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.