changes. Color Color, gamma decay or DgammaDtime measures the rate of change of gamma over the passage of time. Color is a third-order derivative of the option value, twice to underlying asset price and once to time. Color can be an important sensitivity to monitor when maintaining a gamma-hedged portfolio as it can help the trader to anticipate the effectiveness of the hedge as time passes. The mathematical result of the formula for color (see below) is expressed in gamma per year. It is often useful to divide this by the number of days per year to arrive at the change in gamma per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, color itself may change quickly, rendering full day estimates of gamma change inaccurate. Ultima Ultima measures the sensitivity of the option vomma with respect to change in volatility. Ultima has also been referred to as DvommaDvol. Ultima is a third-order derivative of the option value to volatility. Greeks for multi-asset options If the value of a derivative is dependent on two or more underlyings, its Greeks are extended to include the cross-effects between the underlyings. Correlation delta measures the sensitivity of the derivative's value to a change in the correlation between the underlyings. It is also commonly known as cega. Cross gamma measures the rate of change of delta in one underlying to a change in the level of another underlying. Cross vanna measures the rate of change of vega in one underlying due to a change in the level of another underlying. Equivalently, it measures the rate of change of delta in the second underlying due to a change in the volatility of the first underlying. Cross volga measures the rate of change of vega in one underlying to a change in the volatility of another underlying. Formulas for European option Greeks The Greeks of European options (calls and puts) under the Black–Scholes model are calculated as follows, where (phi) is the standard normal probability density function and is the standard normal cumulative distribution function. Note that the gamma and vega formulas are the same for calls and puts. For a given: Stock price , Strike price , Risk-free rate , Annual dividend yield , Time to maturity (represented as a unit-less fraction of one year), and Volatility . where Under the Black