## Oil, Gas & the Log–Log Parabola Options Pricing Model

The LLP option pricing model provides insight to option traders that is particularly helpful in the energy complex. The log-log parabola (LLP) options pricing model has proven itself to be a valuable measure for pricing and trading options across all asset classes. Here we will apply it across the energy futures complex, specifically, crude oil, natural gas, gasoline and heating oil.

If you take a look at “LLP options pricing on Aug. 25, 2017” (below), it shows the information that is produced by the LLP model. For each strike price on the options chain, the model presents five unique metrics:

1. The predicted option price,
2. The dollar difference between the option’s market price and predicted price,
3. The slope (or delta value) of the options price curve at every strike pride,
4. Upper and lower breakeven prices of the underlying futures contract at expiration for a delta neutral trade (selling the number of calls equal to an inverse ratio to delta) offset by a long position in the futures contract, and
5. The option’s premium. For each options price chain the LLP model generates a formula that permits forecasting a put or call price during short time periods (several days) and makes it possible for a comparison between the market price and predicted price at each strike price. Parabola Curve
The LLP model depends on an option price curve being a log-log parabola, thus the LLP title. Predictive formulas are regression equations in which the independent variable (on the horizontal or X axis) is the natural logarithm of the ratio of the futures price to each strike price, and the dependent variable (on the vertical or Y axis) is the natural logarithm of the call or put market price as a ratio to each strike price.

A trader who uses the LLP model need not be concerned with calculations because these are completed by the Microsoft Excel spreadsheet in a convenient size: 27 columns by 42 rows. The only input values are the strike prices and corresponding put or call market prices in the options chain. The user also inputs the number of strike prices in the chain — from 3 to 20. Three is the minimum number because at least three points determine a curve. Restricting the number of strike prices to 20 permits the model’s results to be shown on one page.

In addition to the calculation of a log-log parabola for the pricing of options, the LLP model may be useful in other applications in which analysis of data results in parabolic curves. Instead of converting data into natural logarithms, the program can compute a parabolic regression formula based on non-logarithmic input. In this way, LLP provides a parabolic regression function. The dollar variations from predicted prices for calls on crude oil, gasoline, heating/diesel oil and natural gas are small considering the dollar value of each option point. For example, the largest price variance for gasoline calls on Aug. 25, 2017, is \$1.66 while the value of each option point is \$42,000.

Because of the relatively tight fit between options market prices and prices predicted by the regression formula, it seems safe to say that options price chains may be described as log-log parabolas. Natural logarithms of the ratios of an underlying futures price to several strike prices are related to the ratios of natural logarithms of call or put prices to the same strike prices by a regression formula that describes a parabolic curve.

The LLP options pricing model uses the portion of the parabola that extends from the smallest call or put price (for a call, the largest strike price on the options price chain, and for a put, the smallest strike price on the chain). When the price of the underlying futures contract becomes equal to a strike price, the predicted prices want to follow the parabolic curve instead of becoming closer to the option’s intrinsic value (the option’s value if exercised in terms of the underlying futures contract). “Oil and gas ETFs” (above) shows the current position of energy futures in August 2017. Three of the energies form a relatively tight group between crude oil, heating/diesel oil and gasoline. Natural gas at times moves away from this group, which may present excellent spread trade opportunities when the variations become large — as they did in February 2017 and later in July 2017. The petroleum-natural gas spread appears ready to close up again by fall of 2017.

The LLP model may be used to compare implied volatilities of underlying futures contracts. For example, “December 2017 energy call options” (below) shows the price curves for crude oil, natural gas, USLD heating/diesel oil and gasoline on Aug. 7, 2017. True to its nature as a sometimes outsider, natural gas futures are viewed by the options market with the largest implied volatility. More volatility is rewarded by the options market with the highest price curve. At the low side of implied volatility is heating/diesel oil. Gasoline and crude oil are close together in the center of the options price curves and in terms of implied volatility.

A look at what the options market thinks of crude oil futures is shown in “Crude oil call options” (below). Ratios of call price-to-strike price are related to the futures prices as ratios to the same strike prices for expirations at December 2017, March 2018, June 2018 and September 2018. The chart shows the accelerating decline in options values as the expiration date is approached.

By computing the differences between market prices and the predicted price curve, the LLP model can suggest calendar spreads in cases where call or put options may be over-priced or underpriced. The predictable speeding-up of declines in price for options that are closer to expiration may be part of a calendar spread trading strategy. The LLP model assists by continuously showing the exact height and shape of the options price curves, with the Delta slope value computed at each strike price.

“Crude oil March 2018 puts and calls” (below) shows put and call price curves meeting at the point at which the current futures price for March 2018 crude oil is equal to the strike price. The LLP model can suggest trades in which calls and puts are used on opposite sides of buy or sell when one or the other happens to be overpriced or underpriced versus the predicted price.   