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Finance

 BlackScholesGamma
 compute the Gamma of a European-style option with given payoff

 Calling Sequence BlackScholesGamma(${S}_{0}$, K, T, sigma, r, d, optiontype) BlackScholesGamma(${S}_{0}$, 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 Gamma of an option or a portfolio of options is the sensitivity of the Delta to changes in the value of the underlying asset

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

 • The BlackScholesGamma command computes the Gamma 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 Gamma of a European call option with strike price 100, which matures in 1 year. This will define the Gamma as a function of the risk-free rate, the dividend yield, and the volatility.

 > $\mathrm{BlackScholesGamma}\left(100,100,1,\mathrm{σ},r,d,'\mathrm{call}'\right)$
 $\frac{{1}}{{200}}{}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{{\mathrm{σ}}{}\sqrt{{\mathrm{π}}}}$ (1)

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

 > $\mathrm{BlackScholesGamma}\left(100,100,1,0.3,0.05,0.03,'\mathrm{call}'\right)$
 ${0.01260567542}$ (2)

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

 > $\mathrm{BlackScholesGamma}\left(100,t→\mathrm{max}\left(t-100,0\right),1,\mathrm{σ},r,d\right)$
 $\frac{{1}}{{200}}{}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{{\mathrm{σ}}{}\sqrt{{\mathrm{π}}}}$ (3)
 > $\mathrm{BlackScholesGamma}\left(100,t→\mathrm{max}\left(t-100,0\right),1,0.3,0.05,0.03\right)$
 ${0.01260567513}$ (4)
 > $\mathrm{BSGamma}≔\mathrm{expand}\left(\mathrm{BlackScholesGamma}\left(100,100,1,\mathrm{σ},r,0.03,'\mathrm{call}'\right)\right)$
 ${\mathrm{BSGamma}}{≔}\frac{{0.001965014020}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{σ}}}^{{2}}}{}{{ⅇ}}^{{-}{0.4999999997}{}{r}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{\frac{{0.02999999998}{}{r}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.0004499999997}}{{{\mathrm{σ}}}^{{2}}}}}{{\mathrm{σ}}}{+}\frac{{0.0001179008410}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{σ}}}^{{2}}}{}{{ⅇ}}^{{-}{0.4999999997}{}{r}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{\frac{{0.02999999998}{}{r}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.0004499999997}}{{{\mathrm{σ}}}^{{2}}}}}{{{\mathrm{σ}}}^{{3}}}{-}\frac{{0.003930028034}{}{r}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{σ}}}^{{2}}}{}{{ⅇ}}^{{-}{0.4999999997}{}{r}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{\frac{{0.02999999998}{}{r}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.0004499999997}}{{{\mathrm{σ}}}^{{2}}}}}{{{\mathrm{σ}}}^{{3}}}{-}\frac{{0.0001179008410}{}{{ⅇ}}^{{-}{0.5000000002}{}{r}}{}{{ⅇ}}^{{-}\frac{{0.0004499999998}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{\frac{{0.02999999998}{}{r}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{σ}}}^{{2}}}}{{{\mathrm{σ}}}^{{3}}}{+}\frac{{0.003930028033}{}{r}{}{{ⅇ}}^{{-}{0.5000000002}{}{r}}{}{{ⅇ}}^{{-}\frac{{0.0004499999998}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{\frac{{0.02999999998}{}{r}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{σ}}}^{{2}}}}{{{\mathrm{σ}}}^{{3}}}{+}\frac{{0.001965014018}{}{{ⅇ}}^{{-}{0.5000000002}{}{r}}{}{{ⅇ}}^{{-}\frac{{0.0004499999998}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{\frac{{0.02999999998}{}{r}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{σ}}}^{{2}}}}{{\mathrm{σ}}}$ (5)
 > $\mathrm{plot3d}\left(\mathrm{BSGamma},\mathrm{σ}=0..1,r=0..1,\mathrm{axes}=\mathrm{BOXED}\right)$

Here are similar examples for the European put option.

 > $\mathrm{BlackScholesGamma}\left(100,50,1,\mathrm{σ},r,d,'\mathrm{put}'\right)$
 $\frac{{1}}{{800}}{}\frac{\sqrt{{2}}{}\left({{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}}}}{}{{\mathrm{σ}}}^{{2}}{+}{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}}}}{}{{\mathrm{σ}}}^{{2}}{+}{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}}}}{}{\mathrm{ln}}{}\left({2}\right){-}{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}}}}{}{d}{+}{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}}}}{}{r}{-}{4}{}{{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}}}}{}{\mathrm{ln}}{}\left({2}\right){+}{4}{}{{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}}}}{}{d}{-}{4}{}{{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}}}}{}{r}\right)}{\sqrt{{\mathrm{π}}}{}{{\mathrm{σ}}}^{{3}}}$ (6)
 > $\mathrm{BlackScholesGamma}\left(100,50,1,0.3,0.05,0.03,'\mathrm{put}'\right)$
 ${0.000529595076}$ (7)
 > $\mathrm{BlackScholesGamma}\left(100,t→\mathrm{max}\left(50-t,0\right),1,\mathrm{σ},r,d\right)$
 ${-}\frac{{1}}{{800}}{}\frac{{{ⅇ}}^{{-}{r}{-}{d}}{}\sqrt{{2}}{}\left({-}{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}}}}{}{{\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}}}}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{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}}}}{}{\mathrm{ln}}{}\left({2}\right){-}{4}{}{{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}}}}{}{d}{+}{4}{}{{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}}}}{}{r}{-}{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}}}}{}{\mathrm{ln}}{}\left({2}\right){+}{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}}}}{}{d}{-}{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}}}}{}{r}\right)}{\sqrt{{\mathrm{π}}}{}{{\mathrm{σ}}}^{{3}}}$ (8)
 > $\mathrm{BlackScholesGamma}\left(100,t→\mathrm{max}\left(50-t,0\right),1,0.3,0.05,0.03,d\right)$
 ${0.0005295950845}$ (9)

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

 > $S≔\mathrm{BlackScholesGamma}\left(100,t→\mathrm{piecewise}\left(t<50,50-t,t<100,0,t-100\right),1,\mathrm{σ},r,d\right)$
 ${S}{≔}\frac{{1}}{{800}}{}\frac{{{ⅇ}}^{{-}{r}{-}{d}}{}\sqrt{{2}}{}\left({4}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{-}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{\mathrm{σ}}}^{{2}}{+}{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}}}}{}{{\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}}}}{}{{\mathrm{σ}}}^{{2}}{-}{4}{}{{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}}}}{}{\mathrm{ln}}{}\left({2}\right){+}{4}{}{{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}}}}{}{d}{-}{4}{}{{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}}}}{}{r}{+}{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}}}}{}{\mathrm{ln}}{}\left({2}\right){-}{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}}}}{}{d}{+}{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}}}}{}{r}\right)}{\sqrt{{\mathrm{π}}}{}{{\mathrm{σ}}}^{{3}}}$ (10)
 > $C≔\mathrm{BlackScholesGamma}\left(100,100,1,\mathrm{σ},r,d,'\mathrm{call}'\right)$
 ${C}{≔}\frac{{1}}{{200}}{}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{{\mathrm{σ}}{}\sqrt{{\mathrm{π}}}}$ (11)
 > $P≔\mathrm{BlackScholesGamma}\left(100,50,1,\mathrm{σ},r,d,'\mathrm{put}'\right)$
 ${P}{≔}\frac{{1}}{{800}}{}\frac{\sqrt{{2}}{}\left({{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{\mathrm{σ}}}^{{2}}{+}{2}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{+}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{{\mathrm{σ}}}^{{2}}{+}{2}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{\mathrm{ln}}{}\left({2}\right){-}{2}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}{}{d}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{r}{}{{\mathrm{σ}}}^{{2}}{+}{4}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{8}{}{\mathrm{ln}}{}\left({2}\right){}{d}{+}{8}{}{\mathrm{ln}}{}\left({2}\right){}{r}{+}{4}{}{{d}}^{{2}}{-}{8}{}{d}{}{r}{+}{4}{}{{r}}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}{}{d}{+}{2}{}{{ⅇ}}^{{-}\frac{{1}}{{8}}{}\frac{{{\mathrm{σ}}}^{{4}}{-}{4}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{σ}}}^{{2}}{+}{4}}{}}\right)}{}$