Finance
BlackScholesTheta
compute the Theta of a European-style option with given payoff
Calling Sequence
Parameters
Description
Examples
References
Compatibility
BlackScholesTheta(S0, K, T, sigma, r, d, optiontype)
BlackScholesTheta(S0, P, T, sigma, r, d)
S0
-
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
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:
Θ=−∂∂TS
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.
with(Finance):
r := 0.05;
r≔0.05
d := 0.03;
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.
expand(BlackScholesTheta(100, 100, 1, sigma, r, d, 'call'));
−0.922405261+1.4556683⁢erf⁡0.01414213562σ+0.3535533905⁢σ−1.⁢10−10⁢ⅇ−0.1249999999⁢σ2⁢ⅇ−0.0001999999998σ2σ−19.16497649⁢σ⁢ⅇ−0.1249999999⁢σ2⁢ⅇ−0.0001999999998σ2+2.378073561⁢erf⁡−0.01414213562σ+0.3535533905⁢σ
In this example you will use numeric values for the risk-free rate, the dividend yield, and the volatility.
expand(BlackScholesTheta(100, 100, 1, 0.3, 0.05, 0.03, 'call'));
−6.187329487
You can also use the generic method in which the option is defined through its payoff function.
BlackScholesTheta(100, t -> max(t-100, 0), 1, 0.3, 0.05, 0.03);
−6.187329483
Theta := expand(BlackScholesTheta(100, K, 1, sigma, r, d, 'call'));
Θ≔1.4556683+1.4556683⁢erf⁡3.270489202σ+0.707106781⁢ln⁡1Kσ+0.3535533905⁢σ+8.787467887⁢ⅇ−10.69609962σ2⁢1K−4.625170183σ2⁢ⅇ−0.4999999997⁢ln⁡1K2σ2⁢ⅇ−0.1249999999⁢σ2σ⁢1K0.4999999997−0.9582488255⁢σ⁢ⅇ−10.69609962σ2⁢1K−4.625170183σ2⁢ⅇ−0.4999999997⁢ln⁡1K2σ2⁢ⅇ−0.1249999999⁢σ21K0.4999999997+1.916497650⁢ln⁡1K⁢ⅇ−10.69609962σ2⁢1K−4.625170183σ2⁢ⅇ−0.4999999997⁢ln⁡1K2σ2⁢ⅇ−0.1249999999⁢σ2σ⁢1K0.4999999997−0.02378073561⁢K+0.02378073561⁢K⁢erf⁡−3.270489202σ−0.707106781⁢ln⁡1Kσ+0.3535533905⁢σ−8.787467868⁢K⁢ⅇ−10.69609962σ2⁢1K−4.625170184σ2⁢ⅇ−0.4999999997⁢ln⁡1K2σ2⁢1K0.4999999998⁢ⅇ−0.1249999999⁢σ2σ−0.9582488230⁢σ⁢K⁢ⅇ−10.69609962σ2⁢1K−4.625170184σ2⁢ⅇ−0.4999999997⁢ln⁡1K2σ2⁢1K0.4999999998⁢ⅇ−0.1249999999⁢σ2−1.916497646⁢K⁢ln⁡1K⁢ⅇ−10.69609962σ2⁢1K−4.625170184σ2⁢ⅇ−0.4999999997⁢ln⁡1K2σ2⁢1K0.4999999998⁢ⅇ−0.1249999999⁢σ2σ
plot3d(Theta, sigma = 0..1, K = 70..120, axes = BOXED);
Here are similar examples for the European put option.
BlackScholesTheta(100, 120, 1, sigma, r, d, 'put');
1.398019973⁢σ+2.853688273⁢erf⁡0.1147786735+0.3535533905⁢σ2σ⁢σ+4.606687476⁢ⅇ−1.⁢0.1147786735+0.3535533905⁢σ22σ2−11.38456907⁢ⅇ−1.⁢0.1147786735+0.3535533905⁢σ22σ2⁢σ2+1.4556683⁢erf⁡−0.1147786735+0.3535533905⁢σ2σ⁢σ−3.91645728⁢ⅇ−0.1249999999⁢−0.3246431136+σ22σ2−9.678793854⁢ⅇ−0.1249999999⁢−0.3246431136+σ22σ2⁢σ2σ
BlackScholesTheta(100, 120, 1, 0.3, 0.05, 0.03, 'put');
−2.96788222
BlackScholesTheta(100, t -> max(120-t, 0), 1, sigma, r, d);
1.4556683⁢erf⁡−0.1147786735+0.3535533905⁢σ2σ⁢σ−3.916457279⁢ⅇ−1.279999999⁢10−18⁢3.12500000⁢108⁢σ2−1.01450973⁢1082σ2−9.678793852⁢ⅇ−1.279999999⁢10−18⁢3.12500000⁢108⁢σ2−1.01450973⁢1082σ2⁢σ2−11.38456907⁢ⅇ−1.279999999⁢10−18⁢3.12500000⁢108⁢σ2+1.01450973⁢1082σ2⁢σ2+4.606687474⁢ⅇ−1.279999999⁢10−18⁢3.12500000⁢108⁢σ2+1.01450973⁢1082σ2+1.398019974⁢σ+2.853688274⁢erf⁡0.1147786735+0.3535533905⁢σ2σ⁢σσ
Compare with
BlackScholesTheta(100, t -> max(t-100, 0), 1, sigma, r, d);
1.455668301⁢erf⁡0.01414213562+0.3535533905⁢σ2σ⁢σ−0.9224052608⁢σ+2.378073561⁢erf⁡−0.01414213562+0.3535533905⁢σ2σ⁢σ−9.67879385⁢ⅇ−0.0001999999999⁢25.⁢σ2+1.2σ2⁢σ2−0.387151754⁢ⅇ−0.0001999999999⁢25.⁢σ2+1.2σ2−9.48714089⁢ⅇ−0.0001999999999⁢5.⁢σ−1.2⁢5.⁢σ+1.2σ2⁢σ2+0.3794856356⁢ⅇ−0.0001999999999⁢5.⁢σ−1.2⁢5.⁢σ+1.2σ2σ
BlackScholesTheta(100, t -> max(120-t, 0), 1, 0.3, 0.05, 0.03, d);
−2.967882246
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.
See Also
Finance[AmericanOption]
Finance[BermudanOption]
Finance[BlackScholesDelta]
Finance[BlackScholesGamma]
Finance[BlackScholesPrice]
Finance[BlackScholesRho]
Finance[BlackScholesVega]
Finance[EuropeanOption]
Finance[ImpliedVolatility]
Finance[LatticePrice]
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