On Quadratic Cost Criteria for Option Hedging

Consider an option with maturity time T corresponding to a contingent claim H (in an incomplete market). A fair hedging price for H should take into account an optimal dynamical hedging plan against H. Let C_t be the cumulative cost and ℑ_t be the set of events of the history up to time t. You can choose the plan at time t such that you minimize

(i) E[{C_t+1 − C_t}² ∣ ℑ_t],

(ii) E[{C_T − C_t}² ∣ ℑ_t], or

(iii) E[{C_T − C₀}²].

Sufficient conditions on the underlying stochastic process (in discrete time) are provided such that the fair hedging price does not depend on the choice of (i), (ii), or (iii), which fact should increase its acceptability.