BSM Outputs d1 N(d1) d2 N(d2) BSM Call Price BSM Put Price 0.3100 0.6217 0.1500 0.5596 $4.695 $2.634 Options on Futures The Black model valuation and selected outputs for options on another of Solomons holdings, the GPX 500 Index (GPX), are shown in Exhibit 2. The spot index level for the GPX is 187.95, and the index is assumed to pay a continuous dividend at a rate of 2.2% () over the life of the options being valued, which expire in 0.36 years. A futures contract on the GPX also expiring in 0.36 years is currently priced at 186.73. Exhibit 2 Black Model for European Options on the GPX Index Black Model Inputs GPX Index X r T Yield 187.95 180 0.39% 24% 0.36 2.2% Black Model Call Value Black Model Put Value Market Call Price Market Put Price $14.2089 $7.4890 $14.26 $7.20 Option Greeks Delta (call) Delta (put) Gamma (call or put) Theta (call) daily Rho (call) per % Vega per % (call or put) 0.6232 0.3689 0.0139 0.0327 0.3705 0.4231 After reviewing Exhibit 2, Solomon asks Lee which option Greek letter best describes the changes in an options value as time to expiration declines.