compute the Rho of a European-style option with given payoff - Maple Help

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Finance[BlackScholesRho] - compute the Rho of a European-style option with given payoff

Calling Sequence

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

BlackScholesRho(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 Rho of an option or a portfolio of options is the sensitivity of the option or portfolio to changes in the risk-free rate

Ρ=rS

• 

The BlackScholesRho command computes the Rho 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 you compute the Rho of a European call option with strike price 100, which matures in 1 year. This will define the Rho as a function of the risk-free rate, the dividend yield, and the volatility.

BlackScholesRho100,100,1,σ,r,d,'call'

50ⅇrerf14σ2+2d2r2σ1

(1)

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

BlackScholesRho100,100,1,0.3,0.05,0.03,'call'

44.4027473

(2)

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

BlackScholesRho100,t→maxt100,0,1,σ,r,d

50erf14σ2+2d2r2σ1ⅇr

(3)

BlackScholesRho100,t→maxt100,0,1,0.3,0.05,0.03

44.40274728

(4)

Ρ:=BlackScholesRho100,K,1,σ,0.05,0.03,'call'

Ρ:=38.71517541ⅇ0.49999999974.625170186+ln1K+0.5σ22σ2+0.4756147122Kσ+0.4756147122Kσerf3.270489202+0.707106781ln1K0.3535533905σ2σ0.3794856357Kⅇ1.3.270489202+0.707106781ln1K0.3535533905σ22σ2σ

(5)

plot3dΡ,σ=0..1,K=70..120,axes=BOXED

Here are similar examples for the European put option.

BlackScholesRho50,100,1,σ,r,d,'put'

252ⅇrerf14σ2+2ln2+2d2r2σπσ2ⅇrπσ+ⅇ18σ44ln2σ2+4dσ2+4rσ2+4ln22+8ln2d8ln2r+4d28dr+4r2σ2222ⅇ18σ4+4ln2σ2+4dσ2+4rσ2+4ln22+8ln2d8ln2r+4d28dr+4r2σ2πσ

(6)

BlackScholesRho50,100,1,0.3,0.05,0.03,'put'

94.32991431

(7)

BlackScholesRho50,t→max100t,0,1,σ,r,d

25ⅇrd2erf14σ2+2ln2+2d2r2σⅇdπσ2ⅇdπσ+ⅇ18σ44ln2σ24dσ24rσ2+4ln22+8ln2d8ln2r+4d28dr+4r2σ2222ⅇ18σ4+4ln2σ24dσ24rσ2+4ln22+8ln2d8ln2r+4d28dr+4r2σ2πσ

(8)

BlackScholesRho50,t→max100t,0,1,0.3,0.05,0.03,d

94.32991433

(9)

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

S:=BlackScholesRho100&comma;t&rarr;piecewiset<50&comma;50t&comma;t<100&comma;0&comma;t100&comma;1&comma;&sigma;&comma;r&comma;d

S:=25&ExponentialE;rderf142&sigma;2&plus;2ln22d&plus;2r&sigma;&ExponentialE;d&pi;&sigma;2&ExponentialE;derf14&sigma;2&plus;2d2r2&sigma;&pi;&sigma;&ExponentialE;18&sigma;44ln2&sigma;24d&sigma;24r&sigma;2&plus;4ln228ln2d&plus;8ln2r&plus;4d28dr&plus;4r2&sigma;22&plus;&ExponentialE;d&pi;&sigma;&plus;22&ExponentialE;18&sigma;4&plus;4ln2&sigma;24d&sigma;24r&sigma;2&plus;4ln228ln2d&plus;8ln2r&plus;4d28dr&plus;4r2&sigma;2&pi;&sigma;

(10)

C:=BlackScholesRho100&comma;100&comma;1&comma;&sigma;&comma;r&comma;d&comma;&apos;call&apos;

C:=50erf14&sigma;2&plus;2d2r2&sigma;1&ExponentialE;r

(11)

P:=BlackScholesRho100&comma;50&comma;1&comma;&sigma;&comma;r&comma;d&comma;&apos;put&apos;

P:=25&ExponentialE;rerf142&sigma;2&plus;2ln22d&plus;2r&sigma;&pi;&sigma;&plus;22&ExponentialE;18&sigma;4&plus;4ln2&sigma;2&plus;4d&sigma;2&plus;4r&sigma;2&plus;4ln228ln2d&plus;8ln2r&plus;4d28dr&plus;4r2&sigma;2&ExponentialE;18&sigma;44ln2&sigma;2&plus;4d&sigma;2&plus;4r&sigma;2&plus;4ln228ln2d&plus;8ln2r&plus;4d28dr&plus;4r2&sigma;22&ExponentialE;r&pi;&sigma;&pi;&sigma;

(12)

Check:

simplifySCP

0

(13)

See Also

Finance[AmericanOption], Finance[BermudanOption], Finance[BlackScholesDelta], Finance[BlackScholesGamma], Finance[BlackScholesPrice], Finance[BlackScholesTheta], Finance[BlackScholesVega], Finance[EuropeanOption], Finance[ImpliedVolatility], Finance[LatticePrice]

References

  

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


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