To conclude our conversation on raw expected values, Iâ€™d like to come full circle and discuss the equation that makes all this possible: Black-Scholes.

Weâ€™ll explore how this one formula allows us to calculate probabilities and expected values in multiple ways. Weâ€™ll compare the different methods to show you why Option Alpha chooses to use the annualized 30-day standard deviation of returns as the volatility input to Black-Scholes.

If youâ€™re just joining us, I highly recommend starting with our first two articles in this series: Trade Ideas Probability and Performance and How to Calculate Expected Value.

## Brief Review of the Black-Scholes-Merton Model

The Black-Scholes-Merton (BSM) model is a seminal work in financial mathematics, providing a theoretical framework for pricing European-style options. Developed independently by Fischer Black, Myron Scholes, and Robert Merton, the model is predicated on a set of assumptions, including:Â

- The absence of transaction costs and taxes.
- Constant risk-free interest rate and volatility.
- Log-normally distributed returns.
- The ability to short-sell and purchase fractional shares.

The Black-Scholes formula for a European call option is:

\[C = S \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)\]

and for a European put option:

\[P = K \cdot e^{-rT} \cdot (1-N(d_2)) - S \cdot (1-N(d_1))\]

Here \(S\) is the spot price, \(K\) is the strike price, \(T\) is the time to maturity, \(r\) is the risk-free rate, and \(\sigma\) is the volatility of the underlying asset. \(N(x)\) represents the cumulative distribution function of the standard normal distribution.

The variables \(d_1\) and \(d_2\) are defined as:

\[d_1 = \frac{{\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right) \cdot T}}{{\sigma \sqrt{T}}}\]

\[d_2 = d_1 - \sigma \sqrt{T}\]

While \(N(d_1)\) and \(N(d_2)\) are related, they serve different roles in option pricing and risk assessment. The \(N(d_1)\) term, which represents the delta of the option in the Black-Scholes model, is often considered a "hedge ratio" that tells you how many shares of the underlying asset you would need to hold to create a risk-free hedge. \(N(d_2)\) is more directly interpreted as the risk-neutral probability that the option will expire in-the-money (ITM).

However, \(N(d_1)\) can serve as a proxy for this probability under certain conditions. The delta \(N(d_1)\) can be interpreted as the risk-neutral probability of the option being ITM if you account for the expected growth of the underlying asset. Specifically, \(N(d_1)\) incorporates the risk-free rate, providing a forward-looking measure that adjusts for the anticipated growth of the underlying at that rate. While \(N(d_2)\) measures the probability of being ITM at expiration, \(N(d_1)\) can be considered a measure of the same probability but adjusted for expected growth. The terms \(N(d_1)\) and \(N(d_2)\) can both be interpreted as risk-neutral probabilities, but they are not created equal.

You might see these two variables represented as the Greek Delta and Probability of ITM on a brokerage platform option chain, if available.

## Abstracting \(N(d_1)\) & \(N(d_2)\) into a Risk-Neutral Probability Framework

At Option Alpha, we abstract the BSM probability measures to apply to any pricing scenario, given some target price level. To evaluate the likelihood of the underlying asset expiring above or below a specific target price \(X\) at expiration \(T\), we can modify \(d_1\) and \(d_2\) as follows:

\[d_1' = \frac{{\ln \left( \frac{S}{X} \right) + \left( r + \frac{\sigma^2}{2} \right) T}}{\sigma \sqrt{T}}\]

\[d_2' = d_1' - \sigma \sqrt{T} = \frac{{\ln \left( \frac{S}{X} \right) + \left( r - \frac{\sigma^2}{2} \right) T}}{\sigma \sqrt{T}}\]

\(N(d_1')\) and \(N(d_2')\) will then signify the risk-neutral probabilities of the underlying asset finishing above the target price \(X\) at expiration \(T\). Conversely, \(1 - N(d_1')\) and \(1 - N(d_2')\) would represent the probabilities of the underlying asset finishing below \(X\).Â

In Calculating Probability, we explain how Option Alpha uses a fast polynomial approximation of the standard normal distribution, \(N(x)\).

For different kinds of options where analytical solutions might not be easily obtainable, \(N(d_1')\) and \(N(d_2')\) can offer useful approximations for the likelihood of the option expiring in-the-money with respect to the target price \(X\).

These probabilities can be used conditionally to assess the likelihood of the underlying asset crossing a certain threshold given specific market conditions or levels of other market variables.

By modifying \(N(d_1)\) and \(N(d_2)\) to \(N(d_1')\) and \(N(d_2')\) respectively, we can extend the utility of the Black-Scholes-Merton model to a broader range of financial instruments and scenarios.

## Parameter Selection

Even though a broker may use the same BSM or BSM-like equations to derive probability numbers, you may notice in passing that these numbers differ slightly from what you see on Option Alpha or even between two different broker platforms. Letâ€™s explore why.

To recap the BSM parameters: \(S\) is the spot price, \(K\) is the strike price, \(T\) is the time to maturity, \(r\) is the risk-free rate, and \(\sigma\) is the volatility of the underlying asset.

### Spot Price

The term "spot price" in financial derivatives typically refers to the current market price of the underlying asset at which it can be bought or sold for immediate delivery. In theory, this would refer to the bid or ask price. But in practice, using a fair-value assessment like the mid or last price is usually more practical. At Option Alpha, we use the last price of the asset.Â

### Strike Price

Arguably the least open to interpretation, the strike price(s) are dictated by the position under observation. For our purposes, this translates to any target price \(X\) in the risk-neutral probability framework, e.g., the breakeven price.

### Time to Maturity

The time to maturity is understood to mean the days to expiration, expressed in years. At Option Alpha, weâ€™re required to produce probability numbers regardless of the time remaining on a contract, including zero days to expiration. So, we take this one step further and use fractional days to expiration. Because the time in years is generally expressed as a floating-point number, so too can the number of days. This greatly enhances precision and results in no two probability calculations being equal. Even if all other variables remain constant, time never stops.

### Risk-free Rate

The risk-free rate commonly references the yield on a theoretically risk-free investment, most often associated with the yield on a government bond, such as U.S. Treasuries. The maturity of the bond used for this purpose can vary widely, but the 3-month and 10-year U.S. Treasury yields are frequently cited benchmarks.

The general consensus in financial literature is that the risk-free rate has a relatively minor impact on the valuation of short-dated options. For options with a short time to expiration (all generated Trade Ideas), the time value of money has less time to compound, making the discount factor \(e^{-rT}\) approach 1. Consequently, the term \(X \cdot e^{-rT}\) approximates the strike price \(X\) more closely, rendering the risk-free rate less impactful on this component of the Black-Scholes formula.

Additionally, the spot price \(S\) and implied volatility \(\sigma\) are often the dominant factors influencing option pricing for short-dated options. These factors generally overshadow the risk-free rate's influence, especially in low-interest-rate environments. So, for all practical purposes, as long as the risk-free rate is sufficiently close to the current rate for short-dated options, even if itâ€™s a static approximation, it shouldnâ€™t matter which measure is chosen.

### Volatility

This is the big one. The volatility parameter arguably has the single most significant effect on the output probability.

At Option Alpha, we use the annualized 30-day standard deviation of returns (historical volatility, or HV) as an input to the probability framework.Â

In Probability Theory: Making a Case for HV, we discussed our theoretical argument for why we prefer historical volatility over implied volatility.

Implied volatility often incorporates market sentiments, risk premiums, and other idiosyncratic factors that may not necessarily reflect the true statistical volatility of the underlying asset. In contrast, 30-day annualized HV is purely empirical, derived from past price fluctuations, and devoid of market sentiment. While it assumes that past behavior will predict future volatility, it serves as a more neutral input, making the resulting risk-neutral probabilities less subject to market biases and distortions.

## Visualizing the Effects of Probability Choice in Expected Value

To illustrate exactly how much the choice in both the probability formula and parameters of probability matter (even within the same formula), weâ€™re going to examine six probability types as the basis for Expected Value:

- Black-Scholes \(N(d_1â€™)\) using 30-day HV
- Black-Scholes \(N(d_2â€™)\) using 30-day HV (used inside Option Alpha)
- Black-Scholes \(N(d_1â€™)\) using black-box IV of the underlying
- Black-Scholes \(N(d_2â€™)\) using black-box IV of the underlying
- Simple EV using market-supplied short deltas
- Simple-Partial EV using market-supplied short and long deltas

Weâ€™re testing four variations within the BSM framework. All input parameters are precisely the same for each as described above, save for the choice of \(d_1â€™\) or \(d_2â€™\) as an input to the cumulative density function and choice of IV or HV.Â

We will borrow the Simple and Simple-Partial EV from our article on Calculating Expected Value to see how well they compare to our BSM delta-variants. Both EV metrics are broad approximations using market-supplied delta values, i.e., greek values we didnâ€™t calculate.

Given this understanding, we should see visual similarities in the shape of the EV curve and BSM \(N(d_1â€™)\) using IV. The primary difference is instead of only using the short and/or long deltas, we are calculating the theoretical delta at every price increment instead of just at the strikes. Weâ€™re essentially calculating Real EV (defined in the referenced article above) using only deltas.

The remaining BSM variants are also calculated as Real EV using the indicated parameters. All probabilities within the Option Alpha platform use the BSM \(N(d_2â€™)\) variant with 30-day HV as the volatility parameter.

### Methodology

The methodology criteria remain the same as weâ€™ve used in the previous two studies on EV: Trade Ideas Probability and Performance and How to Calculate Expected Value.

To summarize, weâ€™re analyzing short put spread, short call spread, and iron condor Trade Ideas captured at 3:55 PM ET every market day. The trades span up to 45 days to expiration on 155 different symbols. We assume cash settlement for all trades to capture partial profit/loss numbers. All theoretical trades are taken to expiration.

Our data set is growing. At the time of this study, there were over 600k trade samples under observation.

### Results

We wonâ€™t be able to plot sample counts for every probability line as we did before because of overcrowding. However, letâ€™s take a look at the OA EV line (identical to BSM \(N(d_2â€™) + AV\)) and how itâ€™s progressing with higher sample counts.Â In the following figures we will use AV, or the annualized historical volatility, to represent the use of HVÂ in the context of the Black-Scholes equation.

Weâ€™ve shown the Figure 2 graph in every EV article so far if youâ€™d like to compare and see firsthand how the law of large numbers works when it plays out. The data points are beginning to track the theoretical EV line even more closely than before.

If youâ€™re unfamiliar, the theoretical perfect EV line is the point where the predicted expected value should meet the average actual profit/loss in a perfect world. This is determined by averaging all profit/loss values for all trades whose EV falls within a particular range, the $10-wide EV â€śbucketsâ€ť in our study.

In Figure 3, we plot the 4 Real EV methods from Black-Scholes using combinations of \(N(d_1â€™)\) or \(N(d_2â€™)\) and AV or IV.Â

It is immediately evident that the four methods produced radically different outcomes. This is surprising because, as we saw in Figure 1, HV and IV differ during our observation window but are not dramatically different. And \(d_2â€™\) is different than \(d_1â€™\), but not dramatically different.

But the combination of both choices, coupled with the more nuanced choices in parameter selection, produces the graph below.

These results clearly indicate the deciding factor in the accuracy of tracking the perfect EV line in the positive \(x\) domain is the choice of volatility parameter.Â

While interesting, all this graph shows is one more view of what weâ€™ve been hearing for years: â€śimplied volatility tends to overstate actual or realized volatility.â€ť And thatâ€™s never been more true when it comes to expected value.

But IV is forward-looking; it encompasses more information! Maybe, maybe not. When systematically basing trading decisions on math and probabilities, if the numbers we rely on are not accurate, theyâ€™re not useful.

The theory is sound, but we also wanted proof that weâ€™re correct with high confidence. The BSM calculations are a product of our choices, our math, our code, and our black-box IV parameters. So, to be confident in our findings, we need to compare our results against metrics we have the least influence on.

In Figure 4, we brought back Simple EV and Simple-Partial EV. If you recall, we did not create the input data for both. They were supplied by market deltas and market prices, re-creatable by anyone with options chains on a brokerage platform.

It stands to reason that if IV is *implied* by the market, the calculation made using the market itself should reflect the output of calculations using that theoretical value.

As we can see clearly in the graph, both Simple calculations track the U-shape curves of the two Real EV BSM lines that were created using IV. In fact, the Simple-Partial EV line seems to split the difference between the \(N(d_1â€™)\) and \(N(d_2â€™)\) IV lines, which is fascinating.Â

Simple-Partial EV, which is calculable by anyone with a spreadsheet and an options chain, is no better than the more computationally intensive Real EV BSM methods when using implied volatilities for the given strategy types.

## Conclusion

This study critically evaluates the Black-Scholes-Merton (BSM) model's flexibility in calculating expected values for various financial instruments. Key findings emphasize the importance of volatility parameter selection, specifically contrasting historical volatility (HV) against implied volatility (IV).

Empirical results indicate that HV offers a more neutral, less biased risk-neutral probability measure compared to IV. Moreover, the choice between \(N(d_1')\) and \(N(d_2')\) in the BSM framework leads to substantive differences in expected value calculations, underlining the model's sensitivity to parameter selection.

Option Alphaâ€™s EV calculation, identical to BSM \(N(d_2â€™)\) using the annualized 30-day standard deviation of returns for its volatility input, produces the most accurate representation of the expected value. This challenges the prevailing industry notion that IV-based calculations are inherently superior.

The study also finds that simpler methods, such as Simple-Partial EV, perform comparably to more computationally intensive BSM variants when IV is used.

The research underscores the need for careful parameter selection in expected value calculations, particularly when these serve as the basis for systematic trading strategies based on math and probability.