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BlackScholesUltima

  

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

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

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

BlackScholesUltima(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 Ultima of an option or a portfolio of options measures Vomma's sensitivity to volatility.

Ultima=σVomma

Ultima=3σ3S

• 

The BlackScholesUltima command computes the Ultima 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:

The Vega of an option measures the sensitivity of the option to volatility, sigma. The Vomma of an option measures Vega's sensitivity to volatility. The Ultima of an option measures Vomma's sensitivity to volatility. The following example illustrates the characteristics of the Ultima of an option with respect to volatility as well as the time to maturity.

In this example, the Ultima is defined as a function of volatility, sigma, and time to maturity, T.  For a European call option, we will assume that the strike price is 100 and the risk-free interest rate of 0.05.  We also assume that this option does not pay any dividends.

UltimaBlackScholesUltima100,100,T,σ,0.05,0,'call':

plot3dUltima,T=1.0..0,σ=0..0.5,'labels'=Time to Maturity,Volatility,Value,'colorscheme'=zgradient,Black,White,Red,'thickness'=0

We can also see how the Ultima behaves as a function of the risk-free interest rate, the dividend yield, and volatility.  To compute the Ultima of a European call option with strike price 100 maturing in 1 year, we take:

BlackScholesUltima100,100,1,σ,r,d,'call'

252ⅇσ4+4dσ2+4rσ2+4d28dr+4r28σ2σ88d2σ4+16drσ48r2σ44σ6+16d464d3r+96d2r248d2σ264dr3+96drσ2+16r448r2σ28σ6π

(1)

This can be numerically solved for specific values of the risk-free rate, the dividend yield, and the volatility.

BlackScholesUltima100,100,1,0.3,0.05,0.03,'call'

−14.919796

(2)

It is also possible to use the generic method in which the option is defined through its payoff function:

BlackScholesUltima100,t→maxt100,0,1,σ,r,d

252ⅇσ4+4dσ2+4rσ2+4d28dr+4r28σ2σ88d2σ4+16drσ48r2σ44σ6+16d464d3r+96d2r248d2σ264dr3+96drσ2+16r448r2σ28σ6π

(3)

BlackScholesUltima100,t→maxt100,0,1,0.3,0.05,0.03

−14.9197957

(4)

UltimaBlackScholesUltima100,100,1,σ,r,0.03,'call'

Ultima9.973557002ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r2σ619.947114ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r3σ4+2.493389246ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ8r+19.94711399ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r4σ20.7561551978ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ2r+25.24107807ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ2r2+242.9019915ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ4r281.6532498ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ2r358.04610176ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ4r2+30.51908445ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ6r+0.3590480519ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r30.01077144155ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r2+39.89422799ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r57.234279682ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ45.061580167ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ8+0.007556166257ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ20.9065963ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ6+1.246694625ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2σ10+0.0001615716233ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r5.984134198ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ2r49.69429739810−7ⅇ1.0.5000000002rσ2+0.1249999999σ4+0.01499999999σ2+0.00044999999980.02999999998r+0.4999999997r2σ228.4556539ⅇ0.0000499999999750.σ2+100.r3.2σ2σ6r+0.7296261952ⅇ0.0000499999999750.σ2+100.r3.2σ2σ2r228.7544211ⅇ0.0000499999999750.σ2+100.r3.2σ2σ4r24.28602953ⅇ0.0000499999999750.σ2+100.r3.2σ2σ2r259.81494597ⅇ0.0000499999999750.σ2+100.r3.2σ2σ4r2+268.6833172ⅇ0.0000499999999750.σ2+100.r3.2σ2σ2r39.678793837ⅇ0.0000499999999750.σ2+100.r3.2σ2r2σ6+19.35758768ⅇ0.0000499999999750.σ2+100.r3.2σ2r3σ42.41969846ⅇ0.0000499999999750.σ2+100.r3.2σ2σ8r+19.35758768ⅇ0.0000499999999750.σ2+100.r3.2σ2r4σ2+5.807276302ⅇ0.0000499999999750.σ2+100.r3.2σ2r40.3484365781ⅇ0.0000499999999750.σ2+100.r3.2σ2r3+0.01045309735ⅇ0.0000499999999750.σ2+100.r3.2σ2r238.71517536ⅇ0.0000499999999750.σ2+100.r3.2σ2r5+1.20984923ⅇ0.0000499999999750.σ2+100.r3.2σ2σ100.0001567964602ⅇ0.0000499999999750.σ2+100.r3.2σ2r4.766805971ⅇ0.0000499999999750.σ2+100.r3.2σ2σ8+0.8623805319ⅇ0.0000499999999750.σ2+100.r3.2σ2σ6+6.915943432ⅇ0.0000499999999750.σ2+100.r3.2σ2σ40.007301488501ⅇ0.0000499999999750.σ2+100.r3.2σ2σ2+9.40778760810−7ⅇ0.0000499999999750.σ2+100.r3.2σ2σ8

(5)

plot3dUltima,σ=0..1,r=0..1

Here are similar examples for the European put option:

BlackScholesUltima100,120,1,0.3,0.05,0.03,'put'

−329.853365

(6)

BlackScholesUltima100,t→max120t,0,1,0.3,0.05,0.03,0

−329.853364

(7)

References

  

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

Compatibility

• 

The Finance[BlackScholesUltima] command was introduced in Maple 2015.

• 

For more information on Maple 2015 changes, see Updates in Maple 2015.

See Also

Finance[BlackScholesColor]

Finance[BlackScholesGamma]

Finance[BlackScholesPrice]

Finance[BlackScholesSpeed]

Finance[BlackScholesZomma]

Finance[EuropeanOption]

Finance[ImpliedVolatility]

Finance[LatticePrice]