Why a Discounted Stop-Loss Rule Does Not Establish Option Arbitrage

Summary

The document examines a proposed strategy for writing a call option: hold the stock when its price exceeds a time-adjusted strike threshold and otherwise sell it. The author tries to address the time value of money by lowering the threshold toward expiry and calculates a discounted terminal profit using the Black–Scholes call price. Numerical experimentation is offered as informal support for the belief that this amount is nonnegative, leading to the question of whether the strategy creates arbitrage.

The excerpt contains the question but no answer, so it does not resolve the apparent profit or identify the flaw in the argument. It explicitly sets aside practical execution issues, including the impossibility of trading exactly at the threshold and the possibility of infinitely many threshold crossings under continuous paths. Its tentative calculation therefore is not evidence of an executable, self-financing arbitrage strategy.

Key ideas

• The proposed call-writing rule switches stock ownership when price crosses a time-dependent threshold.
• The author discounts the strike threshold over time to account for interest rates.
• A Black–Scholes-based terminal profit expression motivates the arbitrage question.
• The claim of nonnegative profit rests on parameter experimentation rather than a proof in the excerpt.
• Execution at the exact threshold and repeated crossings are acknowledged but excluded from the analysis.

Tags

# Isn't this modified stop-loss strategy an arbitrage?


# Isn't this modified stop-loss strategy an arbitrage?












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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.