The ImpliedVolatility command computes the implied Black-Scholes volatility for a European-style option given its price.

The parameter price is the option price.

The parameter spot is the initial (current) value of the underlying asset.

The parameter strike specifies the strike price of the option (if this is a call option or a put option). More general payoff can be specified using the payoff parameter. It must be specified in the form of an operator, which accepts one parameter (spot price at maturity) and returns the corresponding payoff.

The riskfreerate and dividendyield parameters are the risk-free rate and the dividend yield. These parameters can be given in either the algebraic form or the operator form.

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

$S≔100$

$S≔100$

$r≔0.05$

$r≔0.05$

$d≔0.03$

$d≔0.03$

$K≔100$

$K≔100$

$\mathrm{ImpliedVolatility}\left(15.0\,S\,100\,1\,r\,d\,'\mathrm{call}'\right)$

$0.3677816494$

$\mathrm{ImpliedVolatility}\left(15.0\,S\,t\mapsto \max \left(t-100\,0\right)\,1\,r\,d\right)$

$0.3677816498$

$\mathrm{ImpliedVolatility}\left(15.0\,S\,100\,1\,r\,d\,'\mathrm{put}'\right)$

$0.4189619392$

$\mathrm{ImpliedVolatility}\left(15.0\,S\,t\mapsto \max \left(100-t\,0\right)\,1\,r\,d\right)$

$0.4189619390$

$\mathrm{\sigma }≔\mathrm{ImpliedVolatility}\left(15.0\&comma;S\&comma;t\mapsto \mathrm{piecewise}\left(t<90\&comma;90-t\&comma;t<110\&comma;0\&comma;t-110\right)\&comma;1\&comma;r\&comma;d\right)$

$\mathrm{\sigma }≔0.3041486797$

$\mathrm{BlackScholesPrice}\left(S\&comma;t\mapsto \mathrm{piecewise}\left(t<90\&comma;90-t\&comma;t<110\&comma;0\&comma;t-110\right)\&comma;1\&comma;\mathrm{\sigma }\&comma;r\&comma;d\right)$

$14.99999999$

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

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

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