compute the Theta of a European-style option with given payoff
BlackScholesTheta(S0, K, T, sigma, r, d, optiontype)
BlackScholesTheta(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 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 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 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.
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.
Θ ≔ expand⁡BlackScholesTheta⁡100,K,1,σ,r,d,'call'
Here are similar examples for the European put option.
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.
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