On reselling of European option

On Black and Scholes market investor buys a European call option. At each moment of time till the maturity, he is allowed to resell the option for the quoted market price. A model is proposed, under which there is no arbitrage possibility. It is shown that the optimal reselling problem is equivalent to constructing nonrandom two dimensional stopping domains. For a modified model of the market price, it is shown that the stopping domains have a threshold structure.