  A pricing principle is introduced for non-attainable $q$-exponential bounded
contingent claims in an incomplete Brownian motion market setting. The buyer
evaluates the contingent claim under the ``distorted Radon-Nikodym derivative''
and adjustment by Tsallis relative entropy over a family of equivalent
martingale measures. The pricing principle is proved to be a time consistent
and arbitrage-free pricing rule. More importantly, this pricing principle is
found to be closely related to backward stochastic differential equations with
generators $f(y)|z|^2$ type. The pricing functional is compatible with prices
for attainable claims. Except translation invariance, the pricing principle
processes lots of elegant properties such as monotonicity and concavity etc.
The pricing functional is showed between minimal martingale measure pricing and
conditional certainty equivalent pricing under $q$-exponential utility. The
asymptotic behavior of the pricing principle for ambiguity aversion coefficient
is also investigated.
