The BlackScholesPrice command computes the price 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.

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

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

$-50{\ⅇ}^{-d}\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)+50{\ⅇ}^{-r}\mathrm{erf}\left(\frac{\left({\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)+50{\ⅇ}^{-d}-50{\ⅇ}^{-r}$

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

$12.44264640$

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

$-50{\ⅇ}^{-r}\left(\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right){\ⅇ}^{r-d}-{\ⅇ}^{r-d}-\mathrm{erf}\left(\frac{\left({\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)+1\right)$

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

$12.44264640$

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

$\mathrm{Price}≔48.52227668+48.52227668\mathrm{erf}\left(\frac{0.7071067810\left(-0.03000000000+r+0.5000000000{\mathrm{\sigma }}^2\right)}{\mathrm{\sigma }}\right)-100.{\ⅇ}^{-1.r}\left(0.5000000000+0.5000000000\mathrm{erf}\left(\frac{0.7071067810\left(-0.03000000000+r+0.5000000000{\mathrm{\sigma }}^2\right)}{\mathrm{\sigma }}-0.7071067810\mathrm{\sigma }\right)\right)$

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

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

$-50{\ⅇ}^{-d}\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(3\right)-2\ln \left(5\right)+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)+60{\ⅇ}^{-r}\mathrm{erf}\left(\frac{\left({\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(3\right)-2\ln \left(5\right)+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)+60{\ⅇ}^{-r}-50{\ⅇ}^{-d}$

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

$22.92329470$

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

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

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

$22.92329473$

$S≔\mathrm{BlackScholesPrice}\left(100\&comma;t\mapsto \mathrm{piecewise}\left(t<90\&comma;90-t\&comma;t<110\&comma;0\&comma;t-110\right)\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\right)$

$S≔5{\&ExponentialE;}^{-r}\left(10\mathrm{erf}\left(\frac{\sqrt{2}\left({\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-2\ln \left(11\right)-2d+2r\right)}{4\mathrm{\sigma }}\right){\&ExponentialE;}^{r-d}+10\mathrm{erf}\left(\frac{\sqrt{2}\left({\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-4\ln \left(3\right)-2d+2r\right)}{4\mathrm{\sigma }}\right){\&ExponentialE;}^{r-d}-11\mathrm{erf}\left(\frac{\sqrt{2}\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-2\ln \left(11\right)-2d+2r\right)}{4\mathrm{\sigma }}\right)-9\mathrm{erf}\left(\frac{\sqrt{2}\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-4\ln \left(3\right)-2d+2r\right)}{4\mathrm{\sigma }}\right)-2\right)$

$C≔\mathrm{BlackScholesPrice}\left(100\&comma;110\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\&comma;'\mathrm{call}'\right)$

$C≔50{\&ExponentialE;}^{-d}\mathrm{erf}\left(\frac{\sqrt{2}\left({\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-2\ln \left(11\right)-2d+2r\right)}{4\mathrm{\sigma }}\right)-55{\&ExponentialE;}^{-r}\mathrm{erf}\left(\frac{\sqrt{2}\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-2\ln \left(11\right)-2d+2r\right)}{4\mathrm{\sigma }}\right)+50{\&ExponentialE;}^{-d}-55{\&ExponentialE;}^{-r}$

$P≔\mathrm{BlackScholesPrice}\left(100\&comma;90\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\&comma;'\mathrm{put}'\right)$

$P≔-45{\&ExponentialE;}^{-r}\mathrm{erf}\left(\frac{\sqrt{2}\left(-{\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-4\ln \left(3\right)-2d+2r\right)}{4\mathrm{\sigma }}\right)+50{\&ExponentialE;}^{-d}\mathrm{erf}\left(\frac{\sqrt{2}\left({\mathrm{\sigma }}^2+2\ln \left(2\right)+2\ln \left(5\right)-4\ln \left(3\right)-2d+2r\right)}{4\mathrm{\sigma }}\right)+45{\&ExponentialE;}^{-r}-50{\&ExponentialE;}^{-d}$

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

$0$

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

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

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