The geometric basis for options pricing provides information that can be used to construct profitable short strangle trades. “Copper calls” (right) shows the pricing model for calls on December 2016 copper futures.

The futures price (2.1285), where each point is worth $25,000, is related to strike prices ranging from 3.20 to 2.15 and corresponding market prices for the calls beginning at 0.001 ($25) and extending to 0.1245 ($3.113).

Predicted prices are shown for each strike price based on the LLP Logarithmic Regression Model. Delta values (movement in a call price for each unit change in the underlying futures price) go from 0.007 for the farthest out-of-the-money strikes to 0.629 for one that is closer to being in-the-money. From a delta of 0.500 (near to the point at which the futures price equals a strike price), the predicted prices gradually approach the intrinsic value (futures price minus strike price), and from that point the call price moves dollar-for-dollar with the price change in the underlying futures contract.

Upper and lower breakeven prices for each strike price are the prices of the futures contract on the call’s expiration date in December that will result in zero gain or loss from a trade on May 27, 2016, selling the number of calls indicated by the inverse of delta while taking a long position in the underlying. For example, sell 1/0.348 or 2.87 calls at strike 2.30. This generates 2.87 times the premium price of $1,663, or a total of $4,773 for the trader’s margin account. At the option’s expiration date, if the futures price is either 2.4935 or 1.9373, the trade will result in zero profit or loss (it will break even).

A net profit will result from an expiration futures price between the upper and lower breakeven prices, while a loss will occur beyond breakeven in either direction.

The breakeven prices are related to volatility of the underlying futures as determined by the options market. Although prices are changing though the day, the structure of option prices at any moment is shown on the chart “Price curve for December copper” (right). The time is the close of trading on May 27, 2016.

In addition to the price curve for December 2016 calls with the underlying futures price at 2.1285, there are two straight lines. One line shows the intrinsic value (futures price less the strike price) increasing as the strike price declines. The call price curve is shown approaching the intrinsic value at the upper right of the chart.

The other straight line is tangent to the call price curve at strike 2.30. On the “Copper calls” chart price model, strike 2.30 corresponds to the predicted call price of 0.0646. The call price occurs on the curve directly above strike 2.30.

From the points on the price curve above 2.30, delta indicates the slope of the line extending left down to the horizontal axis and upward to the right until it intersects the intrinsic value line.

For strike 2.30, the delta is 0.348. On the horizontal axis, the 2.30 strike tangent line intersects at 2.4935, which is the upper breakeven price for a delta trade based on the 2.30 strike.

From the point of intersection of the 2.30 strike tangent line with the intrinsic value line, a vertical line dropped down to the horizontal axis will show the lower breakeven price of 1.9373.

If a sloped line could be computed for every strike price, the predicted call price curve would be shown as a series of tangent points, with one above each strike price. Every call price curve — futures, stocks, or any asset that can be traded at a fixed price for a limited period — is constructed this way. The market predicts a spread of future prices for the underlying and this spread in turn determines the upper and lower breakeven prices at expiration. The more volatile the underlying, the wider the breakeven spread.

### Intrinsic movement

As shown on “Copper calls” the upper breakeven price has more potential for movement than the lower breakeven. This is due to the intrinsic value forming a barrier that prevents the lower breakeven price from going farther along the horizontal axis. Three additional charts look at the upper breakeven price related to underlying futures and stock prices and show that the breakeven price structure is affected by two factors: Volatility of the underlying and the type of underlying asset.

We will look at the upper breakeven related to futures, stocks and stock indexes, listing the volatility of each asset measured by the height of the call price curve at the point where the underlying equals the strike price.

“Futures: Upper breakeven” (below) includes four futures contracts: wheat, live cattle, copper and crude oil. Reflecting the relatively high implied volatility of crude oil, its upper breakeven price shows 60% at the lowest delta value versus 20% for live cattle futures. All of the futures breakeven paths illustrate the left side of a volatility smile, with the lowest point of the curve close to where the call’s price begins its move up toward the intrinsic value. Volatilities are as follows: crude oil, 10.61%; wheat, 7.05%; copper, 6.56%; and live cattle, 4.81%.

“Stocks: Upper breakeven” (below) illustrates the typical curve for common stock call options. **Starbucks** (SBUX) and **Valero** (VLO) show considerable volatility with the breakeven percentage starting at or near 50% at low delta values.

On the futures and stock charts, high breakeven prices give an advantage to an option seller — for either naked call writing or strangles, in which a call and a put are sold short at low delta values. Volatilities are Starbucks at 13.58% and Valero at 11.64%.

Different shapes occur on “Indexes: Upper breakeven” (below). The two top curves for the more volatile Nasdaq 100 mini and Russell 2000 mini contracts’ decline from 18% and 17% are paired in this analysis as well as in other studies showing historical price changes. For these indexes, there is an almost straight line movement down toward the delta of 0.500.

The E-mini S&P 500 and E-mini DJIA indexes start at relatively low percentages, then both form a saucer shape before declining at 0.500 delta. A writer of call options should be aware of the differences in height (determined by implied volatility) and curve shapes that may result in better or worse performance from trading in options or strangles. Volatilities for stock index minis are the Nasdaq with 4.88%, Russell with 4.59%, S&P 500 with 4.33% and DJIA with 4.11%

Because the options market generates upper and lower breakeven prices for every call, one approach to trading short strangles is to select calls for sale at a price higher than the upper breakeven price and to sell puts at prices lower than the lower breakeven.

### Profiting from strangles

A study of short strangles on six currencies described in “Weighing options: new age of volatility,” Modern Trader, June 2016, resulted in all six short strangles being successful. Closing prices on June 3 were between the upper and lower breakeven prices. The seller of options would be able to keep the original put and call premiums. At the same time, the proposed long strangles — buying a put and call at strike prices within the breakeven boundaries — were generally unprofitable although losses might have been reduced by selling long options before the expiration date.

The following table lists the prices on May 27, 2016, and the computed upper and lower breakeven prices for December 2016 expiration. Short strangles initiated on May 27 would sell calls at strike prices above the upper breakeven price and sell puts below the lower breakeven price.

By selling a call just above the upper breakeven price and selling a put below the lower breakeven, we can experimentally test the market’s ability to set boundary prices for short strangles. June to December is a relatively long period, but the market should take time into account in estimating the breakeven spread.