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

The BlackScholesCharm command computes the Charm 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.

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

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

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

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

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

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

$\mathrm{-0.0239148274}$

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

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

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

$\mathrm{-0.02391495679}$

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

$\mathrm{Charm}≔\frac{8.\times {10}^{\mathrm{-8}}\left(181958.5375{\mathrm{\sigma }}^3+181958.5375\mathrm{erf}\left(\frac{0.707106781r-0.02121320343+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right){\mathrm{\sigma }}^3+2.56488037\times {10}^6{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-0.00125{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r-604924.616{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4+2.419698461\times {10}^6{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2-145181.9076{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}r+2177.728615{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}-2.493389254\times {10}^{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}}{\mathrm{\sigma }}^2-2.49338925\times {10}^6{\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+149603.3551r{\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}}-2244.050326{\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}}-2.568190929\times {10}^6{\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 }}^2-623347.3128{\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\right)}{{\mathrm{\sigma }}^3}$

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

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

$\mathrm{-0.1668222722}$

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

$\mathrm{-0.1668223310}$

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

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

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