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Finance

  

BlackScholesRho

  

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

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

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

with(Finance):

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.

BlackScholesRho(100, 100, 1, sigma, r, d, 'call');

50ⅇrerfσ2+2d2r24σ1

(1)

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

BlackScholesRho(100, 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.

BlackScholesRho(100, t -> max(t-100, 0), 1, sigma, r, d);

50ⅇrerfσ2+2d2r24σ1

(3)

BlackScholesRho(100, t -> max(t-100, 0), 1, 0.3, 0.05, 0.03);

44.40274728

(4)

Rho := BlackScholesRho(100, K, 1, sigma, 0.05, 0.03, 'call');

Ρ38.71517541ⅇ0.49999999974.625170186+ln1K+0.5σ22σ2+0.4756147122Kσ+0.4756147122Kσerf3.270489202+0.707106781ln1K0.3535533905σ2σ0.3794856357Kⅇ1.3.270489202+0.707106781ln1K0.3535533905σ22σ2σ

(5)

plot3d(Rho, sigma = 0..1, K = 70..120, axes = BOXED);

Here are similar examples for the European put option.

BlackScholesRho(50, 100, 1, sigma, r, d, 'put');

252ⅇrerfσ2+2ln2+2d2r24σπσ2ⅇrπσ+ⅇσ44ln2σ2+4dσ2+4rσ2+4ln22+8ln2d8ln2r+4d28dr+4r28σ2222ⅇσ4+4ln2σ2+4dσ2+4rσ2+4ln22+8ln2d8ln2r+4d28dr+4r28σ2πσ

(6)

BlackScholesRho(50, 100, 1, 0.3, 0.05, 0.03, 'put');

−94.32991431

(7)

BlackScholesRho(50, t -> max(100-t, 0), 1, sigma, r, d);

25ⅇr2erfσ2+2ln2+2d2r24σπσ+2ⅇσ2+2ln2+2d2r28σ22ⅇσ44ln2σ2+4dσ24rσ2+4ln22+8ln2d8ln2r+4d28dr+4r28σ22+2πσπσ

(8)

BlackScholesRho(50, t -> max(100-t, 0), 1, 0.3, 0.05, 0.03, d);

−94.32991433

(9)

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

S := BlackScholesRho(100, t -> piecewise(t < 50, 50-t, t < 100, 0, t-100), 1, sigma, r, d);

S25&ExponentialE;r22&ExponentialE;σ4+4ln2σ2+4dσ24rσ2+4ln228ln2d+8ln2r+4d28dr+4r28σ2+erf2σ2+2ln22d+2r4σπσ2πerfσ2+2d2r24σσ&ExponentialE;σ2+2ln22d+2r28σ22+πσπσ

(10)

C := BlackScholesRho(100, 100, 1, sigma, r, d, 'call');

C50&ExponentialE;rerfσ2+2d2r24σ1

(11)

P := BlackScholesRho(100, 50, 1, sigma, r, d, 'put');

P25&ExponentialE;rerf2σ2+2ln22d+2r4σπσ+&ExponentialE;rπσ+2&ExponentialE;σ44ln2σ2+4dσ2+4rσ2+4ln228ln2d+8ln2r+4d28dr+4r28σ222&ExponentialE;σ4+4ln2σ2+4dσ2+4rσ2+4ln228ln2d+8ln2r+4d28dr+4r28σ2πσ

(12)

Check:

simplify(S-C-P);

0

(13)

References

  

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

Compatibility

• 

The Finance[BlackScholesRho] 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[BlackScholesTheta]

Finance[BlackScholesVega]

Finance[EuropeanOption]

Finance[ImpliedVolatility]

Finance[LatticePrice]