ImpliedVolatility - Maple Help

Finance

 ImpliedVolatility
 compute the implied volatility for a European-style option

 Calling Sequence ImpliedVolatility(price, spot, strike, timetomaturity, riskfreerate, dividendyield, optiontype) ImpliedVolatility(price, spot, payoff, timetomaturity, riskfreerate, dividendyield)

Parameters

 price - algebraic expression; option price spot - algebraic expression; spot price of the underlying asset strike - algebraic expression; strike price timetomaturity - algebraic expression; time to maturity (in years) riskfreerate - algebraic expression; continuously compounded risk-free rate dividendyield - algebraic expression; continuously compounded dividend yield optiontype - call or put; option type payoff - operator or procedure; payoff function

Description

 • 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.

Examples

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

Compute implied volatilities for an asset with spot price ${S}_{0}=100$. Assume that the risk-free rate is 5% and the dividend yield is 3%.

 > $S≔100$
 ${S}{≔}{100}$ (1)
 > $r≔0.05$
 ${r}{≔}{0.05}$ (2)
 > $d≔0.03$
 ${d}{≔}{0.03}$ (3)

First you consider European call and put options with strike price $K=100$.

 > $K≔100$
 ${K}{≔}{100}$ (4)
 > $\mathrm{ImpliedVolatility}\left(15.0,S,100,1,r,d,'\mathrm{call}'\right)$
 ${0.3677816494}$ (5)
 > $\mathrm{ImpliedVolatility}\left(15.0,S,t↦\mathrm{max}\left(t-100,0\right),1,r,d\right)$
 ${0.3677816498}$ (6)
 > $\mathrm{ImpliedVolatility}\left(15.0,S,100,1,r,d,'\mathrm{put}'\right)$
 ${0.4189619392}$ (7)
 > $\mathrm{ImpliedVolatility}\left(15.0,S,t↦\mathrm{max}\left(100-t,0\right),1,r,d\right)$
 ${0.4189619390}$ (8)

In this example you consider a strangle.

 > $\mathrm{\sigma }≔\mathrm{ImpliedVolatility}\left(15.0,S,t↦\mathrm{piecewise}\left(t<90,90-t,t<110,0,t-110\right),1,r,d\right)$
 ${\mathrm{\sigma }}{≔}{0.3041486797}$ (9)
 > $\mathrm{BlackScholesPrice}\left(S,t↦\mathrm{piecewise}\left(t<90,90-t,t<110,0,t-110\right),1,\mathrm{\sigma },r,d\right)$
 ${14.99999999}$ (10)

References

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

Compatibility

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