The Color of an option or a portfolio of options measures Gamma's sensitivity to movement in the time to maturity.

The BlackScholesColor command computes the Color of a European-style option with the specified payoff function.

The parameter $S_0$ is the initial (current) value of the underlying asset. The parameter T is the time to maturity in years.

The parameter K specifies the strike price if this is a vanilla put or call option. Any payoff function can be specified using the second calling sequence. In this case the parameter P must be given in the form of an operator, which accepts one parameter (spot price at maturity) and returns the corresponding payoff.

The sigma, r, and d parameters are the volatility, the risk-free rate, and the dividend yield of the underlying asset. These parameters can be given in either the algebraic form or the operator form. The parameter d is optional. By default, the dividend yield is taken to be 0.

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$\mathrm{Color}≔\mathrm{BlackScholesColor}\left(S\left[0\right]\,100\,T\,0.1\,0.05\,0\,'\mathrm{call}'\right)\:$

$\mathrm{plot3d}\left(\mathrm{Color}\,T=1.0..0\,S\left[0\right]=0..200\,'\mathrm{labels}'=\left[''Time\; To\; Maturity''\,''Spot\; Price''\,''Value''\right]\,'\mathrm{colorscheme}'=\left[''zgradient''\,\left[''Black''\,''White''\,''Red''\right]\right]\,'\mathrm{thickness}'=0\right)$

$\mathrm{BlackScholesColor}\left(100\,100\,1\,\mathrm{\sigma }\,r\,d\,'\mathrm{call}'\right)$

$-\frac{\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left({\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2+4{\mathrm{\sigma }}^2\right)}{1600{\mathrm{\sigma }}^3\sqrt{\mathrm{\pi }}}$

$\mathrm{BlackScholesColor}\left(100\,100\,1\,0.3\,0.05\,0.03\,'\mathrm{call}'\right)$

$\mathrm{-0.006976891196}$

$\mathrm{BlackScholesColor}\left(100\,t\mapsto \max \left(t-100\,0\right)\,1\,\mathrm{\sigma }\,r\,d\right)$

$-\frac{\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left({\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2+4{\mathrm{\sigma }}^2\right)}{1600{\mathrm{\sigma }}^3\sqrt{\mathrm{\pi }}}$

$\mathrm{BlackScholesColor}\left(100\,t\mapsto \max \left(t-100\,0\right)\,1\,0.3\,0.05\,0.03\right)$

$\mathrm{-0.006976892}$

$\mathrm{Color}≔\mathrm{BlackScholesColor}\left(100\,100\,1\,\mathrm{\sigma }\,r\,0.03\,'\mathrm{call}'\right)$

$\mathrm{Color}≔\frac{-0.001496033551r{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4-0.001934870059r{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-5.385720778\times {10}^{\mathrm{-6}}r{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}-0.001011433957{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4-0.001012316037{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4-0.0000606860374{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+0.001993831532{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r+0.000967879383{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2{r}^2-0.000483939693{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{4}r-0.002992067103r^2{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+0.00006073896209{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-5.226548674\times {10}^{\mathrm{-8}}{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}+5.385720779\times {10}^{\mathrm{-8}}{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}-0.0002419698462{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6+0.001935758768{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^3-0.0001742182891{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2+5.226548674\times {10}^{\mathrm{-6}}{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}r-0.0002493389254{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6-0.001994711398{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{r}^3+0.0001795240259{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998+0.01499999999{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}{r}^2}{{\mathrm{\sigma }}^5}$

$\mathrm{plot3d}\left(\mathrm{Color}\,\mathrm{\sigma }=0..1\,r=0..1\right)$

$\mathrm{BlackScholesColor}\left(100\,120\,1\,0.3\,0.05\,0.03\,'\mathrm{put}'\right)$

$\mathrm{-0.00440899910}$

$\mathrm{BlackScholesColor}\left(100\,t\mapsto \max \left(120-t\,0\right)\,1\,0.3\,0.05\,0.03\,0\right)$

$\mathrm{-0.004408999}$

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

The Finance[BlackScholesColor] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.