The Vanna of an option or a portfolio of options measures Delta's sensitivity to volatility.

The BlackScholesVanna command computes the Vanna 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.

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

$\mathrm{Vanna}≔\mathrm{BlackScholesVanna}\left(S\left[0\right]\,100\,1\,\mathrm{\sigma }\,0.05\,0\,'\mathrm{call}'\right)\:$

$\mathrm{plot3d}\left(\mathrm{Vanna}\,\mathrm{\sigma }=0..0.5\,S\left[0\right]=0..200\,'\mathrm{labels}'=\left[''Volatility''\,''Spot\; Price''\,''Value''\right]\,'\mathrm{colorscheme}'=\left[''zgradient''\,\left[''Black''\,''White''\,''Red''\right]\right]\,'\mathrm{thickness}'=0\right)$

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

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

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

$0.105047298$

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

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

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

$0.1050473823$

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

$\mathrm{Vanna}≔\frac{0.09678793857{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4-0.3755372014{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-0.387151754{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r+0.0003484365784{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}-0.02322910523{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}r+0.3871517538{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2-0.0003590480521{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}+0.02393653681{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}r-0.3989422801{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{r}^2+0.09973557{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4+0.3989422803{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.4999999997r^2\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2}{{\mathrm{\sigma }}^4}$

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

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

$0.826177365$

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

$0.8261766714$

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

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

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