In John Hull's The Book, section 18.3 he briefly discussed a stop-loss strategy for writing a call option: buy one share of stock whenever $S_t>K$ and sell it otherwise (except at time $0$: if $S_0\le K$ we do nothing), and hence the call writer will own the stock in case $S_T>K$ and not otherwise, making an overall profit of $c_0 - \max(S_0 - K, 0)\ge 0$. However, he pointed out two reasons why this isn't the case: 1). it ignores the time value of money; 2). it is impossible to buy/sell exactly at $K$; also, if the stock price follows the Wiener process it can happen that $\#\{t\mid S_t=K, t\in[0,T]\}=\infty$.
Now, let's temporarily ignore the problem 2) and assume we have a super trader who can carry out the order at the exact price we want. The main problem is then the first one. However, I think it can easily be circumvented by "discounting" our stop-loss rule towards time $T$: whenever $S_t> Ke^{-r(T-t)}$, buy one share and sell it otherwise (except at time $0$: if $S_0\le Ke^{-rT}$ we do nothing). Thus, in the absence of transaction costs, it can be verified that our net profit discounted towards $T$ is $$e^{rT}\cdot\text{BS Price}-\max(S_0e^{rT}-K,0)=S_0e^{rT}\Phi(d_1)-K\Phi(d_2)-\max(S_0e^{rT}-K,0).$$ Although its sign is not obvious, by experimenting with several sets of parameters I become fairly convinced that it is non-negative. But then such a strategy would in effect become an arbitrage in this BS world. What goes wrong?
