# The Pricing of Commodity Options

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The present research will prove particularly useful to option traders. The analysis proposed by the **HyperVolatility** Team will explain, in a few bullet points, how the most popular commodity options pricing models behave and what the practical divergences in terms of prices are. The present study is very valuable to anyone interested in trading options because most trading platforms allow the trader to choose the model via which the theoretical value of the options will be calculated and consequently shown. The pricing models that will be analyzed are the Barone–Adesi –Whaley, the Bjerksund & Stensland (the 2002 version), the Black–76, the Binomial Tree and the classic Black–Scholes–Merton one. The models have been tested against each other and the following charts graphically show the divergence of 1 pricing model with respect to all others. The research has been performed assuming that the underlying asset (S) is a WTI crude oil futures contract, that the volatility (σ) is 20%, that the interest rate (r) is 0.5% and that the Cost of Carry is 0 (which is normal when dealing with commodity options).

As previously mentioned, the study will examine 1 pricing model at the time and, in order to avoid confusion and make things simpler, we decided to list the most important aspects below each graph:

1) The Barone-Adesi-Whaley model overprices options when compared to other formulas. The pricing spread with respect to other models is on average between 0.06% and 0.08%

2) The Barone-Adesi-Whaley prices tend to get closer to other models as the expiration increases

3) The Barone-Adesi-Whaley model, on average, tends to overprice options with respect to the Binomial Tree (~ 0.16% higher) for short maturities. The trend is higher for out-of-the-money options and particularly for put options

4) The prices derived from the Bjerksund & Stensland model are always lower than Barone-Adesi-Whaley prices. The difference is bigger for 1 month options (~ 0.16%)

5) The Black-76 performs as well as the Black–Scholes–Merton model, however, their results overlap and that is why the Black-76 curve is not visible

6) The difference with the Black–Scholes–Merton model becomes larger as the expiration increases but it is not higher than 0.1%

1) The Bjerksund & Stensland model under–prices options in respect to other models. On average the difference ranges between 0.05% — 0.06%

2) The under–pricing tends to reduce as the expiration increases

3) The Bjerksund & Stensland model produces prices which are lower than the Barone–Adesi–Whaley one for any expiration

4) The Black–Scholes–Merton model approximates to the Bjerksund & Stensland one from the 8th month onwards

5) The Black–76 performed as well as the Black–Scholes–Merton model and that is why the overlapped curve cannot be seen in the chart

6) The Binomial Tree approach shows the highest differential with respect to the Bjerksund & Stensland model. The divergence in pricing oscillates around 0.15%

1) The Black–76 model over–prices options only with respect to the Bjerksund & Stensland one (almost 0.05%)

2) The divergence between Black–76 and Bjerksund & Stensland attenuates when longer expirations are approached

3) The Barone–Adesi–Whaley model prices are slightly higher than Black–76 ones and the discrepancy augments with the passage of time (between 0.08% and 0.1% for 10 months and 1 year expiring options respectively)

4) The Binomial Tree approach, if we exclude the short term, delivers higher prices than the Black–76 model but the divergence oscillates around the interval 0.03% — 0.04%

5) The Black–76 model performed as well as the Black–Scholes–Merton one and that is why the BSM curve is flat to 0. Needless to say that the Black–Scholes–Merton curve suggests that there is no difference in pricing

1) The Binomial Tree under–prices options with respect to other models in the short term (around 2.5%) but the divergence is much lower for longer dated derivatives

2) The Barone–Adesi–Whaley model and the BSM model perform as well as the Black–76 one therefore their curves are hidden in the chart

3) The Bjerksund & Stensland model provided higher prices for short dated options but in the long term the Binomial Tree approach shows a slight over–pricing tendency with respect to the former. However, the spread is no higher than 0.04% — 0.05%

1) The performances of the Black–Scholes–Merton formula with respect to other pricing models match perfectly well with the outcome generated by the Black–76 model

2) The above reported chart is identical to the graph extrapolated for the Black–76 model, in fact, the green curve does not move from the 0 axis

If you are interested in trading options you might want to read also the HyperVolatility researches entitled **“Options Greeks: Delta, Gamma, Vega, Theta, Rho”****, ****“Options Greeks: Vanna, Charm, Vomma, DvegaDtime”****, ****“Options Greeks and Hedging Strategies”****, ****“Extracting Implied Volatility: Newton-Raphson, Secant and Bisection Approaches”****, ****“The VIX Index: step by step” **and** “The Volatility Smile”****.**

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