Options pricing models include two that are online as free Excel worksheet programs at www.futuresmag.com: The Black-Scholes model and LLP (log-log parabola) model.
Black-Scholes and similar models were critical to the development of trading at facilities such as the Chicago Board Options Exchange (CBOE) in 1973. As theoretical pricing programs that may be computerized for instant delivery, options pricing models were the necessary foundation that enabled options trading to grow from a dozen or so OTC puts and calls advertised by brokers in financial newspapers pre-1973 to the thousands listed for trading at present.
The LLP model, first described in Futures magazine in February 1985, uses market prices for options on commodity futures and equities that basically are valued according to Black-Scholes. The models are not competitive because the LLP program depends on Black-Scholes, expanding its usefulness in several directions including predictive pricing formulas and indications of over- or under-valuation of options market prices.
“Calls on March 2014 softs” (below) shows the LLP model in action. The chart includes calls on five March 2014 softs futures contracts on June 3, 2013, covering futures prices-to-strike price ratios ranging from 0.70 to 1.00.
The height of each options price curve indicates the relative implied volatility attributed by the market to the underlying. For example, calls on coffee and orange juice futures have the highest expected volatility, while cotton and sugar futures are the lowest, and cocoa has midrange implied volatility according to curve heights.
“Softs futures” (below) shows the cumulative price changes for the five March 2014 contracts over the period March 1 to June 3, 2013. It illustrates how call options are priced according to time to expiration and volatility with no directional forecast. For the two higher priced calls — coffee and orange juice —expected volatilities relate to one that has increased by 20% in price while the other has declined by 10%. The lowest options value for cotton futures is supported by a cumulative price change of approximately zero over the three-month period with little up-or-down price variability.
The Black-Scholes options pricing model typically is used to find the implied volatility of an underlying by a trial-and-error process.
While the other pricing details are known or estimated — time to expiration, risk-free rate of interest and the ratio of futures price to strike price — the standard deviation of underlying price movement is unknown. For example, on June 3, 2013, the July 2014 orange juice futures were priced at 148.35. Time to expiration (382 days) equaled 1.06 years, and a risk-free rate of 0.20% was estimated. Using a strike price of 150, the standard deviation of 0.2474 was computed by the process of successive changes until the estimated call price, 14.4499, approximated the call’s actual market price of 14.450.
When the Black-Scholes model is used to estimate implied volatilities for equities, expected dividends are part of the underlying known values. With futures calculations, the dividend is zero. The model tends to undervalue the put at the same strike price, compared to its market price. For the July 2014 orange juice example, at the 150 strike, the put’s estimated price was 15.78 versus the actual market price of 16.10.
That puts are undervalued only slightly is used in one example to indicate a profitable spread trade using puts at different strike prices. On June 3, 2013, the March 2014 cotton futures were priced at 84.25. With 249 days to expiration (0.69 years), 0.20% risk-free interest rate and a strike price equal to 85, the call’s market price of 4.84 was matched by using the standard deviation of the underlying equal to 0.1838. At the same time, the expected put price was 5.4728 vs. a market price of 7.460. The difference of 1.9872 option price points, or $993.60, seemed excessive.
To check the difference between expected and market prices for the put on March 2014 cotton No. 2, the Black-Scholes model was used on the July 2014 futures and options. This showed the expected put price to be 6.3438 vs. a market price of 6.520 — the usual slight undervaluation by the pricing model, with a reasonable market value for the put.
The pricing analysis based on the Black-Scholes model suggested selling the March 85 put on June 3 while buying the July 85 put to protect against adverse price movements in the underlying. Closing out the trade on June 7 resulted in a net gain of $770, summarized as follows:
- Sell March 85 put at 7.460 for $3,730
- Buy July 85 put at 6.520 for $3,260
- Buy March 85 put at 5.990 for $2,995
- Sell July 85 put at 6.590 for $3,295
The net gain before transaction costs included $735 on the March 85 put and $35 on the July 85 strike. As price protection, the July 85 put could have moved in either direction based on a change in the underlying price of cotton futures. This example shows that the Black-Scholes model has predictive pricing uses in addition to estimating implied volatility.
Volatilities derived from calculations on the options market are expected to be ranked the same by Black-Scholes and LLP, because both models are based on the price range forecasts implied by current put and call prices.
“Volatility comparison” (below) shows how the two models compute volatilities for the set of calls on softs futures on June 3, 2013. The percentages are heights of call price curves measured by the LLP model at the point where the futures price equals the strike price, compared at each expiration date with the Black-Scholes standard deviation.
“Dollar variations: Price curves” (below) shows another potential trading technique provided by the LLP pricing model. For the March 2014 calls on softs futures, the difference from the call price curve is computed. All but the coffee calls have price variations peaking at approximately $20 while the highest variation for March 2014 coffee calls is $87.
Of course, the variations shift continuously and any trade based on specific dollar amounts should be based on a longer-term comparison. In this case, selling the 170 strike at 4.73 ($1,773.75) while buying the 210 strike at 1.80 ($675) on June 3 would have resulted in a gain on June 7. The 170 strike would be bought at 4.32 ($1,620) as the high dollar variation declined, while the 210 strike was sold at 1.56 ($585.00) for a net gain of $63.75.
Having two options pricing models that are compatible with each having special abilities in terms of estimating underlying volatilities, recommending spreads between strike prices and expirations, and computing pricing equations would seem to have definite advantages. As indicated by the softs futures example, there are many interesting and potentially profitable aspects revealed by closer inspection of the Black-Scholes and LLP models.
Paul Cretien is an investment analyst and financial case writer. His e-mail is PaulDCretien@aol.com.