The Rho of an option or a portfolio of options is the sensitivity of the option or portfolio to changes in the risk-free rate

The BlackScholesRho command computes the Rho 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.

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

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

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

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

$44.4027473$

$\mathrm{BlackScholesRho}\left(100\,t\→\max \left(t-100\,0\right)\,1\,\mathrm{\σ}\,r\,d\right)$

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

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

$44.40274728$

$\mathrm{\Ρ}≔\mathrm{BlackScholesRho}\left(100\,K\,1\,\mathrm{\σ}\,0.05\,0.03\,\'\mathrm{call}\'\right)$

$\mathrm{\Ρ}≔\frac{38.71517541{\ⅇ}^{-\frac{0.4999999997{\left(4.625170186\+\ln \left(\frac{1}{K}\right)\+0.5{\mathrm{\σ}}^2\right)}^2}{{\mathrm{\σ}}^2}}\+0.4756147122K\mathrm{\σ}\+0.4756147122K\mathrm{\σ}\mathrm{erf}\left(\frac{3.270489202\+0.707106781\ln \left(\frac{1}{K}\right)-0.3535533905{\mathrm{\σ}}^2}{\mathrm{\σ}}\right)-0.3794856357K{\ⅇ}^{-\frac{1.{\left(3.270489202\+0.707106781\ln \left(\frac{1}{K}\right)-0.3535533905{\mathrm{\σ}}^2\right)}^2}{{\mathrm{\σ}}^2}}}{\mathrm{\σ}}$

$\mathrm{plot3d}\left(\mathrm{\Ρ}\,\mathrm{\σ}\=0..1\,K\=70..120\,\mathrm{axes}\=\mathrm{BOXED}\right)$

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

$\frac{25\left(-2{\ⅇ}^{-r}\mathrm{erf}\left(\frac{1}{4}\frac{\left({\mathrm{\σ}}^2\+2\ln \left(2\right)\+2d-2r\right)\sqrt{2}}{\mathrm{\σ}}\right)\sqrt{\mathrm{\π}}\mathrm{\σ}-2{\ⅇ}^{-r}\sqrt{\mathrm{\π}}\mathrm{\σ}\+{\ⅇ}^{-\frac{1}{8}\frac{{\mathrm{\σ}}^4-4\ln \left(2\right){\mathrm{\σ}}^2\+4d{\mathrm{\σ}}^2\+4r{\mathrm{\σ}}^2\+4{\ln \left(2\right)}^2\+8\ln \left(2\right)d-8\ln \left(2\right)r\+4d^2-8dr\+4r^2}{{\mathrm{\σ}}^2}}\sqrt{2}-2\sqrt{2}{\ⅇ}^{-\frac{1}{8}\frac{{\mathrm{\σ}}^4\+4\ln \left(2\right){\mathrm{\σ}}^2\+4d{\mathrm{\σ}}^2\+4r{\mathrm{\σ}}^2\+4{\ln \left(2\right)}^2\+8\ln \left(2\right)d-8\ln \left(2\right)r\+4d^2-8dr\+4r^2}{{\mathrm{\σ}}^2}}\right)}{\sqrt{\mathrm{\π}}\mathrm{\σ}}$

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

$-94.32991431$

$\mathrm{BlackScholesRho}\left(50\,t\→\max \left(100-t\,0\right)\,1\,\mathrm{\σ}\,r\,d\right)$

$\frac{25{\ⅇ}^{-r-d}\left(-2\mathrm{erf}\left(\frac{1}{4}\frac{\left({\mathrm{\σ}}^2\+2\ln \left(2\right)\+2d-2r\right)\sqrt{2}}{\mathrm{\σ}}\right){\ⅇ}^d\sqrt{\mathrm{\π}}\mathrm{\σ}-2{\ⅇ}^d\sqrt{\mathrm{\π}}\mathrm{\σ}\+{\ⅇ}^{-\frac{1}{8}\frac{{\mathrm{\σ}}^4-4\ln \left(2\right){\mathrm{\σ}}^2-4d{\mathrm{\σ}}^2-4r{\mathrm{\σ}}^2\+4{\ln \left(2\right)}^2\+8\ln \left(2\right)d-8\ln \left(2\right)r\+4d^2-8dr\+4r^2}{{\mathrm{\σ}}^2}}\sqrt{2}-2\sqrt{2}{\ⅇ}^{-\frac{1}{8}\frac{{\mathrm{\σ}}^4\+4\ln \left(2\right){\mathrm{\σ}}^2-4d{\mathrm{\σ}}^2-4r{\mathrm{\σ}}^2\+4{\ln \left(2\right)}^2\+8\ln \left(2\right)d-8\ln \left(2\right)r\+4d^2-8dr\+4r^2}{{\mathrm{\σ}}^2}}\right)}{\sqrt{\mathrm{\π}}\mathrm{\σ}}$

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

$-94.32991433$

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

$S≔\frac{25{\&ExponentialE;}^{-r-d}\left(\mathrm{erf}\left(\frac{1}{4}\frac{\sqrt{2}\left(-{\mathrm{\&sigma;}}^2\&plus;2\ln \left(2\right)-2d\&plus;2r\right)}{\mathrm{\&sigma;}}\right){\&ExponentialE;}^d\sqrt{\mathrm{\&pi;}}\mathrm{\&sigma;}-2{\&ExponentialE;}^d\mathrm{erf}\left(\frac{1}{4}\frac{\left({\mathrm{\&sigma;}}^2\&plus;2d-2r\right)\sqrt{2}}{\mathrm{\&sigma;}}\right)\sqrt{\mathrm{\&pi;}}\mathrm{\&sigma;}-{\&ExponentialE;}^{-\frac{1}{8}\frac{{\mathrm{\&sigma;}}^4-4\ln \left(2\right){\mathrm{\&sigma;}}^2-4d{\mathrm{\&sigma;}}^2-4r{\mathrm{\&sigma;}}^2\&plus;4{\ln \left(2\right)}^2-8\ln \left(2\right)d\&plus;8\ln \left(2\right)r\&plus;4d^2-8dr\&plus;4r^2}{{\mathrm{\&sigma;}}^2}}\sqrt{2}\&plus;2\sqrt{2}{\&ExponentialE;}^{-\frac{1}{8}\frac{{\mathrm{\&sigma;}}^4\&plus;4\ln \left(2\right){\mathrm{\&sigma;}}^2-4d{\mathrm{\&sigma;}}^2-4r{\mathrm{\&sigma;}}^2\&plus;4{\ln \left(2\right)}^2-8\ln \left(2\right)d\&plus;8\ln \left(2\right)r\&plus;4d^2-8dr\&plus;4r^2}{{\mathrm{\&sigma;}}^2}}\&plus;{\&ExponentialE;}^d\sqrt{\mathrm{\&pi;}}\mathrm{\&sigma;}\right)}{\sqrt{\mathrm{\&pi;}}\mathrm{\&sigma;}}$

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

$C≔-50{\&ExponentialE;}^{-r}\left(\mathrm{erf}\left(\frac{1}{4}\frac{\left({\mathrm{\&sigma;}}^2\&plus;2d-2r\right)\sqrt{2}}{\mathrm{\&sigma;}}\right)-1\right)$

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

$P≔\frac{25\left({\&ExponentialE;}^{-r}\mathrm{erf}\left(\frac{1}{4}\frac{\sqrt{2}\left(-{\mathrm{\&sigma;}}^2\&plus;2\ln \left(2\right)-2d\&plus;2r\right)}{\mathrm{\&sigma;}}\right)\sqrt{\mathrm{\&pi;}}\mathrm{\&sigma;}\&plus;2\sqrt{2}{\&ExponentialE;}^{-\frac{1}{8}\frac{{\mathrm{\&sigma;}}^4\&plus;4\ln \left(2\right){\mathrm{\&sigma;}}^2\&plus;4d{\mathrm{\&sigma;}}^2\&plus;4r{\mathrm{\&sigma;}}^2\&plus;4{\ln \left(2\right)}^2-8\ln \left(2\right)d\&plus;8\ln \left(2\right)r\&plus;4d^2-8dr\&plus;4r^2}{{\mathrm{\&sigma;}}^2}}-{\&ExponentialE;}^{-\frac{1}{8}\frac{{\mathrm{\&sigma;}}^4-4\ln \left(2\right){\mathrm{\&sigma;}}^2\&plus;4d{\mathrm{\&sigma;}}^2\&plus;4r{\mathrm{\&sigma;}}^2\&plus;4{\ln \left(2\right)}^2-8\ln \left(2\right)d\&plus;8\ln \left(2\right)r\&plus;4d^2-8dr\&plus;4r^2}{{\mathrm{\&sigma;}}^2}}\sqrt{2}-{\&ExponentialE;}^{-r}\sqrt{\mathrm{\&pi;}}\mathrm{\&sigma;}\right)}{\sqrt{\mathrm{\&pi;}}\mathrm{\&sigma;}}$

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

$0$

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

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

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