compute the Vega of a European-style option with given payoff
BlackScholesVega(S0, K, T, sigma, r, d, optiontype)
BlackScholesVega(S0, P, T, sigma, r, d)
algebraic expression; initial (current) value of the underlying asset
algebraic expression; strike price
algebraic expression; time to maturity
algebraic expression; volatility
algebraic expression; continuously compounded risk-free rate
algebraic expression; continuously compounded dividend yield
operator or procedure; payoff function
call or put; option type
The Vega of an option or a portfolio of options is the sensitivity of the option or portfolio to changes in the volatility of the underlying asset.
The BlackScholesVega command computes the Vega of a European-style option with the specified payoff function.
The parameter S0 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.
r ≔ 0.05
d ≔ 0.03
First you compute the Vega of a European call option with strike price 100, which matures in 1 year. This will define the Vega as a function of the risk-free rate, the dividend yield, and the volatility.
In this example you will use numeric values for the risk-free rate, the dividend yield, and the volatility.
You can also use the generic method in which the option is defined through its payoff function.
Vega ≔ expand⁡BlackScholesVega⁡100,K,1,σ,0.05,0.03,'call'
Here are similar examples for the European put option.
In this example, you will compute the Vega of a strangle.
S ≔ expand⁡BlackScholesVega⁡100,t→piecewise⁡t<90,90−t,t<110,0,t−110,1,σ,r,d
C ≔ expand⁡BlackScholesVega⁡100,110,1,σ,r,d,'call'
P ≔ expand⁡BlackScholesVega⁡100,90,1,σ,r,d,'put'
Check that S is sufficiently close to C+P.
Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.
The Finance[BlackScholesVega] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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