BlackScholesCharm - Maple Help

Finance

 BlackScholesCharm
 compute the Charm of a European-style option with given payoff

 Calling Sequence BlackScholesCharm(${S}_{0}$, K, T, sigma, r, d, optiontype) BlackScholesCharm(${S}_{0}$, 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 Charm of an option or a portfolio of options measures Delta's sensitivity to movement in the time to maturity.

$\mathrm{Charm}=-\frac{\partial }{\partial T}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{\Delta }$

$\mathrm{Charm}=-\frac{{\partial }^{2}}{\partial T\partial {S}_{0}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}S$

 • The BlackScholesCharm command computes the Charm 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

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

The Delta of an option measures the sensitivity of the option to price changes in the underlying asset, ${S}_{0}$. The Charm of an option measures Delta's sensitivity to movement in the time of maturity, T. The following example illustrates the characteristics of the Charm of an option with respect to these two variables.

In this example, the Charm is defined as a function of the underlying asset price ${S}_{0}$, and time to maturity, T.  For a European call option, we will assume that the strike price is 100, volatility is 0.10, and the risk-free interest rate of 0.05.  We also assume that this option does not pay any dividends.

 > $\mathrm{Charm}≔\mathrm{BlackScholesCharm}\left(S\left[0\right],100,T,0.1,0.05,0,'\mathrm{call}'\right):$
 > $\mathrm{plot3d}\left(\mathrm{Charm},T=1.0..0,S\left[0\right]=0..200,'\mathrm{labels}'=\left["Time To Maturity","Spot Price","Value"\right],'\mathrm{colorscheme}'=\left["zgradient",\left["Black","White","Red"\right]\right],'\mathrm{thickness}'=0\right)$

We can also see how the Charm behaves as a function of the risk-free interest rate, the dividend yield, and volatility.  To compute the Charm of a European call option with strike price 100 maturing in 1 year, we take:

 > $\mathrm{BlackScholesCharm}\left(100,100,1,\mathrm{\sigma },r,d,'\mathrm{call}'\right)$
 $\frac{{-}{4}{}{{ⅇ}}^{{-}{d}}{}{\mathrm{erf}}{}\left(\frac{\left({-}{{\mathrm{\sigma }}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{4}{}{\mathrm{\sigma }}}\right){}\sqrt{{\mathrm{\pi }}}{}{d}{}{\mathrm{\sigma }}{-}{{ⅇ}}^{{-}\frac{{{\mathrm{\sigma }}}^{{4}}{+}{4}{}{d}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{8}{}{{\mathrm{\sigma }}}^{{2}}}}{}\sqrt{{2}}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{{ⅇ}}^{{-}{d}}{}\sqrt{{\mathrm{\pi }}}{}{d}{}{\mathrm{\sigma }}{+}{2}{}{{ⅇ}}^{{-}\frac{{{\mathrm{\sigma }}}^{{4}}{+}{4}{}{d}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{8}{}{{\mathrm{\sigma }}}^{{2}}}}{}\sqrt{{2}}{}{d}{-}{2}{}{{ⅇ}}^{{-}\frac{{{\mathrm{\sigma }}}^{{4}}{+}{4}{}{d}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{8}{}{{\mathrm{\sigma }}}^{{2}}}}{}\sqrt{{2}}{}{r}}{{8}{}{\mathrm{\sigma }}{}\sqrt{{\mathrm{\pi }}}}$ (1)

This can be numerically solved for specific values of the risk-free rate, the dividend yield, and the volatility.

 > $\mathrm{BlackScholesCharm}\left(100,100,1,0.3,0.05,0.03,'\mathrm{call}'\right)$
 ${-0.0239148274}$ (2)

It is also possible to use the generic method in which the option is defined through its payoff function:

 > $\mathrm{BlackScholesCharm}\left(100,t↦\mathrm{max}\left(t-100,0\right),1,\mathrm{\sigma },r,d\right)$
 ${-}\frac{{{ⅇ}}^{{-}{r}}{}\left({4}{}{{ⅇ}}^{{-}{d}{+}{r}}{}{\mathrm{erf}}{}\left(\frac{\left({-}{{\mathrm{\sigma }}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right){}\sqrt{{2}}}{{4}{}{\mathrm{\sigma }}}\right){}\sqrt{{\mathrm{\pi }}}{}{d}{}{\mathrm{\sigma }}{-}{4}{}{{ⅇ}}^{{-}{d}{+}{r}}{}\sqrt{{\mathrm{\pi }}}{}{d}{}{\mathrm{\sigma }}{+}{{ⅇ}}^{{-}\frac{{\left({{\mathrm{\sigma }}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right)}^{{2}}}{{8}{}{{\mathrm{\sigma }}}^{{2}}}}{}\sqrt{{2}}{}{{\mathrm{\sigma }}}^{{2}}{-}{2}{}{{ⅇ}}^{{-}\frac{{\left({{\mathrm{\sigma }}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right)}^{{2}}}{{8}{}{{\mathrm{\sigma }}}^{{2}}}}{}\sqrt{{2}}{}{d}{+}{2}{}{{ⅇ}}^{{-}\frac{{\left({{\mathrm{\sigma }}}^{{2}}{+}{2}{}{d}{-}{2}{}{r}\right)}^{{2}}}{{8}{}{{\mathrm{\sigma }}}^{{2}}}}{}\sqrt{{2}}{}{r}\right)}{{8}{}{\mathrm{\sigma }}{}\sqrt{{\mathrm{\pi }}}}$ (3)
 > $\mathrm{BlackScholesCharm}\left(100,t↦\mathrm{max}\left(t-100,0\right),1,0.3,0.05,0.03\right)$
 ${-0.02391495679}$ (4)
 > $\mathrm{Charm}≔\mathrm{BlackScholesCharm}\left(100,100,1,\mathrm{\sigma },r,0.03,'\mathrm{call}'\right)$
 ${\mathrm{Charm}}{≔}\frac{{8.}{×}{{10}}^{{-8}}{}\left({181958.5375}{}{{\mathrm{\sigma }}}^{{3}}{+}{181958.5375}{}{\mathrm{erf}}{}\left(\frac{{0.707106781}{}{r}{-}{0.02121320343}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}\right){}{{\mathrm{\sigma }}}^{{3}}{+}{2.56488037}{×}{{10}}^{{6}}{}{{ⅇ}}^{{-}\frac{{0.00004999999997}{}{\left({50.}{}{{\mathrm{\sigma }}}^{{2}}{+}{100.}{}{r}{-}{3.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{-}{0.00125}{}{{ⅇ}}^{{-}\frac{{0.00004999999997}{}{\left({50.}{}{{\mathrm{\sigma }}}^{{2}}{+}{100.}{}{r}{-}{3.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{}{r}{-}{604924.616}{}{{ⅇ}}^{{-}\frac{{0.00004999999997}{}{\left({50.}{}{{\mathrm{\sigma }}}^{{2}}{+}{100.}{}{r}{-}{3.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{4}}{+}{2.419698461}{×}{{10}}^{{6}}{}{{ⅇ}}^{{-}\frac{{0.00004999999997}{}{\left({50.}{}{{\mathrm{\sigma }}}^{{2}}{+}{100.}{}{r}{-}{3.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{r}}^{{2}}{-}{145181.9076}{}{{ⅇ}}^{{-}\frac{{0.00004999999997}{}{\left({50.}{}{{\mathrm{\sigma }}}^{{2}}{+}{100.}{}{r}{-}{3.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{r}{+}{2177.728615}{}{{ⅇ}}^{{-}\frac{{0.00004999999997}{}{\left({50.}{}{{\mathrm{\sigma }}}^{{2}}{+}{100.}{}{r}{-}{3.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{-}{2.493389254}{×}{{10}}^{{6}}{}{r}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5000000002}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.4999999997}{}{{r}}^{{2}}{-}{0.02999999998}{}{r}{+}{0.0004499999998}{+}{0.01499999999}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.1249999999}{}{{\mathrm{\sigma }}}^{{4}}\right)}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{-}{2.49338925}{×}{{10}}^{{6}}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5000000002}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.4999999997}{}{{r}}^{{2}}{-}{0.02999999998}{}{r}{+}{0.0004499999998}{+}{0.01499999999}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.1249999999}{}{{\mathrm{\sigma }}}^{{4}}\right)}{{{\mathrm{\sigma }}}^{{2}}}}{}{{r}}^{{2}}{+}{149603.3551}{}{r}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5000000002}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.4999999997}{}{{r}}^{{2}}{-}{0.02999999998}{}{r}{+}{0.0004499999998}{+}{0.01499999999}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.1249999999}{}{{\mathrm{\sigma }}}^{{4}}\right)}{{{\mathrm{\sigma }}}^{{2}}}}{-}{2244.050326}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5000000002}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.4999999997}{}{{r}}^{{2}}{-}{0.02999999998}{}{r}{+}{0.0004499999998}{+}{0.01499999999}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.1249999999}{}{{\mathrm{\sigma }}}^{{4}}\right)}{{{\mathrm{\sigma }}}^{{2}}}}{-}{2.568190929}{×}{{10}}^{{6}}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5000000002}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.4999999997}{}{{r}}^{{2}}{-}{0.02999999998}{}{r}{+}{0.0004499999998}{+}{0.01499999999}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.1249999999}{}{{\mathrm{\sigma }}}^{{4}}\right)}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{-}{623347.3128}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5000000002}{}{r}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.4999999997}{}{{r}}^{{2}}{-}{0.02999999998}{}{r}{+}{0.0004499999998}{+}{0.01499999999}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.1249999999}{}{{\mathrm{\sigma }}}^{{4}}\right)}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{4}}\right)}{{{\mathrm{\sigma }}}^{{3}}}$ (5)
 > $\mathrm{plot3d}\left(\mathrm{Charm},\mathrm{\sigma }=0..1,r=0..1\right)$

Here are similar examples for the European put option:

 > $\mathrm{BlackScholesCharm}\left(100,120,1,0.3,0.05,0.03,'\mathrm{put}'\right)$
 ${-0.1668222722}$ (6)
 > $\mathrm{BlackScholesCharm}\left(100,t↦\mathrm{max}\left(120-t,0\right),1,0.3,0.05,0.03,0\right)$
 ${-0.1668223310}$ (7)

References

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

Compatibility

 • The Finance[BlackScholesCharm] command was introduced in Maple 2015.
 • For more information on Maple 2015 changes, see Updates in Maple 2015.