Welcome to the second installment of our series exploring all things options math and probabilities. In Trade Ideas Probability & Performance: Expected Value vs. Realized P/L, we analyzed data for over 130k trades to visualize the accuracy of EV using Option Alpha’s Trade Ideas.

But what if you don’t have access to Option Alpha or the resources to calculate Black-Scholes EV? What can you do on your own?

We believe in helping everyone trade smarter, not just traders using Option Alpha’s autotrading platform. So, we will share the EV calculations you can use with the information available in any brokerage platform. Specifically, we will look at the “simple” EV cases from Understanding Alpha and Expected Value.

We’ve introduced how EV is calculated in Trade Ideas 2.0 by showing how Expected Value is built on layers of understanding from multiple EV calculations. Today, we’ll investigate if those simpler metrics can stand on their own.

## Methodology

In Understanding Alpha and Expected Value, we discussed three types of EV:

- Max profit/max loss EV (good)
- EV with an average estimated outcome of the partial profit/loss region (better)
- EV as a discrete random variable (best)

For simplicity and brevity, we’ll refer to those three methods as **Simple**, **Simple-Partial**, and **Real EV** methods, respectively.

First, we’ll describe the formulas for the simple EV methods as they relate to options trading and briefly review the more advanced method of EV as a discrete random variable. Next, we’ll analyze all three distributions and plot their performance against the theoretical EV line, similar to our first Research Insights study on EV. This time, however, we will observe both the negative and positive \(x\) ranges equally by plotting -100 to +100 EV.

Finally, we will analyze the results, attempt to rationalize what the graphs show, and compare the performance of the three methods. Our goal is to determine whether the simple methods of EV are worth using and, more importantly, actionable in our trading.

The data set consists of Trade Ideas captured at 3:55 PM ET every trading day. The sample count under observation in this study has increased to over 400k theoretical trades since mid-July. These trades capture the entry criteria of positions that have technically “expired,” meaning the profit/loss can now be calculated based on the underlying price at expiration, and the Expected Value’s performance can be evaluated.

We assume cash settlement without exercise risk to calculate variable wins or losses at expiration. There is no assumption of position management, assignment, early exercise, or transaction costs. We also omit trades that cross expiration or ex-dividend dates, phenomena that we’ve shown significantly impact probabilistic outcomes.

As of this writing, Trade Ideas is calculated for defined-risk Bull, Bear, and Neutral positions on short put spread, short call spread, and iron condor setups, respectively.

The data set spans 1 - 45 days to expiration over 155 symbols. Sample counts across symbols are not evenly distributed.

The primary objective of Trade Ideas is to identify and present mathematically favorable edges in the market while filtering out unfavorable opportunities. This equates to selection skew toward symbols that exhibit desirable traits. Some symbols appear much more often than others in this regard, but that is outside the scope of this article and a topic of future discussion.

## Simple EV: Maximum Outcomes

Simple EV, as the name implies, is the fundamental approach to calculating Expected Value when there are only two possible outcomes: maximum profit or maximum loss.

This process is most often used to study casino-like games where only losing and winning a fixed dollar amount is possible, usually as a bet and a fixed multiple of a bet, respectively. In that sense, it can also be applied to an investment “game” where the player only expects binary outcomes, such as holding an options trade until expiration.

As promised, we want to use data available to all options traders to figure this out, so we’ll use the Greek delta as a measure of probability. Option delta measures an option price’s sensitivity relative to a change in the underlying asset's price. With this relationship, we can infer the likelihood of various outcomes and calculate an expected return.

If we have a random variable \(X\) with exactly two possible outcomes, maximum profit \(\text{MP}\) and maximum loss \(\text{ML}\), with corresponding probabilities \(p\) and \(1-p\), then the expected value \(E[X]\) can be calculated as follows using delta as a proxy for the probability of expiring out-of-the-money (OTM):

\[E[X] = p \cdot \text{MP} + (1-p) \cdot \text{ML}\]

Here, \(p\) is the probability of achieving the maximum profit, and \(1-p\) is the complementary probability, or the probability of incurring the maximum loss.

### Simple EV: Strategy Formulas

This form of EV assumes a binary outcome, where either full profit or full loss is realized at expiration. Simple EV is what most retail traders have access to (unless you subscribe to data feeds and are executing your own code to calculate probability distributions). It is also the most often cited example when explaining Expected Value, even though it doesn’t fully encompass the nuances of options trading.

The calculations for the three strategy types are relatively straightforward. For a short put spread the expected value \(E[X]\) can be mathematically represented as:

\[E[X] = (1 - |\Delta_{\text{SP}}|) \times \text{MP} + |\Delta_{\text{SP}}| \times \text{ML}\]

where \(|\Delta_{\text{SP}}|\) denotes the absolute value of the delta for the short put, \(\text{MP}\) represents the maximum possible profit, and \(\text{ML}\) represents the maximum possible loss. Recall that delta is an estimate of in-the-money (ITM) probability, so its complement, \(1 - |\Delta_{\text{SP}}|\), is the probability of expiring OTM.

For a short call spread:

\[E[X] = (1 - \Delta_{\text{SC}}) \times \text{MP} + \Delta_{\text{SC}} \times \text{ML}\]

For an iron condor:

\[E[X] = \left( (1 - |\Delta_{\text{SP}}|) \times (1 - \Delta_{\text{SC}}) \right) \times MP + \left( 1 - ((1 - |\Delta_{\text{SP}}|) \times (1 - \Delta_{\text{SC}})) \right) \times ML\]

For iron condors, the probabilities associated with the individual short put and short call options should be multiplied together. This is based on the fundamental principles of probability theory concerning independent events.

When looking at the likelihood of both options expiring OTM, we are interested in the joint probability of two independent events occurring simultaneously. This must be the case because both the put side and the call side must both expire OTM for an iron condor to realize full profit. The joint probability of two independent events \(A\) and \(B\) is the product of their individual probabilities:

\[P(A \text{ and } B) = P(A) \times P(B)\]

In the context of an iron condor strategy:

1. \( (1 - |\Delta_{\text{SP}}|) \) is the probability that the short put expires OTM.

2. \( (1 - \Delta_{\text{SC}}) \) is the probability that the short call expires OTM.

The probability that both occur (i.e., that both options expire OTM) is then \( (1 - |\Delta_{\text{SP}}|) \times (1 - \Delta_{\text{SC}}) \). And if we know the probability of expiring OTM, the probability of expiring ITM for the maximum loss case is simply the complement found by subtracting it from 1.

The Simple EV method applies broadly in scenarios where a straightforward understanding of probability suffices.

## Simple-Partial EV: Estimating the Partial Profit/Loss Region

One of the reasons it is so difficult to be a consistently profitable options trader is because payouts are *not* a binary, win-all or lose-all endeavor. Each position has multiple price points where the trade can win a little or lose a little. If we’re considering these outcomes only at expiration, we can roughly estimate the expected payout of a position expiring for a partial win or loss in the region between the short and long strikes using Simple-Partial EV.

There are two major differences between Simple-Partial EV and Simple EV. First, instead of using the short strike delta(s) as a proxy for the probability of max profit and their complements as a probability of max loss, we use the short strike deltas for max profit and long strike deltas for max loss.

Second, we include the EV of the partial profit/loss region using the arithmetic mean of the max profit and max loss as an estimated payout. This is a common shorthand approach to approximating the intermediate value, especially when dealing with financial derivatives or any model where a linear interpolation is deemed appropriate. The probability for the variable region is the complement to the sum of the max profit and max loss probabilities.

The Simple-Partial EV method introduces more granularity into our EV calculations by considering an additional range of payouts. This inclusion aims to encapsulate a more nuanced picture of the probable outcomes compared to the binary win/loss scenario presented by the Simple EV method.

### Simple-Partial EV: Strategy Formulas

The mathematical representation of this approach can be formulated as follows:

For a short put spread, the expected value \(E[X]\) would be:

\[E[X] = |\Delta_{\text{SP}}| \times \text{MP} + \Delta_{\text{LP}} \times \text{ML} + (1 - |\Delta_{\text{SP}}| - \Delta_{\text{LP}}) \times \text{Mean}\]

where \(\Delta_{\text{LP}}\) denotes the long put delta, \(\text{MP}\) is the maximum possible profit, \(\text{ML}\) is the maximum possible loss, and \(\text{Mean} = \frac{\text{MP} + \text{ML}}{2}\).

For a short call spread:

\[E[X] = \Delta_{\text{SC}} \times \text{MP} + \Delta_{\text{LC}} \times \text{ML} + (1 - \Delta_{\text{SC}} - \Delta_{\text{LC}}) \times \text{Mean}\]

where \(\Delta_{\text{SC}}\) denotes the short call delta and \(\Delta_{\text{LC}}\) denotes the long call delta.

For an iron condor, the formula is a composition of the short put spread and the short call spread combined EV:

\[E[X] = (1 - |\Delta_{\text{SP}}|) \times \text{MP}_{\text{SPS}} + |\Delta_{\text{LP}}| \times \text{ML}_{\text{SPS}} + |\Delta_{\text{SP}} - \Delta_{\text{LP}}| \times \frac{\text{MP}_{\text{SPS}} + \text{ML}_{\text{SPS}}}{2} \]

\[+ (1 - \Delta_{\text{SC}}) \times \text{MP}_{\text{SCS}} + \Delta_{\text{LC}} \times \text{ML}_{\text{SCS}} + (\Delta_{\text{SC}} - \Delta_{\text{LC}}) \times \frac{\text{MP}_{\text{SCS}} + \text{ML}_{\text{SCS}}}{2} \]

In this equation:

- \( \Delta_{\text{SP}} \) and \( \Delta_{\text{LP}} \) are the deltas for the short put and long put options, respectively, in the short put spread.
- \( \Delta_{\text{SC}} \) and \( \Delta_{\text{LC}} \) are the deltas for the short call and long call options, respectively, in the short call spread.
- \( \text{MP}_{\text{SPS}} \) and \( \text{ML}_{\text{SPS}} \) represent the maximum possible profit and maximum possible loss for the short put spread.
- \( \text{MP}_{\text{SCS}} \) and \( \text{ML}_{\text{SCS}} \) represent the maximum possible profit and maximum possible loss for the short call spread.

Here, the probability of the partial profit/loss region for the entire spread can be calculated by taking the complement of the sum of the probabilities of achieving maximum profit and maximum loss, which can be represented as \(1 - (|\Delta_{\text{SP}}| \times \Delta_{\text{SC}}) - (\Delta_{\text{LP}} \times \Delta_{\text{LC}})\).

One of the main advantages of the Simple-Partial EV approach is that it offers a more detailed description of a trade's risk-reward profile. It accommodates the scenario where outcomes are not strictly binary but can fall within a range of values, thereby offering a more realistic representation of market conditions.

However, it's worth noting that this approach remains an approximation. The inclusion of the mean value as an estimated payout for the partial profit/loss region may not be an accurate representation of the true expected value for this range.

Additionally, the method assumes linear interpolation between the two extreme values, which may not hold in all market conditions or option setups. The efficacy of this method can be subject to issues of model risk and assumptions related to the linearity of payouts.

## Real EV: Discrete Random Variable

To briefly summarize what we’ve learned about random variables in Understanding Alpha and Expected Value, Real EV involves a more nuanced approach by considering the *entire* range of potential outcomes rather than simplifying them to maximum profit/loss, and/or a single point between those two extremes.

This method treats each option strategy as a discrete random variable and calculates the EV based on the actual probability density function (PDF) or cumulative density function (CDF) of the underlying asset. This could be a non-parametric distribution obtained through historical data or one modeled using methods such as Black-Scholes, Binomial, or other exotic option models. For our purposes, we're using Black-Scholes probability to combine the EVs of the max loss, the max profit, and every $0.01 increment inside of the partial profit/loss region.

The expected value \(E[X]\) for a discrete random variable \(X\) taking on \(n\) different values \(x_1, x_2, \ldots, x_n\) with corresponding probabilities \(p_1, p_2, \ldots, p_n\) is calculated for a defined-risk options trade as:

\[ E[X] = P(ML) * ML + P(MP) * MP + \sum_{i=1}^{n} p_i \times x_i \]

Here, \(P(x)\) is the probability of event \(x\) occurring. For any payout that is not maximum profit or loss, each \(x_i\) would be a possible value for the strategy payout at expiration, and \(p_i\) would be the probability of that payout. Real EV considers all possible values and payouts at every possible price the underlying may expire at.

The merit of Real EV is its applicability to more complicated situations where multiple variables influence the outcome. Consequently, the method is more computationally intensive and requires a deeper understanding of stochastic calculus and probability theory.

## Results & Data Analysis

A density distribution chart, often represented as a histogram, portrays the frequency or probability density of a continuous or discrete variable. Unlike a regular histogram, which shows the frequency of observations in each bin, a density plot is normalized to show the relative likelihood of observing a value in each bin. The area under the curve in a density plot sums to 1, making it easier to compare distributions with different sample sizes or scales.

### Interpretation of EV Density Distributions

In Figure 1, we can clearly see the differences in the distribution curves of each EV type.

Simple EV has the widest distribution, signaling the highest variability. The curve is approximately Gaussian but has negative skewness.

Simple-Partial EV exhibits a taller spread and is slightly skewed to the left. The mild negative skewness and amplitude may indicate a greater occurrence of higher values.

Real EV is centered near zero with slight positive skewness. It also has a slightly narrower distribution compared to the others. This suggests that most of the Real EV values are near zero and have lower variance.

In Figure 2, we clearly see that Simple EV does not track the theoretical perfect EV line. It actually more closely tracks the zero line, undoubtedly because the EV guesses are grossly overstating the actual risk and the excess profit that’s realized in those buckets buoys the Average P/L in the negative \(x\) domain.

During the time period under observation, we studied approximately 400k trades. However, only 53% of trades expired for a full profit, and 20% of trades expired for a full loss, making simple EV inaccurate, at best, 27% of the time. There were 11% of trades that expired for partial profit and 16% of trades that expired for partial loss, which is statistically significant enough to affect the reliability of this method.

While the accessibility of Simple EV is among its strengths, it's worth noting that this simplicity may sometimes compromise precision, rendering it less suitable for complex scenarios. The method lacks granularity, and this lack of detail might result in misleading conclusions in more complicated financial instruments or volatile market conditions.

Both Simple EV and Simple-Partial EV are more pessimistic versions of actual or real EV. In fact, they make it harder to find an edge, but that’s not necessarily a bad thing for the cautious investor.

In our analysis, we observed that Simple-Partial EV provided increased accuracy. Expiration in the partial profit/loss region accounted for 26% of trades. Because Simple-Partial EV attempts to account for this scenario, we can visibly see the net result of that improvement in Figure 3. The Average P/L line is beginning to tend toward or converge on the perfect EV line. In other words, the U-shape curve shown in the Simple EV graph is starting to flatten because Simple-Partial encompasses more information.

Simple-Partial EV can be considered a more advanced form of Simple EV, providing additional granularity that is advantageous under certain market conditions. However, like its simpler counterpart, it has limitations that traders must be aware of.

This method still overestimates risk in negative \(x\), but it also underestimates profit potential in positive \(x\). Similar to Simple EV, however, even with our current data set there are too few samples in positive \(x\) at this time to rely upon our conclusions as a +EV estimator.

Although the Real EV graph in Figure 4 looks slightly different than when our first article was published, it’s indicative of market movement.

The directional trades that lost the most are dragging down the averages. But we fully expect over many more occurrences that each trade type will normalize to the perfect EV line. Regardless, the Real EV data still tracks perfect EV, even if it’s leading or lagging day-to-day or week-to-week.

It is interesting to note that while Real EV is a good estimator in positive \(x\), it is the only method under observation that underestimates risk in negative \(x\). Since most traders are only concerned about finding +EV, this shouldn't be an issue, but may be valuable information for anyone trading contrarian EV.

If options are viewed as a binary endeavor, winning or losing trades in their entirety like casino betting, it’s reminiscent of a zero-sum game. This is clearly evidenced by the Simple EV line in Figure 5 when compared to the other two methods. Simple EV drastically overestimates the downside risk associated with each trade, which results in the EV tracking the zero line.

This implies that money in options trading is made on the fringes; partial wins and losses matter.

Simple EV and Simple-Partial EV appear to overstate the amount of downside risk, whereas Real EV appears to understate the downside risk, however slightly. But Real EV is the only method that aligns with what we “expect” the Expected Value to be in the positive \(x\) domain. That is, sufficient samples exist through a range of \(x\) such that we, as practitioners, can trust and rely on these numbers.

In our original analysis of Real EV over 130k trades, we noted that the Average P/L was leading the perfect EV line between $0 and $60. But now, with more than triple the data points, Average P/L is lagging the perfect EV line. This is of no real concern. Recall how even in a fair coin flip over medium occurrences we may not see a near-perfect 50/50 ratio of heads to tails, it's only after very large occurrences. I suspect we'll see the same thing here over time for Real +EV.

## Conclusions

After exhaustive data collection and analysis, it's evident that while Simple and Simple-Partial EV methods are easy to understand and implement, they lack the precision offered by Real EV methods. Therefore, for traders with computational capabilities, Real EV stands as the superior method for estimating expected value in options trading. For those with limited resources or mathematical background, Simple or Simple-Partial methods offer an approachable yet less accurate alternative.

The efficacy of any method ultimately depends on the complexity of the trading strategy and the robustness of the underlying model. We believe that an understanding of all three methods gives traders a more holistic view of risk and reward, enabling them to make more informed decisions.

Upon analyzing the results, we observed that:

- Simple EV performed admirably but lacked the nuance to capture more complex outcomes.
- Simple-Partial EV showed better accuracy in capturing variable profits and losses but was still not fully representative.
- Real EV provided the closest match to the actual realized outcomes but requires significantly more computational resources.

The correlation between the EV methods and actual outcomes demonstrated that while Simple and Simple-Partial methods are easier to compute, Real EV provided the most accurate representation of a trade’s true expected value.

When trading on Simple EV information, which always represents the worst-case scenario, any +EV trade should prove reliable, if you can find one. Only 6% of all trades in the data set were rated +EV from the Simple method.

When trading based on Simple-Partial EV information, the $1 - $30 range, and particularly the $1 - $10 range with sufficient sample size, seems to be a reasonably accurate estimator of EV.

When trading based on Real EV information, the sweet spot is still in the > +$20 range, where it should track the perfect EV line (we previously stated $20 - $60 was the “sweet spot,” but there are sufficient data points now to believe accuracy is maintained for higher EV).