The Zomma of an option or a portfolio of options measures Gamma's sensitivity to volatility.

The BlackScholesZomma command computes the Zomma 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.

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

$\mathrm{Zomma}≔\mathrm{BlackScholesZomma}\left(S\left[0\right]\,100\,T\,0.1\,0.05\,0\,'\mathrm{call}'\right)\:$

$\mathrm{plot3d}\left(\mathrm{Zomma}\,T=1.0..0\,S\left[0\right]=0..200\,'\mathrm{labels}'=\left[''Time\; to\; Maturity''\,''Spot\; Price''\,''Value''\right]\,'\mathrm{colorscheme}'=\left[''zgradient''\,\left[''Black''\,''White''\,''Red''\right]\right]\,'\mathrm{thickness}'=0\right)$

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

$\frac{\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left(-{\mathrm{\sigma }}^4+4d^2-8dr+4r^2-4{\mathrm{\sigma }}^2\right)}{800{\mathrm{\sigma }}^4\sqrt{\mathrm{\pi }}}$

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

$\mathrm{-0.04277759297}$

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

$\frac{\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left(-{\mathrm{\sigma }}^4+4d^2-8dr+4r^2-4{\mathrm{\sigma }}^2\right)}{800{\mathrm{\sigma }}^4\sqrt{\mathrm{\pi }}}$

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

$\mathrm{-0.042777628}$

$\mathrm{Zomma}≔\mathrm{BlackScholesZomma}\left(100\,100\,1\,\mathrm{\sigma }\,r\,0.03\,'\mathrm{call}'\right)$

$\mathrm{Zomma}≔\frac{1.045309735\times {10}^{\mathrm{-7}}{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}-1.077144155\times {10}^{\mathrm{-7}}{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}-0.0004839396923{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6-0.00001045309735{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}r+0.0003484365783{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2-0.003871517536{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^3-0.0004986778503{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6+0.00001077144155{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}r-0.0003590480519{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{r}^2+0.003989422799{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{r}^3-0.0003466943955{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+0.0003608432924{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-0.01208795109{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r+0.001994711401{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2{r}^2-0.0009973557002{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}r{\mathrm{\sigma }}^4-0.001964795153{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4-0.001964790729{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4+0.01149840709{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r+0.00193575877{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2{r}^2+0.0009678793843{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}r{\mathrm{\sigma }}^4}{{\mathrm{\sigma }}^6}$

$\mathrm{plot3d}\left(\mathrm{Zomma}\,\mathrm{\sigma }=0..1\,r=0..1\right)$

$\mathrm{BlackScholesZomma}\left(100\,120\,1\,0.3\,0.05\,0.03\,'\mathrm{put}'\right)$

$\mathrm{-0.02908022227}$

$\mathrm{BlackScholesZomma}\left(100\,t\mapsto \max \left(120-t\,0\right)\,1\,0.3\,0.05\,0.03\,0\right)$

$\mathrm{-0.029080244}$

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

The Finance[BlackScholesZomma] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.