 BlackScholesTheta - Maple Help

Finance

 BlackScholesTheta
 compute the Theta of a European-style option with given payoff Calling Sequence BlackScholesTheta(${S}_{0}$, K, T, sigma, r, d, optiontype) BlackScholesTheta(${S}_{0}$, P, T, sigma, r, d) Parameters

 ${S}_{0}$ - algebraic expression; initial (current) value of the underlying asset K - algebraic expression; strike price T - algebraic expression; time to maturity sigma - algebraic expression; volatility r - algebraic expression; continuously compounded risk-free rate d - algebraic expression; continuously compounded dividend yield P - operator or procedure; payoff function optiontype - call or put; option type Description

 • 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:

$\mathrm{\Theta }=-\frac{ⅆS}{ⅆT}$

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{Finance}\right):$
 > $r≔0.05$
 ${r}{≔}{0.05}$ (1)
 > $d≔0.03$
 ${d}{≔}{0.03}$ (2)

First you compute the Theta of a European call option with strike price 100, which matures in 1 year. This will define the Theta as a function of the risk-free rate, the dividend yield, and the volatility.

 > $\mathrm{expand}\left(\mathrm{BlackScholesTheta}\left(100,100,1,\mathrm{σ},r,d,'\mathrm{call}'\right)\right)$
 ${-}{0.922405261}{+}{1.4556683}{}{\mathrm{erf}}{}\left(\frac{{0.01414213562}}{{\mathrm{\sigma }}}{+}{0.3535533905}{}{\mathrm{\sigma }}\right){-}\frac{{1.}{×}{{10}}^{{-10}}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{\sigma }}}^{{2}}}{}{{ⅇ}}^{{-}\frac{{0.0001999999998}}{{{\mathrm{\sigma }}}^{{2}}}}}{{\mathrm{\sigma }}}{-}{19.16497649}{}{\mathrm{\sigma }}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{\sigma }}}^{{2}}}{}{{ⅇ}}^{{-}\frac{{0.0001999999998}}{{{\mathrm{\sigma }}}^{{2}}}}{+}{2.378073561}{}{\mathrm{erf}}{}\left({-}\frac{{0.01414213562}}{{\mathrm{\sigma }}}{+}{0.3535533905}{}{\mathrm{\sigma }}\right)$ (3)

In this example you will use numeric values for the risk-free rate, the dividend yield, and the volatility.

 > $\mathrm{expand}\left(\mathrm{BlackScholesTheta}\left(100,100,1,0.3,0.05,0.03,'\mathrm{call}'\right)\right)$
 ${-6.187329487}$ (4)

You can also use the generic method in which the option is defined through its payoff function.

 > $\mathrm{BlackScholesTheta}\left(100,t→\mathrm{max}\left(t-100,0\right),1,0.3,0.05,0.03\right)$
 ${-6.187329483}$ (5)
 > $\mathrm{Θ}≔\mathrm{expand}\left(\mathrm{BlackScholesTheta}\left(100,K,1,\mathrm{σ},r,d,'\mathrm{call}'\right)\right)$
 ${\mathrm{\Theta }}{≔}{1.4556683}{+}{1.4556683}{}{\mathrm{erf}}{}\left(\frac{{3.270489202}}{{\mathrm{\sigma }}}{+}\frac{{0.707106781}{}{\mathrm{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}{}{{\mathrm{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}{}{{\mathrm{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}{}{\mathrm{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}{}{{\mathrm{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}}}{-}{0.02378073561}{}{K}{+}{0.02378073561}{}{K}{}{\mathrm{erf}}{}\left({-}\frac{{3.270489202}}{{\mathrm{\sigma }}}{-}\frac{{0.707106781}{}{\mathrm{ln}}{}\left(\frac{{1}}{{K}}\right)}{{\mathrm{\sigma }}}{+}{0.3535533905}{}{\mathrm{\sigma }}\right){-}\frac{{8.787467868}{}{K}{}{{ⅇ}}^{{-}\frac{{10.69609962}}{{{\mathrm{\sigma }}}^{{2}}}}{}{\left(\frac{{1}}{{K}}\right)}^{{-}\frac{{4.625170184}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{\mathrm{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}{}{{\mathrm{ln}}{}\left(\frac{{1}}{{K}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{\left(\frac{{1}}{{K}}\right)}^{{0.4999999998}}{}{{ⅇ}}^{{-}{0.1249999999}{}{{\mathrm{\sigma }}}^{{2}}}{-}\frac{{1.916497646}{}{\mathrm{ln}}{}\left(\frac{{1}}{{K}}\right){}{K}{}{{ⅇ}}^{{-}\frac{{10.69609962}}{{{\mathrm{\sigma }}}^{{2}}}}{}{\left(\frac{{1}}{{K}}\right)}^{{-}\frac{{4.625170184}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{ⅇ}}^{{-}\frac{{0.4999999997}{}{{\mathrm{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 }}}$ (6)
 > $\mathrm{plot3d}\left(\mathrm{Θ},\mathrm{σ}=0..1,K=70..120,\mathrm{axes}=\mathrm{BOXED}\right)$ Here are similar examples for the European put option.

 > $\mathrm{BlackScholesTheta}\left(100,120,1,\mathrm{σ},r,d,'\mathrm{put}'\right)$
 $\frac{{1.398019973}{}{\mathrm{\sigma }}{+}{2.853688273}{}{\mathrm{erf}}{}\left(\frac{{0.1147786735}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}\right){}{\mathrm{\sigma }}{+}{4.606687476}{}{{ⅇ}}^{{-}\frac{{1.}{}{\left({0.1147786735}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{-}{11.38456907}{}{{ⅇ}}^{{-}\frac{{1.}{}{\left({0.1147786735}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{+}{1.4556683}{}{\mathrm{erf}}{}\left(\frac{{-}{0.1147786735}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}\right){}{\mathrm{\sigma }}{-}{3.91645728}{}{{ⅇ}}^{{-}\frac{{0.1249999999}{}{\left({-}{0.3246431136}{+}{{\mathrm{\sigma }}}^{{2}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{-}{9.678793854}{}{{ⅇ}}^{{-}\frac{{0.1249999999}{}{\left({-}{0.3246431136}{+}{{\mathrm{\sigma }}}^{{2}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}$ (7)
 > $\mathrm{BlackScholesTheta}\left(100,120,1,0.3,0.05,0.03,'\mathrm{put}'\right)$
 ${-2.96788222}$ (8)
 > $\mathrm{BlackScholesTheta}\left(100,t→\mathrm{max}\left(120-t,0\right),1,\mathrm{σ},r,d\right)$
 $\frac{{-}{3.916457279}{}{{ⅇ}}^{{-}\frac{{1.279999999}{×}{{10}}^{{-18}}{}{\left({3.12500000}{×}{{10}}^{{8}}{}{{\mathrm{\sigma }}}^{{2}}{-}{1.01450973}{×}{{10}}^{{8}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{-}{9.678793852}{}{{ⅇ}}^{{-}\frac{{1.279999999}{×}{{10}}^{{-18}}{}{\left({3.12500000}{×}{{10}}^{{8}}{}{{\mathrm{\sigma }}}^{{2}}{-}{1.01450973}{×}{{10}}^{{8}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{+}{1.4556683}{}{\mathrm{erf}}{}\left(\frac{{-}{0.1147786735}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}\right){}{\mathrm{\sigma }}{-}{11.38456907}{}{{ⅇ}}^{{-}\frac{{1.279999999}{×}{{10}}^{{-18}}{}{\left({3.12500000}{×}{{10}}^{{8}}{}{{\mathrm{\sigma }}}^{{2}}{+}{1.01450973}{×}{{10}}^{{8}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{+}{4.606687474}{}{{ⅇ}}^{{-}\frac{{1.279999999}{×}{{10}}^{{-18}}{}{\left({3.12500000}{×}{{10}}^{{8}}{}{{\mathrm{\sigma }}}^{{2}}{+}{1.01450973}{×}{{10}}^{{8}}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{+}{1.398019974}{}{\mathrm{\sigma }}{+}{2.853688274}{}{\mathrm{erf}}{}\left(\frac{{0.1147786735}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}\right){}{\mathrm{\sigma }}}{{\mathrm{\sigma }}}$ (9)

Compare with

 > $\mathrm{BlackScholesTheta}\left(100,t→\mathrm{max}\left(t-100,0\right),1,\mathrm{σ},r,d\right)$
 $\frac{{1.455668301}{}{\mathrm{erf}}{}\left(\frac{{0.01414213562}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}\right){}{\mathrm{\sigma }}{-}{0.9224052608}{}{\mathrm{\sigma }}{+}{2.378073561}{}{\mathrm{erf}}{}\left(\frac{{-}{0.01414213562}{+}{0.3535533905}{}{{\mathrm{\sigma }}}^{{2}}}{{\mathrm{\sigma }}}\right){}{\mathrm{\sigma }}{-}{9.67879385}{}{{ⅇ}}^{{-}\frac{{0.0001999999999}{}{\left({25.}{}{{\mathrm{\sigma }}}^{{2}}{+}{1.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{-}{0.387151754}{}{{ⅇ}}^{{-}\frac{{0.0001999999999}{}{\left({25.}{}{{\mathrm{\sigma }}}^{{2}}{+}{1.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{-}{9.48714089}{}{{ⅇ}}^{{-}\frac{{0.0001999999999}{}{\left({5.}{}{\mathrm{\sigma }}{-}{1.}\right)}^{{2}}{}{\left({5.}{}{\mathrm{\sigma }}{+}{1.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}{}{{\mathrm{\sigma }}}^{{2}}{+}{0.3794856356}{}{{ⅇ}}^{{-}\frac{{0.0001999999999}{}{\left({5.}{}{\mathrm{\sigma }}{-}{1.}\right)}^{{2}}{}{\left({5.}{}{\mathrm{\sigma }}{+}{1.}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}}{{\mathrm{\sigma }}}$ (10)
 > $\mathrm{BlackScholesTheta}\left(100,t→\mathrm{max}\left(120-t,0\right),1,0.3,0.05,0.03,d\right)$
 ${-2.967882255}$ (11) References

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

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