We introduce a generalized quarter-symmetric metric recurrent connection and study its geometrical properties. We also derive the Schur?s theorem for the generalized quarter-symmetric metric recurrent connection.

The present paper attains a Harnack inequality for the option pricing (or Kolmogorov) equation with gradient estimate arguments. We then perform a no-arbitrage analysis of a financial market.

In this paper, we obtain Chen’s inequalities for submanifolds in (κ,μ)-contact space form endowed with a semi-symmetric metric connection.

The present paper considers a class of financial market with transaction costs and constructs a geometric no-arbitrage analysis frame. Then, this paper arrives at the fact that this financial market is of no-arbitrage if and only if the curvature 2-form of a specific connection is zero. Furthermore, this paper derives the fact that the no-arbitrage condition for the one-period financial market is equivalent to the geometric no-arbitrage condition. Finally, an example states the equivalence between the geometric no-arbitrage condition and the existence of the solutions for a maximization problem of expected utility.

We find a novel and intimate correspondence in the present paper between the martingale and one-parameter transformation group and develop a nonequilibrium geometric no-arbitrage principle to a frictional financial market via this correspondence. Further, we achieve a fundamental pricing theorem via a geometric pricing transform (generator). Finally, we derive that the nonequilibrium geometric no-arbitrage is equivalent to NFLVR in a frictionless financial market. In addition, we apply the nonequilibrium geometric no-arbitrage condition to a frictional financial market. At the end of this paper, a numerical example confirms the effectiveness of the nonequilibrium geometric no-arbitrage condition.