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

The BlackScholesUltima command computes the Ultima 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.

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

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

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

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

$\frac{25\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left({\mathrm{\sigma }}^8-8d^2{\mathrm{\sigma }}^4+16dr{\mathrm{\sigma }}^4-8r^2{\mathrm{\sigma }}^4-4{\mathrm{\sigma }}^6+16d^4-64d^{3}r+96d^2{r}^2-48d^2{\mathrm{\sigma }}^2-64d{r}^3+96dr{\mathrm{\sigma }}^2+16r^4-48r^2{\mathrm{\sigma }}^2\right)}{8{\mathrm{\sigma }}^6\sqrt{\mathrm{\pi }}}$

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

$\mathrm{-14.919796}$

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

$\frac{25\sqrt{2}{\ⅇ}^{-\frac{{\mathrm{\sigma }}^4+4d{\mathrm{\sigma }}^2+4r{\mathrm{\sigma }}^2+4d^2-8dr+4r^2}{8{\mathrm{\sigma }}^2}}\left({\mathrm{\sigma }}^8-8d^2{\mathrm{\sigma }}^4+16dr{\mathrm{\sigma }}^4-8r^2{\mathrm{\sigma }}^4-4{\mathrm{\sigma }}^6+16d^4-64d^{3}r+96d^2{r}^2-48d^2{\mathrm{\sigma }}^2-64d{r}^3+96dr{\mathrm{\sigma }}^2+16r^4-48r^2{\mathrm{\sigma }}^2\right)}{8{\mathrm{\sigma }}^6\sqrt{\mathrm{\pi }}}$

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

$\mathrm{-14.9197957}$

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

$\mathrm{Ultima}≔\frac{0.007556166257{\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 }}^2-0.9065963{\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 }}^6-5.984134198{\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}^4+0.3590480519{\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}^3+39.89422799{\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}^5-0.01077144155{\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+1.246694625{\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 }}^{10}+0.0001615716233{\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-7.234279682{\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 }}^4-5.061580167{\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 }}^8-0.7561551978{\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}r+25.24107807{\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{r}^2+242.9019915{\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}r-281.6532498{\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{r}^3-58.04610176{\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{r}^2+30.51908445{\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 }}^{6}r-19.947114{\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}^3{\mathrm{\sigma }}^4+19.94711399{\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}^4{\mathrm{\sigma }}^2+2.493389246{\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 }}^{8}r-9.973557002{\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{\mathrm{\sigma }}^6-9.694297398\times {10}^{\mathrm{-7}}{\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}}+268.6833172{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2{r}^3+19.35758768{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^3{\mathrm{\sigma }}^4+19.35758768{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^4{\mathrm{\sigma }}^2-2.41969846{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{8}r-9.678793837{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2{\mathrm{\sigma }}^6-28.4556539{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{6}r+0.7296261952{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{2}r-228.7544211{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{4}r-24.28602953{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2{r}^2-59.81494597{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4{r}^2+5.807276302{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^4-0.3484365781{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^3-38.71517536{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^5+0.01045309735{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{r}^2+1.20984923{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^{10}-0.0001567964602{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}r-4.766805971{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^8+0.8623805319{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^6+6.915943432{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^4-0.007301488501{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+9.407787608\times {10}^{\mathrm{-7}}{\ⅇ}^{-\frac{0.00004999999997{\left(50.{\mathrm{\sigma }}^2+100.r-3.\right)}^2}{{\mathrm{\sigma }}^2}}}{{\mathrm{\sigma }}^8}$

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

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

$\mathrm{-329.853365}$

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

$\mathrm{-329.853364}$

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

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

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