The Black/Scholes option pricing model and the log-log parabola (LLP) system are two methods that may be used to compute option price curves and to forecast option prices through the short-term. Although the formulas behind the two models are fairly complex, the calculations may be completed on brief computer spreadsheets with results such as those shown in “Black/Scholes” and “LLP calls”.
To show how the two pricing methods work, we will use euro futures options listed by the Chicago Mercantile Exchange. The trade unit for euro futures is 125,000 euros, with each point equal to $0.0001 per euro or $12.50 per contract. Euro futures have a large number of strike prices traded for puts and calls. This feature makes them ideal for illustrating option price comparisons.
HOW BLACK/SCHOLES WORKS
The Black/Scholes model is a theoretical option pricing method that is based on riskless arbitrage between underlying assets such as stocks or futures contracts and options on the assets’ prices. There are seven fundamental inputs for the model to compute a theoretical option price when the asset has a future cost and when cash in the form of dividends or interest is received before the option’s expiration.
For futures options the variables are asset price, strike price, standard deviation or variance of asset returns, and time to expiration as a proportion of a year. When Black/Scholes is applied to pricing options on futures contracts the dividend and risk-free rate may be set at zero or omitted because no cash is received before expiration and the asset has no cost to discount.
Without Black/Scholes and similar computer-based option pricing models, the market for exchange-traded options on all assets could not exist as it does today. The theoretical price equations were developed in the early 1970s coincidentally with the Chicago Board Options Exchange initializing exchange-traded options, which before had traded only over-the-counter. Following the beginning trades with about a dozen equity options, the option market expanded along with derivative securities of all types.
For the euro example shown in “Black/Scholes,” the strike price selected for illustration purposes on April 28, 2006, is 1.27. The closing price for September 2006 euro futures on that date was 1.2720. The time to expiration is estimated at 0.42 of one year.
Typically, the only unknown input variable is the standard deviation of asset returns. An educated guess or more detailed historical analysis could be used to find this value, but it is easier to estimate variability from current market prices. The standard deviation and variance figures in the Euro example were found by trial and error resulting in the variability measures implied by the option market —0.0805 and 0.0065 respectively.
The hedge ratio on “Black/Scholes,” 0.5224, is equal to the slope of the option price curve at a futures price of 1.2720. The inverse of the slope indicates the number of call options with strike prices of 1.2700 that should be bought to hedge against one short September 2006 Euro futures contract. In this example, each short September 2006 euro futures would be hedged by holding 1.914 call options that have a strike price of 1.27.
This trade is a “delta-neutral” hedge in which the ratio of options to futures is determined only by the slope of the option price curve instead of being influenced by the trader’s opinion regarding future price changes. The hedge ratio nearest the futures price of 1.2720 on the “LLP calls” price curve is the ratio for a 1.27 strike price, or 0.5096.
Retaining the standard deviation of 0.0805 and substituting new strike prices in place of 1.27, Black/Scholes option prices are computed for the 16-strike prices shown on “LLP calls.” The resulting comparison shows how closely the actual market prices of September 2006 call options match the prices generated by Black/Scholes and LLP calculations.
The relationship is so close because most option market prices are created by Black/Scholes calculations and LLP is a regression model that produces an average price curve based on the relationship between market prices and corresponding strike prices. Comparative prices for the market, Black/Scholes and LLP, also are charted in “LLP calls.”
Formulas in “Black/Scholes” compute a theoretical put value along with each estimated call price. Black/Scholes put prices are listed on “LLP puts”. Both LLP spreadsheets show strike prices descending from high to low so the call option price curve rises with higher futures prices and put prices decrease with higher futures prices. Comparative market, Black/Scholes and LLP prices again are closely related, as shown in the in LLP puts.”
LLP option price models calculate upper and lower breakeven prices for the futures contract at expiration when the neutral hedge is traded. Breakeven futures prices for September 2006 euro calls with a strike price of 1.27 are 1.3240 and 1.2160. The difference between the current strike price and upper breakeven, 0.054, is subtracted from 1.2700 to obtain the lower breakeven futures price of 1.2160. The breakeven prices are shown on “Breakeven prices” where the slope of the option price intersects the horizontal axis and intrinsic value line.
Both option pricing methods have advantages. Black/Scholes is the fundamental pricing model for the option market. Once reasonably correct input variables are determined, the resulting prices computed by the model should be close to actual market values. This method gives a trader a way to compare market prices with alternative values based on different inputs. It also provides a means to compute the implied variances for any assets that have options traded, assisting in predicting price movements for investments other than options.
LLP is a regression analysis that produces a predictive price equation from any number of futures strike price combinations, with a minimum of three observations required to compute a curve. Additional price data may be added to create a moving average model of futures and option price relationships. The LLP regression curve is able to predict option prices for an array of strike prices or asset prices at one time.
Predicted prices by either Black/Scholes or LLP should be useful throughout periods of days or weeks, depending on the stability of pricing variables. Changes can be significant and sudden, such as the market’s perception of variability following events that impact underlying asset prices. Time to expiration becomes an increasingly important factor as the expiration date nears; however, the ratio of asset price-to-strike price is the most dependable day-to-day predictive variable.
The closing price for the euro September 2006 futures contract was 1.2840 on May 5, five trading days after the original calculations on April 28. The following table of results indicates close approximations between actual market prices for the options and predicted prices at the 1.27 strike price, with only small differences between Black/Scholes and LLP.
The delta-neutral hedge on April 28 recommended 1.914 calls for each short euro forex September 2006 futures contract. Throughout the one-week period, the futures increased from 1.2720 to 1.2842, for a gain of 122 points or $1,525 and a corresponding loss on the short position. The 1.27 strike price call options increased from 0.0274 to 0.0339 (65 points). The futures loss of $1,525 was offset by a total call option gain of 1.914 x 65 x $12.50 ($1,555).
In this example, the option gain was slightly larger than that of the futures contract. Because the option price curve increases faster than the straight-line slope and falls more slowly, a trade specified as a delta-neutral hedge should be expected to perform somewhat better than the slope would indicate.
Looking at “Breakeven prices,” it is clear that option price curves have an advantage similar to the convexity of bond prices related to increasing or decreasing bond yields.
The example hedge could have been rounded off to buy two calls for each short futures contract. A further advantage of euro futures options is that with the larger number of strike prices, it is likely that one of them will be close to the futures price. Where the futures price is approximately equal to the strike price of any option, the hedge ratio should be about 0.5000, making a two-to-one call to futures hedge appropriate.
Black/Scholes and LLP option price models produce price curves, predicted prices and hedge ratios on computer spreadsheets with a minimum number of formulas. With continuous flows of new information on pricing methods, they are valid and useful instruments for helping traders understand current price relationships in the futures option market.
To download full spreadsheets used for this article, complete with all the calculations necessary to implement both the Black/Scholes and LLP options pricing models, go to the downloads page at www.futuresmag.com.
Paul D. Cretien is a retired professor of finance at Baylor University and a chartered financial analyst. He is the author of the book The Basics of Bank Investments (Graduate School of Banking at LSU, 2004).