Comparing Option Pricing Models: The Black/Scholes Option Pricing Model Is the Best-Known, but It Isn't Always the Best. Another Pricing Model, the LLP or Log-Log Parabola, Can Give Useful Results Too
Cretien, Paul D., Modern Trader
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" (right).
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. …