Stops and targets can’t create an edge: the optional stopping theorem in plain English

Educational only, not financial advice. Simulations and backtests are models, not promises. Check every rule with your own firm.

"Use a 1:3 reward to risk and you only need to win 25% of the time." True, and also exactly the problem. If your entries have no edge, every choice of stop and target leaves your average result at zero before costs and below zero after. Here’s why.

A coin-flip market

Imagine a price that moves up or down one tick at a time with equal probability. You buy, set a stop 10 ticks below and a target 30 ticks above, and wait. How often do you hit the target?

For a symmetric random walk, the chance of reaching +a before −b is b ÷ (a + b). So:

The win rate moves exactly enough to cancel the payoff. That isn’t a coincidence of these numbers. It’s a theorem.

The optional stopping theorem

A martingale is a process whose expected next value is its current value: a fair game. Doob’s optional stopping theorem says that if you stop a martingale using any rule that can’t see the future (a stop-loss, a profit target, a trailing stop, a time exit, "close if RSI crosses 70"), and the rule is bounded in a sensible way, then your expected value when you stop equals your starting value.

In trading terms: on a driftless price, no exit rule has positive expectancy. Exits change the shape of your results (win rate, size of wins and losses, skew, how long trades last), not the mean.

Then add costs

Every round trip pays a spread, commission and slippage. If your cost is c in price and your stop is S, your expectancy per trade is about −c ÷ S in R. A 0.5-point spread on a 10-point US500 stop is −0.05R a trade, whatever the target. Tighter stops make it worse: the same spread on a 5-point stop is −0.10R.

That’s why small-stop scalping systems look great before costs and bleed after them, and why the pass-odds calculator asks for costs in R.

What the research says

• Kaminski and Lo (2014) prove that under a random walk, a stop-loss rule always lowers expected return. Stops can only add value when returns are predictable, for example under momentum.
• Lo and Remorov (2017) extend this: with transaction costs, tight stops lose, and the benefit needs strong serial correlation.
• Warburton and Zhang (2006) model stop-loss, take-profit and breakout rules on a random walk: expected profit is set by drift; with zero drift the exits only trade win probability against win size.
• Lei and Li (2009) find stop-loss strategies neither reduce nor increase losses relative to buy-and-hold; any value is risk reduction, not return.
• Biondo et al. (2013) find random-entry strategies on major indices did about as well as standard technical rules, with less variability.

What this doesn’t say

• Exits still matter for risk. A stop caps the damage of one trade, and in a prop challenge that’s what keeps you inside the daily limit. Stops are risk management, not edge.
• If your entries have an edge, exits can help or hurt it. With momentum, letting winners run captures more of the drift; with mean reversion, a near target fits better. But the edge has to come from the entry, the market or a structural flow first.
• Markets aren’t perfect random walks. There are documented effects, such as intraday momentum into the close (Gao et al. 2018; Baltussen et al. 2021). The point is narrower: you can’t manufacture an edge from exit rules alone.

In our own testing

When we backtested Donchian channel exits, ATR and chandelier trailing stops, fixed 1:1 targets and midline exits on stocks and ETFs, the exit choice moved win rates around a lot and average results very little. After costs, the day-only versions were all negative. That’s what the theorem predicts. (Our tests are our own and not peer-reviewed.)

The practical takeaway

Pick stops for risk control and targets that fit how your edge plays out. Don’t pick them hoping to create one. If a backtest only works with one magic target, suspect curve fitting. And in a challenge, remember that a high win rate from a wide stop and a close target is still the same expectancy, now with a larger loss when it comes.

Sources

1. Williams, D. (1991). Probability with Martingales. Cambridge University Press. [Book]
2. Kaminski, K. M. & Lo, A. W. (2014). When do stop-loss rules stop losses? Journal of Financial Markets 18, 234–254. doi:10.1016/j.finmar.2013.07.001 [Peer-reviewed]
3. Lo, A. W. & Remorov, A. (2017). Stop-loss strategies with serial correlation, regime switching, and transaction costs. Journal of Financial Markets 34, 1–15. doi:10.1016/j.finmar.2017.02.003 [Peer-reviewed]
4. Warburton, A. J. & Zhang, Z. G. (2006). A simple computational model for analyzing the properties of stop-loss, take-profit, and price breakout trading strategies. Computers & Operations Research 33(1), 32–42. doi:10.1016/j.cor.2004.06.001 [Peer-reviewed]
5. Lei, A. Y. C. & Li, H. (2009). The value of stop loss strategies. Financial Services Review 18(1), 23–51. doi:10.61190/fsr.v18i1.4933 [Peer-reviewed]
6. Biondo, A. E., Pluchino, A., Rapisarda, A. & Helbing, D. (2013). Are random trading strategies more successful than technical ones? PLoS ONE 8(7), e68344. doi:10.1371/journal.pone.0068344 [Peer-reviewed]
7. Gao, L., Han, Y., Li, S. Z. & Zhou, G. (2018). Market intraday momentum. Journal of Financial Economics 129(2), 394–414. doi:10.1016/j.jfineco.2018.05.009 [Peer-reviewed]
8. Baltussen, G., Da, Z., Lammers, S. & Martens, M. (2021). Hedging demand and market intraday momentum. Journal of Financial Economics 142(1), 377–403. doi:10.1016/j.jfineco.2021.04.029 [Peer-reviewed]

"Peer-reviewed" means published in a refereed journal. Working papers and books aren’t peer-reviewed. Figures from "our own tests" are our backtests and simulations, not peer-reviewed research.