The Theta of an option or a portfolio of options is the rate of change of the option price or the portfolio price with time. As time progresses, the time to maturity decreases; this explains the minus sign in the following definition:

The BlackScholesTheta command computes the Theta 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.

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

$r≔0.05$

$r≔0.05$

$d≔0.03$

$d≔0.03$

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

$-0.922405261+1.4556683\mathrm{erf}\left(\frac{0.01414213562}{\mathrm{\sigma }}+0.3535533905\mathrm{\sigma }\right)-\frac{1.\times {10}^{\mathrm{-10}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\ⅇ}^{-\frac{0.0001999999998}{{\mathrm{\sigma }}^2}}}{\mathrm{\sigma }}-19.16497649\mathrm{\sigma }{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}{\ⅇ}^{-\frac{0.0001999999998}{{\mathrm{\sigma }}^2}}+2.378073561\mathrm{erf}\left(-\frac{0.01414213562}{\mathrm{\sigma }}+0.3535533905\mathrm{\sigma }\right)$

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

$\mathrm{-6.187329487}$

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

$\mathrm{-6.187329483}$

$\mathrm{\Theta }≔\mathrm{expand}\left(\mathrm{BlackScholesTheta}\left(100\,K\,1\,\mathrm{\sigma }\,r\,d\,'\mathrm{call}'\right)\right)$

$\mathrm{\Theta }≔1.4556683+1.4556683\mathrm{erf}\left(\frac{3.270489202}{\mathrm{\sigma }}+\frac{0.707106781\ln \left(\frac{1}{K}\right)}{\mathrm{\sigma }}+0.3535533905\mathrm{\sigma }\right)+\frac{8.787467887{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{-\frac{4.625170183}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{\mathrm{\sigma }{\left(\frac{1}{K}\right)}^{0.4999999997}}-\frac{0.9582488255\mathrm{\sigma }{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{-\frac{4.625170183}{{\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)}^{0.4999999997}}+\frac{1.916497650\ln \left(\frac{1}{K}\right){\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{-\frac{4.625170183}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{\mathrm{\sigma }{\left(\frac{1}{K}\right)}^{0.4999999997}}-0.02378073561K+0.02378073561K\mathrm{erf}\left(-\frac{3.270489202}{\mathrm{\sigma }}-\frac{0.707106781\ln \left(\frac{1}{K}\right)}{\mathrm{\sigma }}+0.3535533905\mathrm{\sigma }\right)-\frac{8.787467868K{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{-\frac{4.625170184}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{0.4999999998}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}-0.9582488230\mathrm{\sigma }K{\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{-\frac{4.625170184}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{0.4999999998}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}-\frac{1.916497646K\ln \left(\frac{1}{K}\right){\ⅇ}^{-\frac{10.69609962}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{-\frac{4.625170184}{{\mathrm{\sigma }}^2}}{\ⅇ}^{-\frac{0.4999999997{\ln \left(\frac{1}{K}\right)}^2}{{\mathrm{\sigma }}^2}}{\left(\frac{1}{K}\right)}^{0.4999999998}{\ⅇ}^{-0.1249999999{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}$

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

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

$\frac{1.398019973\mathrm{\sigma }+2.853688273\mathrm{erf}\left(\frac{0.1147786735+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }+4.606687476{\ⅇ}^{-\frac{1.{\left(0.1147786735+0.3535533905{\mathrm{\sigma }}^2\right)}^2}{{\mathrm{\sigma }}^2}}-11.38456907{\ⅇ}^{-\frac{1.{\left(0.1147786735+0.3535533905{\mathrm{\sigma }}^2\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+1.4556683\mathrm{erf}\left(\frac{-0.1147786735+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }-3.91645728{\ⅇ}^{-\frac{0.1249999999{\left(-0.3246431136+{\mathrm{\sigma }}^2\right)}^2}{{\mathrm{\sigma }}^2}}-9.678793854{\ⅇ}^{-\frac{0.1249999999{\left(-0.3246431136+{\mathrm{\sigma }}^2\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2}{\mathrm{\sigma }}$

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

$\mathrm{-2.96788222}$

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

$\frac{1.4556683\mathrm{erf}\left(\frac{-0.1147786735+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }-3.916457279{\ⅇ}^{-\frac{1.279999999\times {10}^{\mathrm{-18}}{\left(3.12500000\times {10}^8{\mathrm{\sigma }}^2-1.01450973\times {10}^8\right)}^2}{{\mathrm{\sigma }}^2}}-9.678793852{\ⅇ}^{-\frac{1.279999999\times {10}^{\mathrm{-18}}{\left(3.12500000\times {10}^8{\mathrm{\sigma }}^2-1.01450973\times {10}^8\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-11.38456907{\ⅇ}^{-\frac{1.279999999\times {10}^{\mathrm{-18}}{\left(3.12500000\times {10}^8{\mathrm{\sigma }}^2+1.01450973\times {10}^8\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+4.606687474{\ⅇ}^{-\frac{1.279999999\times {10}^{\mathrm{-18}}{\left(3.12500000\times {10}^8{\mathrm{\sigma }}^2+1.01450973\times {10}^8\right)}^2}{{\mathrm{\sigma }}^2}}+1.398019974\mathrm{\sigma }+2.853688274\mathrm{erf}\left(\frac{0.1147786735+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }}{\mathrm{\sigma }}$

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

$\frac{1.455668301\mathrm{erf}\left(\frac{0.01414213562+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }-0.9224052608\mathrm{\sigma }+2.378073561\mathrm{erf}\left(\frac{-0.01414213562+0.3535533905{\mathrm{\sigma }}^2}{\mathrm{\sigma }}\right)\mathrm{\sigma }-9.67879385{\ⅇ}^{-\frac{0.0001999999999{\left(25.{\mathrm{\sigma }}^2+1.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2-0.387151754{\ⅇ}^{-\frac{0.0001999999999{\left(25.{\mathrm{\sigma }}^2+1.\right)}^2}{{\mathrm{\sigma }}^2}}-9.48714089{\ⅇ}^{-\frac{0.0001999999999{\left(5.\mathrm{\sigma }-1.\right)}^2{\left(5.\mathrm{\sigma }+1.\right)}^2}{{\mathrm{\sigma }}^2}}{\mathrm{\sigma }}^2+0.3794856356{\ⅇ}^{-\frac{0.0001999999999{\left(5.\mathrm{\sigma }-1.\right)}^2{\left(5.\mathrm{\sigma }+1.\right)}^2}{{\mathrm{\sigma }}^2}}}{\mathrm{\sigma }}$

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

$\mathrm{-2.967882246}$

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

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

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