CSI & Black-Scholes: A Deeper Dive

The table below illustrates how CSI can be applied to the forecasting of short term price movements in the market and how these forecasts are more accurate than the commonly used Black Scholes model.

Black-Schole is a pricing model used to determine the theoretical value for a call or a put option. One of the outputs of the Black-Schole model is 'option Delta', which is often used as an informal measure of the probability that an option’s strike price will meet or exceed this value on expiration date.

A commonly cited flaw in the framework is its continuous assumption of normally distributed future returns.

As previously mentioned, the distribution of future returns varies significantly based on which quadrant of CSI the market is presently in. By modifying the Black Scholes model to, instead, use the distribution of returns dictated by CSI, we can achieve much more accurate price forecasts (i.e., Delta values) than would be achieved using the standard Black Scholes model.


Taking a closer look at the chart , above, there are five particularly notable values:

  1. Expiration Date (i.e., the date we are aiming to make a price forecast for)
  2. Strike (i.e., the forecasted price on the expiration date)
  3. Delta (i.e., the likelihood of the S&P500 reaching or exceeding the forecasted price by the expiration date as predicted by the standard Black Scholes model)
  4. CSI Delta (i.e., the likelihood of the S&P500 reaching or exceeding the forecasted price by the expiration date as predicted by substituting our CSI distribution of future return into the Black Scholes model)
  5. Difference (i.e., the difference in prediction between the conventional Black Scholes and CSI valuation methods)

A 'Difference' reading of .10 means that the CSI valuation methodology is forecasting a 10% greater likelihood of the S&P500 reaching or exceeding the strike price on the expiration date. Notably, the 'Difference' column also tells us where the market is undervaluing and overvaluing various stock options at the present moment in time.

How much more accurate is this methodology versus the traditional Black Scholes model?

To answer this question, it is valuable to study both the historical forecasting capabilities of the 'CSI modified' Black Scholes model - incorporating CSI’s estimated distribution of likely returns- and how those results stack up against the forecasting accuracy of the traditional Black Scholes model (without modification).

To accomplish this, we back-tested the historical Delta values of SPX call options to measure the accuracy of their forecasts against the resulting price of SPX on expiration.

The dataset consisted of millions of observed daily SPX option prices from Jan 1, 2015 – May 1, 2021.

The option prices were measured at market close, and the delta corresponded to the quote mid-price. We also studied the corresponding “CSI Deltas” at the same moment of time for each measurement.

The results are displayed in the corresponding chart – where:

  1. The green line shows our expected accuracy whereas a delta of .5 should result in the S&P500 reaching or exceeding the option’s strike price 50% of the time by option expiration.
  2. The orange line represents the measured accuracy of conventional Black Scholes deltas in forecasting SPX to reach or exceed the option’s strike price.
  3. The blue line represents the measured accuracy of our CSI Black Scholes in forecasting SPX to reach or exceed the option’s strike price.

The primary takeaways of this show that for the measurement period:

  1. Delta has notably underestimated the probability that S&P500 will meet or exceed an options strike price.
  2. The CSI modified forecasting is far more accurate in measuring deltas and therefore forecasting short-term expected price movements.

The following charts the historical differences in forecasts between CSI and Black Scholes delta values.strong> As previously explained, each value charted is the difference between the CSI delta and Black Scholes delta for the stock option expiring 1 week out within a strike price 4% below the S&P500 price on that day.

The difference between CSI and Black Scholes delta values can be paired with CSI quadrant to forecast near-term price and volatility expectations.

Below, a large positive difference between these delta values during Quadrant 1 implies 2x better returns than average over the following week. Meanwhile, a large negative difference between these deltas during Quadrant 2 implies 5x worse performance over the following week (versus typical 1-week returns).