The Lambda of an option or a portfolio of options is the percentage change in option value per percentage change in the price of the underlying asset, which is used as a measure of leverage.

The BlackScholesLambda command computes the Lambda 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.

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

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

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

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

$\frac{{\ⅇ}^{-d}\left(\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-1\right)}{{\ⅇ}^{-d}\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-{\ⅇ}^{-r}\mathrm{erf}\left(\frac{\left({\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-{\ⅇ}^{-d}+{\ⅇ}^{-r}}$

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

$4.568593519$

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

$\frac{{\ⅇ}^{r-d}\left(\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)-1\right)}{\mathrm{erf}\left(\frac{\left(-{\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right){\ⅇ}^{r-d}-{\ⅇ}^{r-d}-\mathrm{erf}\left(\frac{\left({\mathrm{\sigma }}^2+2d-2r\right)\sqrt{2}}{4\mathrm{\sigma }}\right)+1}$

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

$4.568593411$

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

$\mathrm{\Lambda }≔\frac{-48.52227668\mathrm{\sigma }-48.52227668\mathrm{erf}\left(\frac{-0.02121320343+0.707106781r+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }-38.7151754{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}+39.89422803{\ⅇ}^{-\frac{1.\left(0.5000000002r{\mathrm{\sigma }}^2+0.0004499999998-0.02999999998r+0.01499999999{\mathrm{\sigma }}^2+0.4999999997r^2+0.1249999999{\mathrm{\sigma }}^4\right)}{{\mathrm{\sigma }}^2}}}{\left(-48.52227668-48.52227668\mathrm{erf}\left(\frac{-0.02121320343+0.707106781r+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)+50.{\ⅇ}^{-1.r}+50.{\ⅇ}^{-1.r}\mathrm{erf}\left(\frac{-0.02121320343+0.707106781r-0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\right)\mathrm{\sigma }}$

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

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

$\mathrm{-2.760749063}$

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

$\mathrm{-2.760749005}$

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

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

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