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BlackScholesTheta

  

compute the Theta of a European-style option with given payoff

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

BlackScholesTheta(S0, K, T, sigma, r, d, optiontype)

BlackScholesTheta(S0, P, T, sigma, r, d)

Parameters

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

Description

• 

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.

Examples

withFinance:

r0.05

r:=0.05

(1)

d0.03

d:=0.03

(2)

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.

expandBlackScholesTheta100,100,1,σ,r,d,'call'

0.922405261+1.4556683erf0.01414213562σ+0.3535533905σ1.10-10ⅇ0.1249999999σ2ⅇ0.0001999999998σ2σ19.16497649σⅇ0.1249999999σ2ⅇ0.0001999999998σ2+2.378073561erf0.01414213562σ+0.3535533905σ

(3)

In this example you will use numeric values for the risk-free rate, the dividend yield, and the volatility.

expandBlackScholesTheta100,100,1,0.3,0.05,0.03,'call'

6.187329487

(4)

You can also use the generic method in which the option is defined through its payoff function.

BlackScholesTheta100,t→maxt100,0,1,0.3,0.05,0.03

6.187329483

(5)

ΘexpandBlackScholesTheta100,K,1,σ,r,d,'call'

Θ:=1.4556683+1.4556683erf3.270489202σ+0.707106781ln1Kσ+0.3535533905σ+8.787467889ⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ2σ1K0.49999999970.9582488253σⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ21K0.4999999997+1.916497650ln1Kⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ2σ1K0.49999999970.02378073561K+0.02378073561Kerf3.270489202σ0.707106781ln1Kσ+0.3535533905σ8.787467868Kⅇ10.69609962σ21K4.625170184σ2ⅇ0.4999999997ln1K2σ21K0.4999999998ⅇ0.1249999999σ2σ0.9582488230σKⅇ10.69609962σ21K4.625170184σ2ⅇ0.4999999997ln1K2σ21K0.4999999998ⅇ0.1249999999σ21.916497646Kln1Kⅇ10.69609962σ21K4.625170184σ2ⅇ0.4999999997ln1K2σ21K0.4999999998ⅇ0.1249999999σ2σ

(6)

plot3dΘ,σ=0..1,K=70..120,axes=BOXED

Here are similar examples for the European put option.

BlackScholesTheta100,120,1,σ,r,d,'put'

1.398019973σ+2.853688273erf0.1147786735+0.3535533905σ2σσ+4.606687476ⅇ1.0.1147786735+0.3535533905σ22σ211.38456907ⅇ1.0.1147786735+0.3535533905σ22σ2σ2+1.4556683erf0.1147786735+0.3535533905σ2σσ3.91645728ⅇ0.12499999990.3246431136+σ22σ29.678793854ⅇ0.12499999990.3246431136+σ22σ2σ2σ

(7)

BlackScholesTheta100,120,1,0.3,0.05,0.03,'put'

2.96788222

(8)

BlackScholesTheta100,t→max120t,0,1,σ,r,d

1.4556683erf0.1147786735+0.3535533905σ2σσ3.916457279ⅇ1.27999999910-183.12500000108σ21.014509731082σ29.678793852ⅇ1.27999999910-183.12500000108σ21.014509731082σ2σ211.38456907ⅇ1.27999999910-183.12500000108σ2+1.014509731082σ2σ2+4.606687474ⅇ1.27999999910-183.12500000108σ2+1.014509731082σ2+1.398019974σ+2.853688274erf0.1147786735+0.3535533905σ2σσσ

(9)

Compare with

BlackScholesTheta100,t→maxt100,0,1,σ,r,d

1.455668301erf0.01414213562+0.3535533905σ2σσ9.67879385ⅇ0.000199999999925.σ2+1.2σ2σ20.387151754ⅇ0.000199999999925.σ2+1.2σ20.9224052608σ9.48714089ⅇ0.00019999999995.σ1.25.σ+1.2σ2σ2+0.3794856356ⅇ0.00019999999995.σ1.25.σ+1.2σ2+2.378073561erf0.01414213562+0.3535533905σ2σσσ

(10)

BlackScholesTheta100,t→max120t,0,1,0.3,0.05,0.03,d

2.967882245

(11)

References

  

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

Compatibility

• 

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[BlackScholesPrice]

Finance[BlackScholesRho]

Finance[BlackScholesVega]

Finance[EuropeanOption]

Finance[ImpliedVolatility]

Finance[LatticePrice]

 


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