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.

with(Finance):

r := 0.05;

$r≔0.05$

d := 0.03;

$d≔0.03$

expand(BlackScholesTheta(100, 100, 1, sigma, r, d, 'call'));

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

expand(BlackScholesTheta(100, 100, 1, 0.3, 0.05, 0.03, 'call'));

$\mathrm{-6.187329487}$

BlackScholesTheta(100, t -> max(t-100, 0), 1, 0.3, 0.05, 0.03);

$\mathrm{-6.187329483}$

Theta := expand(BlackScholesTheta(100, K, 1, sigma, r, d, 'call'));

$\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 }}$

plot3d(Theta, sigma = 0..1, K = 70..120, axes = BOXED);

BlackScholesTheta(100, 120, 1, sigma, r, d, 'put');

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

BlackScholesTheta(100, 120, 1, 0.3, 0.05, 0.03, 'put');

$\mathrm{-2.96788222}$

BlackScholesTheta(100, t -> max(120-t, 0), 1, sigma, r, d);

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

BlackScholesTheta(100, t -> max(t-100, 0), 1, sigma, r, d);

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

BlackScholesTheta(100, t -> max(120-t, 0), 1, 0.3, 0.05, 0.03, d);

$\mathrm{-2.967882246}$

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.