Welcome to the first installment in a series on all things Probability—a field that epitomizes the mathematical understanding of uncertainty. Probability is more than a set of mathematical rules; it's the study of randomness and patterns that exist within that randomness. In this series, we’ll explore all facets of Probability as they relate to options trading.

Expected Value (EV) is a core concept in probability theory. In options trading, it is a practical tool to understand market expectations and optimize trading decisions, which is why we’ve made it central to Trade Idea selection. By quantifying potential outcomes and assessing the potential risk of a trading strategy, EV serves as a mathematical compass aiding traders in decision-making.

Given its recent addition to Trade Ideas in both raw form and use in the Alpha rank, I’d like to start this series by visualizing EV and studying its effectiveness. We analyzed approximately 130k Trade Ideas generated over a 30-day period across 155 different underlying symbols. Enjoy!

## Expected Value: From Theory to Practice

Expected Value serves as a mathematical abstraction representing future possibilities and outcomes. It encapsulates the mean of a probability distribution, providing an aggregate view of potential results. Its applications in finance, economics, and risk management make it a cornerstone concept. Whether predicting market trends or assessing insurance risks, EV's role extends across disciplines, offering a unified approach to understanding uncertainty.

### Mathematical Definition of EV

Expected Value is defined as the sum of all possible values, each weighted by its probability of occurrence. Mathematically, it is expressed as:

##### \[EV = \sum (x_i \cdot p_i)\]

where \(x_i\) is a possible value, and \(p_i\) is its probability. This equation elegantly captures the essence of long-term averages, allowing for the modeling of complex random processes. It forms the basis for many statistical procedures and plays a pivotal role in areas like actuarial science and operations research.

In financial context, EV elucidates the long-term average profit or loss one can anticipate, reflecting the law of large numbers. This law states that as the number of occurrences grows, the sample mean approaches the population mean, or EV. It's a concept deeply rooted in infinite occurrences, providing a foundation for investment strategies. By considering all potential outcomes and their probabilities, investors and economists can forecast future trends, minimize risks, and optimize returns.

## Methodology

We performed an analysis of Trade Ideas from approximately mid-July 2023 to mid-August 2023. The set of all Trades Ideas at the end of each trading day was retained in a separate database (the last calculation is performed at 3:55 PM EST). The underlying’s closing price was recorded and the Profit/Loss was calculated for each trade at expiration. Days to expiration ranged from 1 to 30 days.

It is important to note that the Profit/Loss figures represented here assume cash settlement, determined by where the underlying price expired relative to the strikes. There is no assumption of position management, assignment, early exercise, or transaction costs.

For example, if the underlying price expired below the breakeven price of a short put spread but above the long put strike, it registered as a partial loss. At the time of this study, only iron condor, short put spread, and short call spreads are included in Trade Ideas.

Any Trade Idea that crossed earnings (excluded by default in the UI) was excluded from this study – we’ll see why later.

In order to compare Profit/Loss to EV, we had to develop a method that would make them relatable over many occurrences. Each trade in the study has a theoretical EV as well as a P/L, but in order to study them both we had to look at them independently but relative to their value.

In other words, it doesn’t make sense to compare an individual trade’s expected value against its profit or loss because, by definition, EV is telling us what happens over many similar trades and is not directly relevant to a single trade.

The other difficulty is that we don’t have infinite occurrences of apples to compare with apples. Almost no trade setups are identical. For example, all other things being equal, advancing just one day in time completely changes the probability calculation used for EV.

To compare Profit/Loss to theoretical EV, our only choice is to look at the aggregate results for all occurrences. If we divide the profit range into “buckets,” say $10 wide, we can see how many trades were theoretically expected to fall within that range and compare it to how many trade's profit/loss values actually ended in that range.

We will first examine the raw trade counts to get an idea of what the distributions look like. Then, we will plot the real average profit for a subset of the “tradable” EV range to determine Trade Ideas’ effectiveness at calculating EV.

## Results: Comparing Profit/Loss to EV

Although they’re not directly comparable in this way, this graph plots trade counts that fall within $10 ranges for both EV and P/L to gain an understanding of their distribution.

For example, nearly 20k trades had a pre-trade EV between +$10 and +$20, while almost 10k trades realized a profit in the same range.

Plotting high counts of predicted EV values clearly shows central tendency, with a concentration around zero. This makes sense because the Efficient Market Hypothesis (EMH) states that all information is already priced into the market and it is therefore impossible to find excess value. What that means for options traders is for *most *trade setups chosen at random, the EV will most likely be less than or very close to zero.

In the aggregate, the total profits of all winning traders are theoretically balanced by the total losses of all losing traders. This creates a dynamic reminiscent of a zero-sum game, where the total gains across all traders are offset by the total losses, resulting in a net zero effect.

But in practice, we as options traders don’t actually believe that. We believe the market is not perfectly priced and that there are inefficiencies we can exploit. The Profit/Loss distribution does not appear to be normally distributed, which we might have expected. But these are not just any random trades; they’re Trade Ideas. While Trade Ideas does not exclusively list +EV trades, the results are definitely skewed that way.

And if the returns are *not *normally distributed, this suggests perhaps the EV is understating the true profit potential and there is merit to the notion that Trade Ideas are indeed isolating trade setups that are market-beating.

However, we must also acknowledge the influence of many factors. According to the Central Limit Theorem, if many independent and identically distributed random variables contribute to Profit/Loss, their sum will tend toward a normal distribution, regardless of the original distribution of the individual variables.

The acknowledgment of the influence of many factors is a fundamental aspect of complex systems and decision-making processes, recognizing that outcomes are often the result of a complex interplay of variables rather than a singular cause-and-effect relationship.

Translated, this roughly means that maybe we just don’t have enough data yet and no matter what we think we see in the graphs, the Profit/Loss will just end up being normally distributed anyway. But for now, we press on.

### Trading Positive Expected Value (+EV)

For the next part of this study, we will focus on the positive \(x\) domain, or the set of all \(x\)-values greater than zero, \(\{x | x > 0\}\). This is not a completely objective study, since we are only evaluating the output and performance of Trade Ideas and not the market as a whole. Trade Ideas is heavily skewed toward finding +EV trades because we, as traders, are rarely going to be looking for trades that exhibit negative return behavior.

A benefit of looking at the set of \(x\) greater than zero means we’re not limited to choosing a distribution type to best represent the data that is capable of representing all real numbers for \(x\).

We used the Akaike information criterion (AIC) estimator of prediction error to assess a collection of models for the data. AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection (the lower the number, the better).

For Profit/Loss:

**Gamma Distribution: AIC = 855.994 (best fit)**- Exponential Distribution: AIC = 862.105
- Beta Distribution: AIC = 862.818
- Laplace Distribution: AIC = 907.471
- Normal Distribution: AIC = 921.995

For Expected Value:

**Gamma Distribution: AIC = 746.468 (best fit)**- Beta Distribution: AIC = 795.705
- Exponential Distribution: AIC = 891.229
- Laplace Distribution: AIC = 959.089
- Normal Distribution: AIC = 1043.895

When data follows a gamma distribution pattern, it conveys several specific mathematical and statistical properties: positive values, right-hand skewness, and asymmetry (among others). This is interesting, but not incredibly useful outside of confirming the +EV skew. It does, however, show us that there is an intersection or convergent local maxima between EV and Profit/Loss somewhere between $0 and $50. Let’s see if we can find it.

## Results: Average P/L vs. Perfect EV

Assume the profit range is divided into $10 “buckets.” If EV represents the average outcome over many occurrences, then it stands to reason if you averaged the theoretical EV for every trade that falls within each bucket, the average Profit/Loss would be equal to the average EV. Stated another way, assume there are infinite trades made within each EV bucket range. Assuming EV values are evenly distributed throughout that range, the average of all EVs in the bucket will be its midpoint. We can then very simply plot the theoretically perfect expected value line as \(f(x) = x\).

We don’t have infinite trades, but we do have a sufficiently large dataset. If we take the average Profit/Loss of all trades whose EV falls within the range of each bucket, we can compare the Trade Idea outcome against the theoretically perfect expected value and see if Option Alpha's EV calculation is in the ballpark.

We took a snapshot of a $160 EV range, from -$30 to $130 (the sample counts really start to thin out above the $130 range). Although we are only interested in the domain where \(x\) is positive, we included a handful of data points to illustrate how the theoretical progression continues into negative-\(x\). Included above each data point is the sample count, or how many trades had an EV within the lower and upper bounds to its left and right, respectively.

The graph suggests that the average P/L predicted by Trade Ideas is in fact following the theoretical EV line, which means it is accurate at least for this sample data and this market environment. For reference, the VIX remained under $20 during the duration of this study and the market was relatively quiet.

Notice, too, how the spline extends above the perfect EV line between $20 and $60. This appears to be right at the intersection of the EV and Profit/Loss gamma distributions, or at the very least the local maxima in the positive-\(x\) domain. It is the “sweet spot” for calculated EV.

One key thing to note is that these trades do not include any Trade Ideas that crossed earnings or ex-dividend dates. We’ve seen time and time again how either of these events can significantly impact probabilities, so we choose to ignore them altogether.

Watch what happens when we run the same study only using Trade Ideas that have an earnings event or ex-div prior to expiration:

The sample counts increase across some buckets and we’re able to populate the rightmost data points, but the Profit/Loss landscape is *dramatically *different. The Average P/L line is not correlated to the perfect EV line whatsoever. Expected Value, and probabilities in general, cannot be relied upon for accurately estimating future returns.

## Conclusions

In this study, we examined Expected Value (EV) as it applies to Trade Ideas. The analysis spans from theoretical exploration to practical implications, leveraging mathematical concepts and empirical data.

The data analysis reveals a nuanced relationship between EV and Profit/Loss (P/L). Through the utilization of various statistical distributions, the study finds that the gamma distribution best fits both Profit/Loss and Expected Value for positive \(x\)-values. This results highlight the inherent skewness and asymmetry within trading decisions, confirming the positive Expected Value skew, but the application and interpretation extend beyond mere statistical compliance.

One of the most salient findings is the "sweet spot" identified for calculated EV exists in the $20-60 calculated EV range, the bounds where trades are outperforming theoretical expected value. This illustrates not only the accuracy of Trade Ideas in predicting average P/L but also underscores a region of optimal investment decision-making, a potential guideline for traders.

However, this study is not without its caveats. The dynamic and complex nature of the market implies that these conclusions might be specific to the observed data and market conditions at the time of this research. Additionally, the exclusion of specific market events, such as earnings or ex-dividend dates, brings forth a significant divergence in the Profit/Loss landscape. This variation emphasizes the critical influence of market events and the intricate factors that contribute to the trading decision process.

In conclusion, our investigation into the EV illuminates the multifaceted relationship between theory and practice. It offers traders a glimpse at a valuable mathematical tool that, when applied judiciously, can enhance their ability to predict trends, minimize risks, and optimize returns. The findings present a promising avenue for further research and development of more adaptive and nuanced trading strategies.