The Vega of an option or a portfolio of options is the sensitivity of the option or portfolio to changes in the volatility of the underlying asset.

The BlackScholesVega command computes the Vega 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.

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

$r≔0.05$

$r≔0.05$

$d≔0.03$

$d≔0.03$

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

$38.32995297{\ⅇ}^{-\frac{0.0001999999998}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}-\frac{1.\times {10}^{\mathrm{-10}}{\ⅇ}^{-\frac{0.0001999999998}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{{\mathrm{\sigma }}^2}$

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

$37.81702623$

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

$38.32995296{\ⅇ}^{-\frac{0.0001999999998}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}-\frac{2.0000\times {10}^{\mathrm{-10}}{\ⅇ}^{-\frac{0.0001999999998}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{{\mathrm{\sigma }}^2}$

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

$37.81702620$

$\mathrm{Vega}≔\mathrm{expand}\left(\mathrm{BlackScholesVega}\left(100\,K\,1\,\mathrm{\sigma }\,0.05\,0.03\,'\mathrm{call}'\right)\right)$

$\mathrm{Vega}≔\frac{1.916497650{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\left(\frac{1}{K}\right)}^{-\frac{4.625170183}{{\mathrm{\sigma }}^2}}}{{\left(\frac{1}{K}\right)}^{0.4999999997}}-\frac{17.72825559{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\left(\frac{1}{K}\right)}^{-\frac{4.625170183}{{\mathrm{\sigma }}^2}}}{{\left(\frac{1}{K}\right)}^{0.4999999997}{\mathrm{\sigma }}^2}-\frac{3.832995301{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\left(\frac{1}{K}\right)}^{-\frac{4.625170183}{{\mathrm{\sigma }}^2}}\ln \left(\frac{1}{K}\right)}{{\left(\frac{1}{K}\right)}^{0.4999999997}{\mathrm{\sigma }}^2}+\frac{17.72825555{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}K{\left(\frac{1}{K}\right)}^{-\frac{4.625170184}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{0.4999999998}}{{\mathrm{\sigma }}^2}+\frac{3.832995293{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}K{\left(\frac{1}{K}\right)}^{-\frac{4.625170184}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{0.4999999998}\ln \left(\frac{1}{K}\right)}{{\mathrm{\sigma }}^2}+1.916497646{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}K{\left(\frac{1}{K}\right)}^{-\frac{4.625170184}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{0.4999999998}$

$\mathrm{plot3d}\left(\mathrm{Vega}\,\mathrm{\sigma }=0..1\,K=70..120\,\mathrm{axes}=\mathrm{BOXED}\right)$

$\mathrm{expand}\left(\mathrm{BlackScholesVega}\left(100\,120\,1\,\mathrm{\sigma }\,r\,d\,'\mathrm{put}'\right)\right)$

$41.98835973{\ⅇ}^{-\frac{0.01317414389}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}$

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

$35.86504172$

$\mathrm{expand}\left(\mathrm{BlackScholesVega}\left(100\,t\mapsto \max \left(120-t\,0\right)\,1\,\mathrm{\sigma }\,r\,d\right)\right)$

$\frac{1.0\times {10}^{\mathrm{-9}}{\ⅇ}^{-\frac{0.01317414389}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{{\mathrm{\sigma }}^2}+41.98835974{\ⅇ}^{-\frac{0.01317414389}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}$

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

$35.86504186$

$S≔\mathrm{expand}\left(\mathrm{BlackScholesVega}\left(100\&comma;t\mapsto \mathrm{piecewise}\left(t<90\&comma;90-t\&comma;t<110\&comma;0\&comma;t-110\right)\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\right)\right)$

$S≔36.36298620{\&ExponentialE;}^{-\frac{0.007857629439}{{\mathrm{\sigma }}^2}}{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}+\frac{5.\times {10}^{\mathrm{-9}}{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.007857629439}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^2}+40.20079382{\&ExponentialE;}^{-\frac{0.002835811589}{{\mathrm{\sigma }}^2}}{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}$

$C≔\mathrm{expand}\left(\mathrm{BlackScholesVega}\left(100\&comma;110\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\&comma;'\mathrm{call}'\right)\right)$

$C≔20.10039692{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.002835811588}{{\mathrm{\sigma }}^2}}+\frac{3.027529011{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.002835811588}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^2}+20.10039691{\&ExponentialE;}^{-\frac{0.002835811589}{{\mathrm{\sigma }}^2}}{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}-\frac{3.027529011{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.002835811589}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^2}$

$P≔\mathrm{expand}\left(\mathrm{BlackScholesVega}\left(100\&comma;90\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\&comma;'\mathrm{put}'\right)\right)$

$P≔\frac{4.558482700{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.007857629432}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^2}+18.18149310{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.007857629432}{{\mathrm{\sigma }}^2}}-\frac{4.558482699{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.007857629430}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^2}+18.18149310{\&ExponentialE;}^{-0.1249999999{\mathrm{\sigma }}^2}{\&ExponentialE;}^{-\frac{0.007857629430}{{\mathrm{\sigma }}^2}}$

$\mathrm{plot}\left(\left[S\&comma;C+P\right]\&comma;\mathrm{\sigma }=0..1\&comma;\mathrm{color}=\left[\mathrm{red}\&comma;\mathrm{blue}\right]\&comma;\mathrm{thickness}=3\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}\right)$

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

The Finance[BlackScholesVega] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.