The Delta of an option or a portfolio of options is the sensitivity of the option or portfolio to changes in the value of the underlying asset

The BlackScholesDelta command computes the Delta 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.

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

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

$-\frac{{\ⅇ}^{-d}\left(\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-1\right)}{2}$

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

$0.568453937$

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

$-\frac{{\ⅇ}^{-d}\left(\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-1\right)}{2}$

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

$0.5684539378$

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

$\mathrm{\Delta }≔\frac{0.4852227668\mathrm{\sigma }+0.4852227668\mathrm{erf}\left(\frac{-0.02121320343+0.707106781r+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }+0.387151754{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}-0.3989422803{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}}{\mathrm{\sigma }}$

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

$\mathrm{BlackScholesDelta}\left(100\,120\,1\,\mathrm{\sigma }\,r\,d\,'\mathrm{put}'\right)$

$\frac{{\ⅇ}^{-d}\left(\mathrm{erf}\left(\frac{\left(-2\ln \left(\frac{6}{5}\right)+{\mathrm{\sigma }}^2-2d+2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-1\right)}{2}$

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

$\mathrm{-0.632854644}$

$\mathrm{BlackScholesDelta}\left(100\,t\mapsto \max \left(120-t\,0\right)\,1\,\mathrm{\sigma }\,r\,d\right)$

$\frac{{\ⅇ}^{-r}\left(2^{-\frac{\ln \left(\frac{3}{5}\right)-{\mathrm{\sigma }}^2+d-r}{{\mathrm{\sigma }}^2}}{5}^{\frac{{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}{3}^{\frac{\frac{{\mathrm{\sigma }}^2}{2}+\ln \left(5\right)-d+r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(3\right)}^2+4{\ln \left(5\right)}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}+5\mathrm{erf}\left(\frac{\left(-2\ln \left(\frac{6}{5}\right)+{\mathrm{\sigma }}^2-2d+2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right){\ⅇ}^{r-d}\sqrt{\mathrm{\pi }}\mathrm{\sigma }-5{\ⅇ}^{r-d}\sqrt{\mathrm{\pi }}\mathrm{\sigma }-6{\ⅇ}^{-\frac{{\left(-2\ln \left(\frac{6}{5}\right)-{\mathrm{\sigma }}^2-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}\sqrt{2}\right)}{10\sqrt{\mathrm{\pi }}\mathrm{\sigma }}$

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

$\mathrm{-0.6328546388}$

$S≔\mathrm{BlackScholesDelta}\left(100\&comma;t\mapsto \mathrm{piecewise}\left(t<50\&comma;50-t\&comma;t<100\&comma;0\&comma;t-100\right)\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\right)$

$S≔-\frac{{\&ExponentialE;}^{-r}\left(-2\mathrm{erf}\left(\frac{\sqrt{2}\left({\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}{4\mathrm{\sigma }}\right){\&ExponentialE;}^{r-d}\sqrt{\mathrm{\pi }}\mathrm{\sigma }+2\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right){\&ExponentialE;}^{r-d}\sqrt{\mathrm{\pi }}\mathrm{\sigma }+{\&ExponentialE;}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}\sqrt{2}-2^{\frac{{\mathrm{\sigma }}^2+d-r}{{\mathrm{\sigma }}^2}}{\&ExponentialE;}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\right)}{4\sqrt{\mathrm{\pi }}\mathrm{\sigma }}$

$C≔\mathrm{BlackScholesDelta}\left(100\&comma;100\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\&comma;'\mathrm{call}'\right)$

$C≔-\frac{{\&ExponentialE;}^{-d}\left(\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-1\right)}{2}$

$P≔\mathrm{BlackScholesDelta}\left(100\&comma;50\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\&comma;'\mathrm{put}'\right)$

$P≔\frac{{\&ExponentialE;}^{-d}\left(\mathrm{erf}\left(\frac{\sqrt{2}\left({\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}{4\mathrm{\sigma }}\right)-1\right)}{2}$

$\mathrm{simplify}\left(S-C-P\right)$

$0$

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

The Finance[BlackScholesDelta] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.