The Vomma of an option or a portfolio of options measures Vega's sensitivity to volatility.

The BlackScholesVomma command computes the Vomma 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.

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

$\mathrm{Vomma}≔\mathrm{BlackScholesVomma}\left(100\,100\,T\,\mathrm{\sigma }\,0.05\,0\,'\mathrm{call}'\right)\:$

$\mathrm{plot3d}\left(\mathrm{Vomma}\,T=1.0..0\,\mathrm{\sigma }=0..0.5\,'\mathrm{labels}'=\left[''Time\; to\; Maturity''\,''Volatility''\,''Value''\right]\,'\mathrm{colorscheme}'=\left[''zgradient''\,\left[''Black''\,''White''\,''Red''\right]\right]\,'\mathrm{thickness}'=0\right)$

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

$\frac{25\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4{\mathrm{\sigma }}^{2}d+4{\mathrm{\sigma }}^{2}r+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left(-{\mathrm{\sigma }}^4+4d^2-8dr+4r^2\right)}{2{\mathrm{\sigma }}^3\sqrt{\mathrm{\pi }}}$

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

$\mathrm{-2.2760247}$

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

$\frac{25\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4{\mathrm{\sigma }}^{2}d+4{\mathrm{\sigma }}^{2}r+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left(-{\mathrm{\sigma }}^4+4d^2-8dr+4r^2\right)}{2{\mathrm{\sigma }}^3\sqrt{\mathrm{\pi }}}$

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

$\mathrm{-2.27602480}$

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

$\mathrm{Vomma}≔\frac{3.484365784{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2-38.71517538{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^3-4.839396921{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6-0.2903638153{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4+2.411606085{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+0.1077144156{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}r-3.590480521{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{r}^2+39.89422801{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{r}^3-4.986778501{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6-2.305488695{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-0.1045309736{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}r+0.29920671{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4+76.26889555{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r+19.35758769{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2{\mathrm{\sigma }}^2+9.678793843{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{4}r-80.9852829{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+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+19.94711401{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{r}^2{\mathrm{\sigma }}^2-9.973557006{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{4}r+0.001045309735{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}-0.001077144156{\ⅇ}^{-\frac{1.\left(0.5000000002{\mathrm{\sigma }}^{2}r+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^5}$

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

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

$32.3094718$

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

$32.3094713$

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

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

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