From the June 01, 2008 issue of Futures Magazine • Subscribe!

A Black-Scholes peek at futures prices

For futures contracts, option-pricing models provide a significant source of information on the market consensus regarding price movements until expiry. Given the quantity of price information available through option price data — and the relatively small investment in time needed to perform the calculations described here — the results of viewing future price predictions included in basic option pricing models seem well worth the effort.

The Black-Scholes and LLP (log-log parabola) option pricing models are available for download at futuresmag.com. They can be accessed by going to Free Tools, then to Downloads/Formulas and Excel Spreadsheets. Select from the three pricing models.

Here, we’ll use June 2008 euro forex futures call prices to illustrate option price predictions (see “Euro FX calls”).

PRICE LINES

Looking at the chart, a tangent line touching the curve at the 1.565 strike price is added to the option price curve, along with an up-sloping line of intrinsic values (equal to the euro futures price of 1.5655 less strike prices extending from 1.565 to 1.495). The upper and lower breakeven prices are found at the point where the tangent line intersects the horizontal axis (zero option value) at the left side of the chart and the intrinsic value line on the right side. As shown on the table “Price breakdown,” the 1.565 strike price corresponds to the underlying price of 1.5655, while the upper and lower breakeven prices are 1.6368 and 1.4999, respectively.

The breakeven prices are those that will permit a trader who uses a delta-neutral trade, using the delta slope (hedge ratio) to determine how many call options to buy in hedging a short sale of one futures contract. In this example, the slope of the curve at the 1.565 strike is 0.5247. The reciprocal of this number, 1.90585, is how many options to buy for each short futures contract. In this neutral hedge, the trader is not taking a bullish or bearish position. Keep in mind that it is understood that the fraction indicating the number of options bought will be rounded in an actual trade.

“Breakeven spreads” shows the differences between the upper and lower breakeven prices over the given range of strike prices. As the strike price moves higher and lower than the current underlying futures price, the spreads increase. Because it might normally be assumed that the market consensus of future price spreads would be constant for all strike prices, the phenomena of increased spreads in both directions are referred to as volatility skews. This is related to the effect noted by Galen Burghardt in “The Eurodollar Futures and Options Handbook” (McGraw-Hill, 2003), which looks at the variation among implied volatilities at different strike prices.

It is easy to see the relationship of the upper and lower breakeven prices with volatility measures. Traders in the option market make predictions on future price movements as they estimate the chances of profits or losses on various combinations of puts and calls. The higher volatility is, the wider the spread between breakeven prices.

In the example described above, a short sale of the underlying at 1.5655 is accompanied by a purchase of 1.90585 call options at 0.0344. The cost of the trade is 0.06556. At expiration, if the futures price is 1.6368, the loss on the short sale is 0.0713. Each call gains 0.718 (1.6368 - 1.5650). This gain multiplied by 1.90585 less the initial cost of 0.06556 equals 0.0713 for a breakeven trade at the upper breakeven price.

If the underlying price is 1.4999 at expiration, the calls will expire without value while the short futures contract gains 0.656 (1.5655 - 1.4999), approximating the 0.06556 initial cost and achieving zero profit or loss at the lower breakeven price.

FINDING BREAKEVEN

It is possible to use the call option-pricing model to find upper and lower breakeven prices when ratios of options bought against the short sale of the underlying asset are used. For example, suppose the trader decided not to use the delta-neutral ratio but to calculate breakeven prices at a ratio of 0.50 for all strike prices. Upper and lower breakeven prices will then be adjusted to correspond with the trader’s position of a bullish or bearish view of expected future underlying prices.

By purchasing a larger than neutral number of options, a bullish sentiment is expressed. Both the upper and lower breakeven prices will fall. The resulting effect on breakeven price spreads is shown on “Breakeven spreads.” While both the delta-neutral and 0.50 ratio spreads increase as strike prices are higher or lower than the current underlying price, the adjusted ratio causes greater spread differences.

Possibly the most important observation here is that the upper and lower breakeven prices correspond to the delta-neutral hedge and are not simply derived from the option price curve but are determinants of the option prices across all of the strike prices.

The sequence of generating option prices is:

1. The market predicts the future price of the underlying, expressed as a spread from high to low depending on expected price volatility and the time remaining to expiration

2. Straight, sloped lines are computed between the points at which the predicted high and low prices intersect the horizontal axis and intrinsic value line — with the slope of the line equal to delta of the option price curve being developed

3. The point on the sloped line above the strike price is a point of tangency on the option price curve

4. The option price curve is the series of tangent points corresponding to all of the current strike prices

This process, progressing from breakeven prices to computed option prices, shows that without a price spread being forecast for the time of expiration, the option price could not exist. As the price spread widens or narrows with a change in perceived volatility or time to expiration, the tangent points push the option price curve higher or lower.

Put and call prices may be located by using a purely geometric process. The option price is a point on a straight line drawn from the lower breakeven price to the intrinsic value above the upper breakeven price. The point representing the option price is above the underlying price.

The geometric solution, with the process reversed for put prices, shows that the option price “curve” is composed of a series of points on straight lines whose slopes determine the delta value for each put and call price.

Upper and lower breakeven prices and the breakeven price spread should be valuable information for traders in any underlying future, even when an option trade is not contemplated. Because they reflect current forecasts of price changes by the market of options, the price predictions from option-pricing models can help anyone who is trying to assess possible future trends.

EURODOLLAR LINK

Although breakeven prices computed from option pricing models may be helpful information for traders in any instrument, they are particularly valuable for those who trade Eurodollar futures with maturities longer than one year.

Eurodollar quarterly futures prices are focused on the present U.S. Treasury yield curve in an attempt to make the Eurodollar yield curve approximately parallel to Treasury yields. Thus, Eurodollar futures of all maturities are about the present. On the other hand, the upper and lower breakeven price spreads associated with options on Eurodollars show the market’s predictions of quarterly prices and interest rates through the expiration.

“Breakeven prices” shows upper and lower breakeven prices at strikes near the underlying market prices of several futures contracts on March 27. For Eurodollar futures and options, the price and rate spread shown is approximately 70 basis points, with quarterly interest rates in June ranging from 1.9697% to 2.6587% when the breakeven prices are subtracted from 100. These forecasts may be reviewed in June to see how accurate the market consensus was about three months ahead of expiration.

Option breakeven price predictions are available because of the broad acceptance of the Black-Scholes model and variations of it. Before its introduction in 1973, option traders were left to their own estimates of option values. Black-Scholes, with its complete theoretical pricing, made it possible for thousands of options with various strike prices and expirations to be valued simultaneously. Now these same concepts can be used beyond simple pricing calculations to analyze market action and even make predictions for future price direction.

The close correlations of market prices with pricing formula predictions show that an option trader is locked into a system that permits little variation from the theoretical model. Another footprint of Black-Scholes is the mathematical quirk referred to above as volatility skews, in which implied volatilities for the same underlying futures contract vary among strike prices in a predictable manner.

Use of option breakeven prices is an easy way for futures traders to see what the market is forecasting. The predictions are a natural consequence of the math involved with option price models — and a collateral benefit to every futures trader.

Paul Cretien, CFA, is an investment analyst and financial case writer. He may be e-mailed at PaulDCretien@aol.com.

Comments
comments powered by Disqus
Check out Futures Magazine - Polls on LockerDome on LockerDome