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Finance

  

BlackScholesDelta

  

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

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

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

BlackScholesDelta(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 Delta of an option or a portfolio of options is the sensitivity of the option or portfolio to changes in the value of the underlying asset

Δ=S0S

• 

The BlackScholesDelta command computes the Delta 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:

First compute the Delta of a European call option with strike price 100, which matures in 1 year. This will define the Delta as a function of the risk-free rate, the dividend yield, and the volatility.

BlackScholesDelta100,100,1,σ,r,d,'call'

ⅇderfσ2+2d2r24σ12

(1)

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

BlackScholesDelta100,100,1,0.3,0.05,0.03,'call'

0.568453937

(2)

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

BlackScholesDelta100,t→maxt100,0,1,σ,r,d

ⅇderfσ2+2d2r24σ12

(3)

BlackScholesDelta100,t→maxt100,0,1,0.3,0.05,0.03

0.5684539378

(4)

ΔBlackScholesDelta100,100,1,σ,r,0.03,'call'

Δ0.4852227668σ+0.4852227668erf0.02121320343+0.707106781r+0.3535533905σ2σσ+0.387151754ⅇ0.0000499999999750.σ2+100.r3.2σ20.3989422803ⅇ1.0.5000000002rσ2+0.00044999999980.02999999998r+0.01499999999σ2+0.4999999997r2+0.1249999999σ4σ2σ

(5)

plot3dΔ,σ=0..1,r=0..1,axes=BOXED

Here are similar examples for the European put option.

BlackScholesDelta100,120,1,σ,r,d,'put'

5ⅇderf2ln65σ2+2d2r24σπσ+6ⅇσ2ln1296625+4ln32+ln5ln16561+ln35ln256+dln1679616390625+rln3906251679616+4ln52+4ln22+σ4+4dσ2+4rσ2+4d28dr+4r28σ225ⅇσ2ln6251296+4ln32+ln5ln16561+ln35ln256+dln1679616390625+rln3906251679616+4ln52+4ln22+σ4+4dσ2+4rσ2+4d28dr+4r28σ22+5ⅇdπσ10πσ

(6)

BlackScholesDelta100,120,1,0.3,0.05,0.03,'put'

−0.632854644

(7)

BlackScholesDelta100,t→max120t,0,1,σ,r,d

ⅇr5ⅇrderf2ln65σ2+2d2r24σπσ+5ⅇrdπσ+6ⅇ2ln65+σ2+2d2r28σ225ⅇσ2ln6251296+4ln32+ln5ln16561+ln35ln256+dln1679616390625+rln3906251679616+4ln52+4ln22+σ4+4dσ24rσ2+4d28dr+4r28σ2210πσ

(8)

BlackScholesDelta100,t→max120t,0,1,0.3,0.05,0.03,d

−0.6328546388

(9)

In this example, you will compute the Delta of a strangle.

SBlackScholesDelta100&comma;t&rarr;piecewiset<50&comma;50t&comma;t<100&comma;0&comma;t100&comma;1&comma;&sigma;&comma;r&comma;d

S&ExponentialE;r2&ExponentialE;rderf2σ2+2ln22d+2r4σπσ2&ExponentialE;rderfσ2+2d2r24σπσ+22&ExponentialE;σ2ln16+4ln22+dln1256+rln256+σ4+4dσ24rσ2+4d28dr+4r28σ2&ExponentialE;σ2+2ln22d+2r28σ224πσ

(10)

CBlackScholesDelta100&comma;100&comma;1&comma;&sigma;&comma;r&comma;d&comma;&apos;call&apos;

C&ExponentialE;derfσ2+2d2r24σ12

(11)

PBlackScholesDelta100&comma;50&comma;1&comma;&sigma;&comma;r&comma;d&comma;&apos;put&apos;

P2&ExponentialE;derf2σ2+2ln22d+2r4σπσ+22&ExponentialE;σ2ln16+4ln22+dln1256+rln256+σ4+4dσ2+4rσ2+4d28dr+4r28σ2&ExponentialE;σ2ln116+4ln22+dln1256+rln256+σ4+4dσ2+4rσ2+4d28dr+4r28σ222&ExponentialE;dπσ4πσ

(12)

Check:

simplifySCP

0

(13)

References

  

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

Compatibility

• 

The Finance[BlackScholesDelta] 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[BlackScholesGamma]

Finance[BlackScholesPrice]

Finance[BlackScholesPrice]

Finance[BlackScholesRho]

Finance[BlackScholesTheta]

Finance[BlackScholesVega]

Finance[EuropeanOption]

Finance[ImpliedVolatility]

Finance[LatticePrice]