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.

with(Finance):

BlackScholesRho(100, 100, 1, sigma, r, d, 'call');

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

BlackScholesRho(100, 100, 1, 0.3, 0.05, 0.03, 'call');

$44.4027473$

BlackScholesRho(100, t -> max(t-100, 0), 1, sigma, r, d);

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

BlackScholesRho(100, t -> max(t-100, 0), 1, 0.3, 0.05, 0.03);

$44.40274728$

Rho := BlackScholesRho(100, K, 1, sigma, 0.05, 0.03, 'call');

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

plot3d(Rho, sigma = 0..1, K = 70..120, axes = BOXED);

BlackScholesRho(50, 100, 1, sigma, r, d, 'put');

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

BlackScholesRho(50, 100, 1, 0.3, 0.05, 0.03, 'put');

$\mathrm{-94.32991431}$

BlackScholesRho(50, t -> max(100-t, 0), 1, sigma, r, d);

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

BlackScholesRho(50, t -> max(100-t, 0), 1, 0.3, 0.05, 0.03, d);

$\mathrm{-94.32991433}$

S := BlackScholesRho(100, t -> piecewise(t < 50, 50-t, t < 100, 0, t-100), 1, sigma, r, d);

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

C := BlackScholesRho(100, 100, 1, sigma, r, d, 'call');

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

P := BlackScholesRho(100, 50, 1, sigma, r, d, 'put');

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

simplify(S-C-P);

$0$

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.