed, we get inefficient equilibrium  = [m21/(1-v), m12/(1-v)], where v = m11 – m21 may be derived as the eigenvalue of M and  is a generalized version of p whereby the elements of the market vector are no longer restricted to their efficient equilibrium values. Though this proves that the generalized financial market cannot possibly get reduced to pure random walk if we do away with the assumption of normality, it does not necessarily rule out the possibility of mean reversion as M itself undergoes transition over time implying a probable re-establishment of the condition {m11 = m22, m12 = m21} at some point of time in the foreseeable future. The temporal drift rate may be viewed as the mean reversion parameter k such that kjMt  Mt+j. In particular, the options market demonstrates a rather perplexing departure from efficiency. In a Black-Scholes type world, if stock price volatility is known a priori, the option prices are completely determined and any deviations are quickly arbitraged away. Therefore, statistically significant mispricings in the options market are somewhat unique as the only nondeterministic variable in option pricing theory is volatility. Moreover, given the knowledge of implied volatility on the short-term options, the miscalibration in implied volatility on the longer term options seem odd as the parameters of the process driving volatility over time can simply be estimated by an AR1 model (Stein, 1993). [2] Clearly, the process is not quite as straightforward as a simple parameter estimation routine from an autoregressive process. Something does seem to affect the market players‟ collective pricing of longer term options, which clearly overshadows the straightforward considerations of implied volatility on the shortterm options. One clear reason for inefficiencies to exist is through overreaction of the market players to new information. Some inefficiency however may also be attributed to purely random white noise unrelated to any coherent market information. If the process driving volatility is indeed mean reverting then a low implied volatility on an option with a shorter time to expiration will be indicative of a higher implied volatility on an option with a longer time to expiration. Again, a high implied volatility on an option with a shorter time to expiration will be indicative of a lower implied volatility on an option with a longer time to expiration. However statistical evidence often contradicts this rational expectations hypothesis for the implied volatility term structure. Denoted by ‟t (t), (where the symbol ‟ indicates first derivative) the implied volatility at time t of an option expiring at time T is given in a Black-Scholes type world as follows: