Probability Theory: How Many Trades Does it Take to be Successful?

If you’ve been around options trading long enough, you’ll inevitably hear phrases like, “Let the probabilities play out,” or, “Trade small, trade often.” The rationale given is often the Law of Large Numbers, meaning a sufficient number of trades is required to realize those probabilities. But exactly how many trades do you need to make? 

Trade small, Trade often: Does it work?

This research was inspired by a question from our very own Community. I was directly asked how many trades it takes for “probabilities to play out” when trading a strategy through Trade Ideas. I didn’t have a great answer at the time, but the one I gave related the answer to flipping a fair coin:

When flipping a fair coin (binomial trials with a 50% chance of landing heads or tails), to be within 1% accuracy of the 50/50 average with 95% confidence, approximately 9,603 coin flips are needed.

• 99% confidence, ~16,587 flips
• 95% confidence, ~9,603 flips 
• 90% confidence, ~6,763 flips 
• 75% confidence, ~3,308 flips
• 60% confidence, ~1,770 flips 

There are approximately 252 trading days in a year. A trader would need to enter 38 trades per day to achieve that many trades. In practice, if we’re willing to give up a little bit of confidence and accuracy to be “close enough” to the average, how many trades do we actually have to make?

The tastytrade team estimates ~1,000 occurrences. That number roughly reflects the accuracy of being within 3% of the average with 95% confidence:

• 99% confidence, ~1,843 flips
• 95% confidence, ~1,067 flips
• 90% confidence, ~751 flips
• 75% confidence, ~367 flips
• 60% confidence, ~196 flips

In Figure 1, we demonstrate a computer-generated model of flipping a fair coin. We flipped 10 coins 1,000 times, represented as 10 unique lines on the graph. Our “win rate” on the y-axis is represented by the percentage of flips that landed on heads. 

The graph above was generated using pseudo-random number generation with entropy. It’s as close to real, near-perfect randomness as we can reasonably hope to achieve.  

The results are very interesting because even though we implicitly know and understand a fair coin has a 50% chance of landing on heads or tails, the sequence and flip count it takes to achieve convergence to 50% is very different between seemingly identical trials.

Would you ever have guessed that it’s possible to have a 60% win rate after 100 flips or trades? Or that it’s possible to have a 38% win rate after 250 trades when the probability of winning any individual toss is nearly guaranteed to be 50%? That’s what this graph shows.

Around 500 flips is where we begin to see convergence and at 1,000 flips we start to see tightening around 50%. Let’s increase the flip count to 5,000 (5x) and re-run the test.

Figure 2 shows us 10 new, independently run tests that differ from Figure 1. Visually, the statistics math seems spot on at +/- 3% around 1,000 trades. But even so, the banding around 50% looks very different in this graph than the previous one. It does appear, however, that beyond 2,000 flips there’s no dramatic improvement in convergence. So we should expect our own options probability model to also exhibit similar behavior between 1,000 to 2,000 trades.

The more we trade targeting a specific probability, the closer we will get to our target on average. However, there is a major implicit assumption that the probability model being used is accurate and reflective of the environment. Let’s see if we can translate this into trading terms and validate both Trade Ideas and its probabilities.

Options Trading Probability of Success

We already know our probability scores assigned to options trades are imperfect because they’re using an underlying model (Black-Scholes) on a market that doesn’t perfectly mimic a normal distribution.

Even so, we’ve demonstrated in previous research on the "Moment in Time" problem that Trade Ideas probabilities are reasonably accurate in the majority of cases. So let’s investigate if we’re experiencing similar probabilistic convergence in Trade Ideas as we are with the fair coin.

In mid-July 2023, we started capturing an end-of-day snapshot of Trade Ideas as it’s presented at 3:55 PM ET each trading day inside of Option Alpha. The database is massive and now contains over 3 million trades. It’s unrealistic to think a retail trader can trade all of them to achieve Law of Large Numbers probabilities in a few short months. 

We need to distill Trade Ideas down to something more realistic that’s achievable at a retail level while remaining as unbiased as possible in trade selection.

Trade Selection Criteria

In an attempt to demonstrate convergence to an average probability, expressed as a win percentage, we’re going to borrow from the results of our past research on optimal trade criteria while remaining as broad and generalized as possible. We’re going to select:

• Days to expiration between 21 and 45
• Probability of Max Profit between 65% and 75%
• Alpha greater than zero
• No trades that cross earnings or ex-dividend events

The probability range chosen here is somewhat arbitrary and not necessarily a trading recommendation for profitability. It was chosen in part because of the high sample counts we have in that range and also because it exceeds 50% with enough margin that a score in that range must be non-arbitrary, better than random chance. Any probability model must be accurate to demonstrate convergence at a 70% midline.

The sample size is ~200k unfiltered trades meeting the above criteria. Although we recently added debit spreads to Trade Ideas, the accumulated data in this study consists of iron condors (IC), short put spreads (SPS), and short call spreads (SCS). The breakdown of IC, SPS, and SCS meeting our criteria is 9%, 44%, and 47% respectively.

We will randomly select, at maximum, 1 trade per symbol per expiration, which yields somewhere between 2,000 and 2,500 tradeable opportunities across ~150 symbols during our time horizon. We’re attempting to provide the fairest assessment we can, so the criteria for trade selection are minimal. 

Using this methodology, we will demonstrate what happens during random selection over 100, 250, 500, 1,000, and 2,000 trades evenly distributed between July 2023 and December 2023.

Temporal Diversification

Sequencing risk in options trading is predominantly concerned with the concentration of risk at specific future points in time. In options trading, where contracts have fixed expiration dates, the risk is that market conditions at or near these expiration dates may not align with the anticipated strategy outcome. This concentrated temporal risk can significantly amplify losses, especially if multiple positions are set to expire or require action simultaneously under unfavorable market conditions. 

To mitigate this, it's crucial to diversify not just in terms of asset selection but also in timing, ensuring that all options positions don't converge on the same expiry or decision point. By distributing trades across different time horizons, traders can reduce the impact of adverse market movements at any single point, thereby diluting the concentration of risk and creating a more resilient trading strategy over time.

To put it simply, we want to evenly distribute the trade counts of our experiment over time. Trade Ideas can (and often does) choose the same strategy setup in subsequent days: same underlying, same strikes, same everything, just one day closer to expiration. 

By allowing only 1 trade per symbol per expiration that matches our criteria, chosen randomly, we are distributing the time risk. It does us no good to look at the probability of 100 or 1000 trades that all expire on the same day. Instead, we want to observe what happens over many “coin flips” in ever-changing market conditions.

Law of Large Numbers: Trade Count Impacts Win Rate

Each line in each graph below represents an independent sampling. Five tests with 10 independent samples each represent 50 total trials. Similar to our coin demonstration, the y-axis shows the running win rate percentage.

Does Options Trading Work?

Let’s talk about convergence to the 70% win rate line. These graphs bear a striking resemblance to the coin-flipping example, which is a good thing. 

The most telling graph, in my opinion, is Figure 3 (100 trades). In a single image, it demonstrates that depending on the “luck of the draw” a trader can either have an 85% win rate or a 55% win rate after 50 trades when targeting a 70% probability. Or, similarly, a sequence of 100 trades could track a 70% win rate from the outset. Any of these scenarios are possible.

I imagine many new options traders who end up falling into the 55% win rate bucket may become disheartened and want to quit altogether, especially if that extra 10-15% of losses they weren’t expecting were large drawdowns.

I can sympathize with that rationale, especially when solvency is an issue. No one wants to continue making deposits to a brokerage account when we should be withdrawing our winnings.

However, the results from Figures 5, 6, and 7 are very clear and compelling: unwavering consistency pays off in the end. One thousand trades is absolutely within +/- 3% of the desired 70% average. To achieve that in 1 year would be approximately 4 trades per day over 252 trading days.

Consistent + Persistent = Success

After staring at these graphs for the past week, I keep thinking about the number of times I’ve read or heard someone say: “I made [some small number of] trades over the past month and I lost money. This doesn’t work.”

It’s like hearing someone say, “I went to the gym every day this month and didn’t lose any weight. This doesn’t work!” Or, “I started a new diet this past week but gained weight instead. This doesn’t work!” How many times have you heard those statements in your life?

It’s easy to tell those people to stick with it, you’ll see results in time – we all understand that it’s possible because we’ve witnessed it. But just as there’s a difference between working out and working out intelligently, or following a diet vs. eating well and living a healthy lifestyle, there’s a massive difference between trading more and trading well.

At Option Alpha, we try to drastically reduce the barrier to entry for trading well and finding good trades by filtering Trade Ideas. Accurate probabilities are only one piece of the puzzle, but a very important piece.

How Many Options Trades Does it takes to be Successful?

In this article, we’ve demonstrated that temporally diversified, randomly selected Trade Ideas converge on a desired probability with sufficient confidence after approximately 1,000 trades. We’ve attempted to highlight the relationship between the number of trades and the realization of probabilistic outcomes in options trading.

Our research indicates that convergence to a desired win rate, modeled in this study as a 70% probability of success, occurs after approximately 1,000 trades. This finding aligns with the theoretical underpinnings of the Law of Large Numbers, suggesting that a sufficiently large number of trades is required to average out variability and converge to the expected probability. This outcome also aligns with our statistical expectations for accuracy and confidence. 

The inclusion of temporal diversification in trade selection highlights the importance of distributing trades across various time horizons. This approach mitigates sequencing risk and contributes to the robustness of the strategy, ensuring that our findings are reflective of a real-world trading environment with varying market conditions.

The study underscores the necessity of patience and persistence, demonstrating that early variances in win rates are to be expected. The results advocate for a disciplined approach to trading, emphasizing the importance of sticking to a strategy long enough to allow the probabilistic models to manifest their predictive power.

The convergence observed in our tests validates the probability model employed by Trade Ideas. This suggests that the model, despite its inherent assumptions and limitations, is capable of providing a reasonably accurate reflection of the market dynamics, proving useful for traders seeking to make informed decisions based on probabilistic assessments.

In conclusion, this research provides a quantitative foundation for understanding the relationship between trade volume and probabilistic outcomes. It underscores the importance of a disciplined, well-informed approach to trading, emphasizing the need for a substantial number of trades to realize the expected probabilistic outcomes.