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

The BlackScholesGamma command computes the Gamma 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.

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

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

$\frac{\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}}{200\mathrm{\sigma }\sqrt{\mathrm{\pi }}}$

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

$0.01260567542$

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

$\frac{\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}}{200\mathrm{\sigma }\sqrt{\mathrm{\pi }}}$

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

$0.01260567513$

$\mathrm{BSGamma}≔\mathrm{expand}\left(\mathrm{BlackScholesGamma}\left(100\,100\,1\,\mathrm{\sigma }\,r\,0.03\,'\mathrm{call}'\right)\right)$

$\mathrm{BSGamma}≔\frac{0.001965014020{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\ⅇ}^{-0.4999999997r}{\ⅇ}^{-\frac{0.4999999997r^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{0.02999999998r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.0004499999997}{{\mathrm{\sigma }}^2}}}{\mathrm{\sigma }}+\frac{0.0001179008410{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\ⅇ}^{-0.4999999997r}{\ⅇ}^{-\frac{0.4999999997r^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{0.02999999998r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.0004499999997}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^3}-\frac{0.003930028034r{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\ⅇ}^{-0.4999999997r}{\ⅇ}^{-\frac{0.4999999997r^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{0.02999999998r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.0004499999997}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^3}-\frac{0.0001179008410{\ⅇ}^{-0.5000000002r}{\ⅇ}^{-\frac{0.0004499999998}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{0.02999999998r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997r^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{{\mathrm{\sigma }}^3}+\frac{0.003930028033r{\ⅇ}^{-0.5000000002r}{\ⅇ}^{-\frac{0.0004499999998}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{0.02999999998r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997r^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{{\mathrm{\sigma }}^3}+\frac{0.001965014018{\ⅇ}^{-0.5000000002r}{\ⅇ}^{-\frac{0.0004499999998}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{0.02999999998r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997r^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}$

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

$\mathrm{BlackScholesGamma}\left(100\,50\,1\,\mathrm{\sigma }\,r\,d\,'\mathrm{put}'\right)$

$\frac{2^{\frac{d-r}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}}{200\mathrm{\sigma }\sqrt{\mathrm{\pi }}}$

$\mathrm{BlackScholesGamma}\left(100\,50\,1\,0.3\,0.05\,0.03\,'\mathrm{put}'\right)$

$0.000529595076$

$\mathrm{BlackScholesGamma}\left(100\,t\mapsto \max \left(50-t\,0\right)\,1\,\mathrm{\sigma }\,r\,d\right)$

$\frac{{\ⅇ}^{-r}\sqrt{2}\left(2^{\frac{{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}-2^{\frac{3{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}\ln \left(2\right){\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}+2^{\frac{3{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}d{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}-2^{\frac{3{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}r{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}+{\ⅇ}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+2\ln \left(2\right){\ⅇ}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}-2{\ⅇ}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}d+2{\ⅇ}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}r\right)}{800\sqrt{\mathrm{\pi }}{\mathrm{\sigma }}^3}$

$\mathrm{BlackScholesGamma}\left(100\,t\mapsto \max \left(50-t\,0\right)\,1\,0.3\,0.05\,0.03\,d\right)$

$0.0005295950875$

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

$S≔\frac{{\&ExponentialE;}^{-r}\sqrt{2}\left(2^{\frac{{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2{\&ExponentialE;}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}-2^{\frac{3{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}\ln \left(2\right){\&ExponentialE;}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}+2^{\frac{3{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}d{\&ExponentialE;}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}-2^{\frac{3{\mathrm{\sigma }}^2+2d-2r}{2{\mathrm{\sigma }}^2}}r{\&ExponentialE;}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2-4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}+{\&ExponentialE;}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+4{\&ExponentialE;}^{-\frac{{\left({\mathrm{\sigma }}^2+2d-2r\right)}^2}{8{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+2\ln \left(2\right){\&ExponentialE;}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}-2{\&ExponentialE;}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}d+2{\&ExponentialE;}^{-\frac{{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)-2d+2r\right)}^2}{8{\mathrm{\sigma }}^2}}r\right)}{800\sqrt{\mathrm{\pi }}{\mathrm{\sigma }}^3}$

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

$C≔\frac{\sqrt{2}{\&ExponentialE;}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}}{200\mathrm{\sigma }\sqrt{\mathrm{\pi }}}$

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

$P≔\frac{2^{\frac{d-r}{{\mathrm{\sigma }}^2}}{\&ExponentialE;}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4{\ln \left(2\right)}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}}{200\mathrm{\sigma }\sqrt{\mathrm{\pi }}}$

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

$0$

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

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

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