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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=diffS,σ

• 

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

r:=0.05

(1)

d0.03

d:=0.03

(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,t→maxt100,0,1,σ,r,d

38.32995296ⅇ0.1249999999σ2ⅇ0.0001999999998σ2

(5)

BlackScholesVega100,t→maxt100,0,1,0.3,0.05,0.03

37.81702620

(6)

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

Vega:=1.916497650ⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ21K0.499999999717.72825559ⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ2σ21K0.49999999973.832995301ln1Kⅇ10.69609962σ21K4.625170183σ2ⅇ0.4999999997ln1K2σ2ⅇ0.1249999999σ2σ21K0.4999999997+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,t→max120t,0,1,σ,r,d

41.98835974ⅇ0.01317414389σ2ⅇ0.1249999999σ26.10-9ⅇ0.01317414389σ2ⅇ0.1249999999σ2σ2

(10)

BlackScholesVega100,t→max120t,0,1,0.3,0.05,0.03,d

35.86504179

(11)

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

SexpandBlackScholesVega100&comma;t&rarr;piecewiset<90&comma;90t&comma;t<110&comma;0&comma;t110&comma;1&comma;&sigma;&comma;r&comma;d

S:=36.36298620&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.007857629439&sigma;2&plus;5.10-9&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.007857629439&sigma;2&sigma;2&plus;40.20079382&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.002835811589&sigma;21.10-9&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.002835811589&sigma;2&sigma;2

(12)

CexpandBlackScholesVega100&comma;110&comma;1&comma;&sigma;&comma;r&comma;d&comma;&apos;call&apos;

C:=20.10039692&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.002835811588&sigma;2&plus;3.027529011&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.002835811588&sigma;2&sigma;23.027529011&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.002835811589&sigma;2&sigma;2&plus;20.10039691&ExponentialE;0.1249999999&sigma;2&ExponentialE;0.002835811589&sigma;2

(13)

PexpandBlackScholesVega100&comma;90&comma;1&comma;&sigma;&comma;r&comma;d&comma;&apos;put&apos;

P:=4.558482700&ExponentialE;0.007857629432&sigma;2&ExponentialE;0.1249999999&sigma;2&sigma;2&plus;18.18149310&ExponentialE;0.007857629432&sigma;2&ExponentialE;0.1249999999&sigma;24.558482699&ExponentialE;0.007857629430&sigma;2&ExponentialE;0.1249999999&sigma;2&sigma;2&plus;18.18149310&ExponentialE;0.007857629430&sigma;2&ExponentialE;0.1249999999&sigma;2

(14)

Check that S is sufficiently close to C&plus;P.

plotS&comma;C&plus;P&comma;&sigma;&equals;0..1&comma;color&equals;red&comma;blue&comma;thickness&equals;3&comma;axes&equals;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]

 


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