The Veta of an option or a portfolio of options measures Vega's sensitivity to movement in the time to maturity.

The BlackScholesVeta command computes the Veta 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.

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

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

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

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

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

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

$16.88635268$

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

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

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

$16.88635268$

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

$\mathrm{Veta}≔\frac{20.54552743r{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+0.05226548675{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^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}^3-1.742182892{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2-2.419698461{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6-0.589437219{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-0.05385720783{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}r-19.947114{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{r}^3+1.79524026{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{r}^2-2.49338925{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6+9.82395365{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4+9.243248131{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4+0.5545948879{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-18.77686007{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r+9.678793844{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2{\mathrm{\sigma }}^2-4.839396921{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{4}r-29.92067102r^2{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-14.96033552r{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4-0.0005226548675{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}+0.0005385720783{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.1249999999{\mathrm{\sigma }}^4+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2-0.02999999998r+0.0004499999998\right)}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^4}$

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

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

$22.63804429$

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

$22.63804426$

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

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

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