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.

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

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

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

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

$44.4027473$

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

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

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

$44.4027473$

$\mathrm{{\rm P}}≔\mathrm{BlackScholesRho}\left(100\,K\,1\,\mathrm{\sigma }\,0.05\,0.03\,'\mathrm{call}'\right)$

$\mathrm{{\rm P}}≔\frac{3.832995302{\left(\frac{1}{K}\right)}^{-\frac{1.\left(0.4999999997{\mathrm{\sigma }}^2+4.625170183\right)}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{-10.69609962-0.4999999997{\ln \left(\frac{1}{K}\right)}^2-0.1249999999{\mathrm{\sigma }}^4}{{\mathrm{\sigma }}^2}}+0.4756147122K\mathrm{\sigma }+0.4756147122K\mathrm{\sigma }\mathrm{erf}\left(\frac{3.270489202+0.707106781\ln \left(\frac{1}{K}\right)-0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)-3.832995293K{\left(\frac{1}{K}\right)}^{\frac{0.4999999998{\mathrm{\sigma }}^2-4.625170184}{{\mathrm{\sigma }}^2}}{\ⅇ}^{\frac{-10.69609962-0.4999999997{\ln \left(\frac{1}{K}\right)}^2-0.1249999999{\mathrm{\sigma }}^4}{{\mathrm{\sigma }}^2}}}{\mathrm{\sigma }}$

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

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

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

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

$\mathrm{-94.32991431}$

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

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

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

$\mathrm{-94.32991433}$

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

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

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

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

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

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

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

$0$

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

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

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