With its multiple uses, the Log-Log Parabola (LLP) options-pricing model can be a valuable tool for traders interested in a non-theoretical approach to the options market. One application permits the user to see minute-by-minute the options market’s prediction of breakeven underlying prices for profitable delta neutral trades. These are the upper and lower prices at which call and put adjustment trades may be needed.

The LLP options-pricing model is a Microsoft Excel spreadsheet that permits a number of analytical methods to be applied to options on futures and equities. Input data are any options chain showing the underlying futures or stock price and a set of strike prices with their corresponding options market prices. By using existing prices, the LLP model recognizes that the options market generates a finely tuned set of data on a continuous basis. That is, the LLP does not calculate option prices, but uses listed market prices to produce a regression equation that permits the user to accomplish several types of analysis. These functions include:

- Comparing each options market price in an option chain to its expected price on the LLP price curve.
- Generating the slope of the options price curve, or delta value, at each strike price.
- Showing the height of the options price curve, a measure of relative volatility, at the point at which the underlying asset equals the strike price.
- Computing the upper and lower expiration breakeven prices for delta trades — selling options against long futures or equities — at each strike price.

The results generated by the LLP model are based on market price input. Traders in the market forecast future prices and price variability. The LLP takes this information, analyzes it and shows the results.

“LLP: Crude oil, Swiss francs” (below) is the analysis for March 2013 call options on crude oil and Swiss franc futures on Dec. 27, 2012. Twenty strike prices were used with their corresponding options market prices to calculate the regression equations that generate predicted prices, delta values and breakeven prices for delta trades between the calls and March futures contracts.

Of the 20 strike prices, three are shown in the LLP exhibit. Delta trades based on the LLP values would be based on the slopes of the options price curves at specific strike prices.

For example, at the 92 strike for crude oil futures, 1.0/0.542 or 1.845 calls are sold short for each long futures contract. If the March futures price at expiration equals $92, the trade will be closed out at the maximum profit, $6,442, the credit from sale of calls, $5,812, with a gain on futures equal to the difference between $92 and $91.37 times $1,000, or $630. Lower profits will be made at expiration futures prices between $99.624 and $85.558 as the loss on call short sales is reduced by a gain in the futures price, or the short sale credit offsets losses on the futures contract.

Upper and lower breakeven prices for March 2013 crude oil calls on Dec. 27, 2012 are shown in ““Breakeven prices” (below). The typical pattern for breakeven prices is for the upper breakevens to follow a curve that descends from higher strike prices, hitting the lowest levels near the current futures price then increasing rapidly for calls that are in-the-money. Lower breakeven prices describe a relatively level curve for all strike prices.

Beyond breakeven prices the trade will generate losses. Protective trades between Dec 27 and expiration would include purchase of calls to offset potential losses on sold calls and the sale of additional calls or purchase of puts to reduce losses on a futures price declining beyond the lower breakeven price.

The original neat triangular profit diagram with the profit peaking near the midpoint between the upper and lower breakeven prices will be changed into a more complex shape by successive protective trades.

**Making adjustments**

“Swiss franc delta neutral” (below) is the profit diagram for a trade originated on Dec. 27 at a strike price of 1.095. The red dot at $2,000 profit occurred on Jan. 6, 2013. By this time, the futures price had fallen by $2,000, leaving the trade profit at expiration at $1.973 — down from the maximum profit of $3,786 and suggesting buying puts or selling calls if the futures price continued to decline.

Delta trades might be constructed at any strike price, with the delta values and upper and lower breakeven prices changing along the options price curve. Selecting the 94 strike for March crude oil futures will shift the expiration futures price for maximum profit further from the current price, increasing the gain from the futures contract if it expires at the price of $94, but reducing the credit from sale of calls to $5,523.

“March 2013 call price curves” (below) shows the options prices for March 2013 crude oil and Swiss franc futures on Dec. 27. It is obvious from this chart that crude oil futures are deemed by the options market as far more valuable than Swiss franc futures near year-end 2012. With the futures price equal to the strike price, the curve height for crude oil is approximately 4% compared with Swiss francs at slightly more than 1%.

The height of an options price curve depends on two factors — volatility of the underlying as perceived by the market, and time to expiration of the options. On Dec. 27, Swiss franc futures had 71 days remaining to expiration, while crude oil futures had 49 days. The positive difference in time to expiration for Swiss franc futures makes an even stronger case for higher crude oil volatility.

**Bucking expectations**

“Variations” (below) shows sequences of variations from expected values on the options price curves for March 2013 crude oil and Swiss franc futures over seven trading days: Dec. 17 through Dec. 27, 2012. Four strike prices are selected for each futures contract to illustrate the possible gains or losses from spreads between strike prices.

Along with the lower volatility shown by the call price curve, Swiss franc calls have relatively low variability around the LLP regression curve. For the measurement dates the Swiss franc variations oscillate between plus and minus $10 from the price curve. This variation seems small when the $125,000 per option point is considered, and the crude oil variations are between $100 and –$20 at $1,000 per option point.

Trading on the basis of dollar variations may include buying or selling individual strikes that are undervalued or overvalued, and spread trades between strikes. Because the price curves rise and fall in response to the underlying, it always is less risky to use spread trades. For example, a short sale of the 93 crude oil call on Dec. 20 resulted in a $578 gain when the call price fell from $2,550 to $1,980 the following day. Spreading with the $97 strike price would have reduced the gain to $260, but both changes occurred while the underlying March 2013 crude oil futures fell from $90.40 to $89.23.

The current model is available at www.futuresmag.com. It is a free Excel download file and is set up to analyze from three to 20 pairs of strike prices and options market prices.

*Paul Cretien is an investment analyst and financial case writer. His e-mail is PaulDCretien@aol.com. *