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BlackScholesVega

  

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

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

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

BlackScholesVega(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 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.

Vega=σS

• 

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.

Examples

withFinance:

r0.05

r0.05

(1)

d0.03

d0.03

(2)

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.

expandBlackScholesVega100,100,1,σ,r,d,call

38.32995297ⅇ0.1249999999σ2ⅇ0.0001999999998σ21.10−10ⅇ0.1249999999σ2ⅇ0.0001999999998σ2σ2

(3)

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

BlackScholesVega100,100,1,0.3,0.05,0.03,call

37.81702623

(4)

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

expandBlackScholesVega100,tmaxt100,0,1,σ,r,d

38.32995296ⅇ0.1249999999σ2ⅇ0.0001999999998σ2

(5)

BlackScholesVega100,tmaxt100,0,1,0.3,0.05,0.03

37.81702620

(6)

VegaexpandBlackScholesVega100,K,1,σ,0.05,0.03,call

Vega1.916497650ⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ21K0.499999999717.72825559ⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ2σ21K0.49999999973.832995301ln1Kⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ2σ21K0.4999999997+17.72825555Kⅇ10.69609962σ21K4.625170184σ2ⅇ0.4999999997ln1K2σ21K0.4999999998ⅇ0.1249999999σ2σ2+3.832995293Kln1Kⅇ10.69609962σ21K4.625170184σ2ⅇ0.4999999997ln1K2σ21K0.4999999998ⅇ0.1249999999σ2σ2+1.916497646Kⅇ10.69609962σ21K4.625170184σ2ⅇ0.4999999997ln1K2σ21K0.4999999998ⅇ0.1249999999σ2

(7)

plot3dVega,σ=0..1,K=70..120,axes=BOXED

Here are similar examples for the European put option.

expandBlackScholesVega100,120,1,σ,r,d,put

41.98835974ⅇ0.01317414389σ2ⅇ0.1249999999σ2

(8)

BlackScholesVega100,120,1,0.3,0.05,0.03,put

35.86504172

(9)

expandBlackScholesVega100,tmax120t,0,1,σ,r,d

41.98835973ⅇ0.01317414389σ2ⅇ0.1249999999σ2

(10)

BlackScholesVega100,tmax120t,0,1,0.3,0.05,0.03,d

35.86504186

(11)

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

SexpandBlackScholesVega100&comma;tpiecewiset<90&comma;90t&comma;t<110&comma;0&comma;t110&comma;1&comma;σ&comma;r&comma;d

S36.36298620&ExponentialE;0.1249999999σ2&ExponentialE;0.007857629439σ2+2.10−9&ExponentialE;0.1249999999σ2&ExponentialE;0.007857629439σ2σ2+40.20079383&ExponentialE;0.1249999999σ2&ExponentialE;0.002835811589σ2

(12)

CexpandBlackScholesVega100&comma;110&comma;1&comma;σ&comma;r&comma;d&comma;call

C20.10039692&ExponentialE;0.1249999999σ2&ExponentialE;0.002835811588σ2+3.027529011&ExponentialE;0.1249999999σ2&ExponentialE;0.002835811588σ2σ23.027529011&ExponentialE;0.1249999999σ2&ExponentialE;0.002835811589σ2σ2+20.10039691&ExponentialE;0.1249999999σ2&ExponentialE;0.002835811589σ2

(13)

PexpandBlackScholesVega100&comma;90&comma;1&comma;σ&comma;r&comma;d&comma;put

P4.558482700&ExponentialE;0.007857629432σ2&ExponentialE;0.1249999999σ2σ2+18.18149310&ExponentialE;0.007857629432σ2&ExponentialE;0.1249999999σ24.558482699&ExponentialE;0.007857629430σ2&ExponentialE;0.1249999999σ2σ2+18.18149310&ExponentialE;0.007857629430σ2&ExponentialE;0.1249999999σ2

(14)

Check that S is sufficiently close to C+P.

plotS&comma;C+P&comma;σ=0..1&comma;color=red&comma;blue&comma;thickness=3&comma;axes=BOXED&comma;gridlines

References

  

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

Compatibility

• 

The Finance[BlackScholesVega] 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[BlackScholesTheta]

Finance[EuropeanOption]

Finance[ImpliedVolatility]

Finance[LatticePrice]