One option spread strategy that requires a trader to estimate relevant high and low strike prices between the trade date and expiration is the short option strangle. As defined by James Cordier and Michael Gross in “The Complete Guide to Option Selling” (McGraw-Hill, 2005), this is the process of selling a naked out-of-the-money call and a naked out-of-the-money put on the same underlying asset and expiration date. Future price ranges may be estimated based on charts of past price movements or by various forms of fundamental analysis.

Another method — one that relies on current market consensus regarding the probabilities of price changes — is the LLP (log-log parabola) option-pricing model. This method is compatible with the Black-Scholes option-pricing model that is the basis for most valuations in the option market. The LLP and Black-Scholes models are available here.

“O.J. call options” shows an LLP analysis example that results in estimates of upper and lower breakeven prices at expiration. These are prices at expiration of a futures contract that result in zero profit or loss on a delta-neutral hedge. On Dec. 3, 2008, the March 2009 futures contract closed at 76.35¢ per lb. For 10 strike prices, the analysis provides option prices (market prices and those predicted by a regression equation), intrinsic values (futures price minus strike price), delta values for the slope of the price curve at each strike price, and breakeven prices at expiration for hedge trades — purchase of the number of calls indicated by the delta value against a short-sale of the futures contract. For example, a trader could sell one March 2009 futures short at the 75 strike price and hedge the short sale by holding 1/0.528, or 1.89, calls per short futures.

These are the final products of the LLP analysis shown on “O.J. call options” for the purpose of finding high and low breakeven prices. Because breakeven prices vary as strike prices move away from the current underlying futures price, it is best to use the upper and lower prices along the 75 strike price row. These prices, 91.815 and 59.966, suggest a range surrounding the current price of 76.35 for the trader entering a short option spread. This spread would be designed to sell a naked call on March 2009 orange juice futures at a strike price above 91.815 and at the same time sell a naked put on the March 09 futures at a strike below 59.966.

PROBABILITIES MATTER

The regression equation computed by the LLP model generates predicted call prices that closely match current market prices. Since the early 1970s, option markets have been priced by the Black-Scholes model or similar logarithm-based models. LLP call and put price calculations reflect market pricing and compute accurate results by describing price curves that are parabolic log-log equations.

For March 2009 orange juice calls, it is noted that the 75 strike price has a delta value of 0.528, slightly higher than the 0.50 that might be expected for a strike price equal to the current futures price. As the underlying price increases relative to strike price (as it is here, 76.35 vs. 75), the delta value continues to increase toward a value of 1.00 where the option premium equals the intrinsic value.

Breakeven prices produced by the LLP model are related to Black-Scholes through calculations that imply the probabilities of futures contract price movements, as shown on “O.J. futures” (above). Data entered for the orange juice futures analysis include the futures price, 76.35, strike price, 75, 80 days to expiration in March 2009, and estimated risk-free interest rate of 1.00%.

Probabilities enter the picture when the model calls for an estimate of the standard deviation of returns on the underlying futures contract. Because the option’s actual market price, 8.65, is already known, the standard deviation is entered as a trial-and-error number until the computed Black-Scholes (B-S) price is equal to the market price. The standard deviation found by the trial-and-error process is the implied volatility measure for March 2009 orange juice futures on Dec. 3, 2008.

SEPARATE BUT EQUAL

The Black-Scholes pricing model computes a delta value for the slope of the option price curve at a given strike price. On the orange juice analysis, the delta value is 0.582. This delta is close to the one computed by the LLP model on “O.J. call options.” The two pricing models show similar results being generated by separate calculation methods.

The chart “March 2009 calls” (above) shows how the delta slope is related to upper and lower breakeven prices. For each strike price, there is a line that has the slope computed by the LLP or Black-Scholes models. These straight lines are tangent to the option price curve at points directly above the strike prices. Each tangent line intersects the horizontal axis and the intrinsic value line at upper and lower breakeven prices. The diagonal tangent line at strike price 75 is shown for March 2009 orange juice futures, as well. (The diagonal line has been lowered a bit to avoid masking out the option price curve. Otherwise it will intersect the horizontal axis near 91.815 and the intrinsic value line above 59.966.)

“March 2009 futures” lists 10 futures contracts with upper and lower breakeven prices computed during the first week of December 2008. The upper and lower prices result from regression analysis by the LLP option-pricing model, and as described above they are the points of intersection with the horizontal axis and intrinsic value line for each strike price. The breakeven prices are generated by delta slope values and are related to Black-Scholes probabilities for the various futures call options.

Strike prices that the seller of a call and put would receive in carrying out a short-option strangle are approximately equal to the upper and lower breakeven prices, with some adjustment to allow the strike price to be slightly higher than an upper breakeven price or just under the lower breakeven.

For most of the put and call pairs on “March 2009 futures,” the premiums of selling puts and calls are similar in dollar amount because the distances between the strike prices and current futures price are not too different. In a strangle trade, the option prices for the put and call should move in opposite directions. This price action provides some hedging protection; however, because of the varying slope of the option price curve as futures prices increase or decrease — as well as variations caused by dwindling time to expiration — the opposing price changes are not expected to be equal.

NOT A PREDICTION

Although the breakeven prices are related to price spreads generated by Black-Scholes probability distributions, it would be a mistake to view them as accurate predictions of future prices or of certain profits on option spreads. The probabilities show what the market believes at a given point, and they provide a consensus of opinion on possible price variations. However, it is not unusual for market prices to exceed the upper and lower breakeven prices well before expiration. For the hedger, price movements toward and beyond the original boundaries present the need for risk management trades such as repurchase of calls or puts having increasing premiums, or sale of additional options to offset losses on the original spread.

One advantage of the short-option spread is the lower margin required because of the natural hedge between the put and call. As the futures price moves closer to either strike price, the margin will increase, reducing the initial advantage.

An option’s volatility, expressed by the standard deviation of returns on the underlying futures contract, corresponds with the spread between upper and lower breakeven prices. As the breakeven spread increases, the tangent points for all strike prices move up and the higher option price curve reflects a larger volatility.

Because the market’s assessment of the highest volatility value determines the height of the price curve and the breakeven prices at each strike price, the distance between upper and lower breakeven prices should be related to the standard deviation (implied volatility) computed by the Black-Scholes option.

The probability spread is computed by multiplying three times the standard deviation by the futures price, and then multiplying this product by the proportion of a year to expiration. Dividing the breakeven spread by the probability spread results in the spread ratio, which for this sample of futures averages is slightly less than 1.15.

Volatility of futures prices can produce exceptional option premiums for sellers of puts and calls in short-option spreads, but it also can be the cause of problems. One company that specializes in spreads of this type mentioned the need for more than 30 risk-management trades for a single spread during the decrease in the price of crude oil in the summer and fall of 2008. Calculating implied volatility and breakeven prices may be the easiest part of making profits with option spreads.

Paul Cretien, CFA, is an investment analyst and financial case writer. E-mail him at PaulDCretien@aol.com.