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Finance

 BlackScholesRho
 compute the Rho of a European-style option with given payoff

 Calling Sequence BlackScholesRho(${S}_{0}$, K, T, sigma, r, d, optiontype) BlackScholesRho(${S}_{0}$, P, T, sigma, r, d)

Parameters

 ${S}_{0}$ - algebraic expression; initial (current) value of the underlying asset K - algebraic expression; strike price T - algebraic expression; time to maturity sigma - algebraic expression; volatility r - algebraic expression; continuously compounded risk-free rate d - algebraic expression; continuously compounded dividend yield P - operator or procedure; payoff function optiontype - call or put; option type

Description

 • 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

$\mathrm{Ρ}=\frac{\partial }{\partial r}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}S$

 • 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.

Examples

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

First you compute the Rho of a European call option with strike price 100, which matures in 1 year. This will define the Rho as a function of the risk-free rate, the dividend yield, and the volatility.

 > $\mathrm{BlackScholesRho}\left(100,100,1,\mathrm{σ},r,d,'\mathrm{call}'\right)$
 ${-}{50}{}{{ⅇ}}^{{-}{r}}{}\left({\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\left({{\mathrm{σ}}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{\mathrm{σ}}}\right){-}{1}\right)$ (1)

In this example you will use numeric values for the risk-free rate, the dividend yield, and the volatility.

 > $\mathrm{BlackScholesRho}\left(100,100,1,0.3,0.05,0.03,'\mathrm{call}'\right)$
 ${44.4027473}$ (2)

We can also use the generic method in which the option is defined through its payoff function.

 > $\mathrm{BlackScholesRho}\left(100,t→\mathrm{max}\left(t-100,0\right),1,\mathrm{σ},r,d\right)$
 ${-}{50}{}{{ⅇ}}^{{-}{r}}{}\left({\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\left({{\mathrm{σ}}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{\mathrm{σ}}}\right){-}{1}\right)$ (3)
 > $\mathrm{BlackScholesRho}\left(100,t→\mathrm{max}\left(t-100,0\right),1,0.3,0.05,0.03\right)$
 ${44.40274728}$ (4)
 > $\mathrm{Ρ}≔\mathrm{BlackScholesRho}\left(100,K,1,\mathrm{σ},0.05,0.03,'\mathrm{call}'\right)$
 ${\mathrm{Ρ}}{≔}\frac{{38.71517541}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{\left({4.625170186}{+}{\mathrm{ln}}{}\left(\frac{{1}}{{K}}\right){+}{0.5}{}{{\mathrm{σ}}}^{{2}}\right)}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{+}{0.4756147122}{}{K}{}{\mathrm{σ}}{+}{0.4756147122}{}{K}{}{\mathrm{σ}}{}{\mathrm{erf}}{}\left(\frac{{3.270489202}{+}{0.707106781}{}{\mathrm{ln}}{}\left(\frac{{1}}{{K}}\right){-}{0.3535533905}{}{{\mathrm{σ}}}^{{2}}}{{\mathrm{σ}}}\right){-}{0.3794856357}{}{K}{}{{ⅇ}}^{{-}\frac{{1.}{}{\left({3.270489202}{+}{0.707106781}{}{\mathrm{ln}}{}\left(\frac{{1}}{{K}}\right){-}{0.3535533905}{}{{\mathrm{σ}}}^{{2}}\right)}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{{\mathrm{σ}}}$ (5)
 > $\mathrm{plot3d}\left(\mathrm{Ρ},\mathrm{σ}=0..1,K=70..120,\mathrm{axes}=\mathrm{BOXED}\right)$ Here are similar examples for the European put option.

 > $\mathrm{BlackScholesRho}\left(50,100,1,\mathrm{σ},r,d,'\mathrm{put}'\right)$
 $\frac{{25}{}\left({-}{2}{}{{ⅇ}}^{{-}{r}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\left({{\mathrm{σ}}}^{{2}}{+}{2}{}{\mathrm{ln}}{}\left({2}\right){+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{\mathrm{σ}}}\right){}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}{-}{2}{}{{ⅇ}}^{{-}{r}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}{+}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}\sqrt{{2}}{-}{2}{}\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}\right)}{\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}}$ (6)
 > $\mathrm{BlackScholesRho}\left(50,100,1,0.3,0.05,0.03,'\mathrm{put}'\right)$
 ${-}{94.32991431}$ (7)
 > $\mathrm{BlackScholesRho}\left(50,t→\mathrm{max}\left(100-t,0\right),1,\mathrm{σ},r,d\right)$
 $\frac{{25}{}{{ⅇ}}^{{-}{r}{-}{d}}{}\left({-}{2}{}{\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\left({{\mathrm{σ}}}^{{2}}{+}{2}{}{\mathrm{ln}}{}\left({2}\right){+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{\mathrm{σ}}}\right){}{{ⅇ}}^{{d}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}{-}{2}{}{{ⅇ}}^{{d}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}{+}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{-}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{-}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}\sqrt{{2}}{-}{2}{}\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{-}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{-}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}\right)}{\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}}$ (8)
 > $\mathrm{BlackScholesRho}\left(50,t→\mathrm{max}\left(100-t,0\right),1,0.3,0.05,0.03,d\right)$
 ${-}{94.32991433}$ (9)

In this example, you will compute the Rho of a strangle.

 > $S≔\mathrm{BlackScholesRho}\left(100,t→\mathrm{piecewise}\left(t<50,50-t,t<100,0,t-100\right),1,\mathrm{σ},r,d\right)$
 ${S}{≔}\frac{{25}{}{{ⅇ}}^{{-}{r}{-}{d}}{}\left({\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}\left({-}{{\mathrm{σ}}}^{{2}}{+}{2}{}{\mathrm{ln}}{}\left({2}\right){-}{2}{}{d}{+}{2}{}{r}\right)}{{\mathrm{σ}}}\right){}{{ⅇ}}^{{d}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}{-}{2}{}{{ⅇ}}^{{d}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\left({{\mathrm{σ}}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{\mathrm{σ}}}\right){}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}{-}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{-}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{-}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}\sqrt{{2}}{+}{2}{}\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{-}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{-}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{+}{{ⅇ}}^{{d}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}\right)}{\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}}$ (10)
 > $C≔\mathrm{BlackScholesRho}\left(100,100,1,\mathrm{σ},r,d,'\mathrm{call}'\right)$
 ${C}{≔}{-}{50}{}{{ⅇ}}^{{-}{r}}{}\left({\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\left({{\mathrm{σ}}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{\mathrm{σ}}}\right){-}{1}\right)$ (11)
 > $P≔\mathrm{BlackScholesRho}\left(100,50,1,\mathrm{σ},r,d,'\mathrm{put}'\right)$
 ${P}{≔}\frac{{25}{}\left({{ⅇ}}^{{-}{r}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}\left({-}{{\mathrm{σ}}}^{{2}}{+}{2}{}{\mathrm{ln}}{}\left({2}\right){-}{2}{}{d}{+}{2}{}{r}\right)}{{\mathrm{σ}}}\right){}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}{+}{2}{}\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{-}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}\sqrt{{2}}{-}{{ⅇ}}^{{-}{r}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}\right)}{\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}}$ (12)

Check:

 > $\mathrm{simplify}\left(S-C-P\right)$
 ${0}$ (13)

References

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

Compatibility

 • The Finance[BlackScholesRho] command was introduced in Maple 15.
 • For more information on Maple 15 changes, see Updates in Maple 15.