Finance
BlackScholesRho
compute the Rho of a European-style option with given payoff
Calling Sequence
Parameters
Description
Examples
References
Compatibility
BlackScholesRho(S0, K, T, sigma, r, d, optiontype)
BlackScholesRho(S0, P, T, sigma, r, d)
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
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=ⅆSⅆr
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.
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,σ,r,d,call
−50⁢ⅇ−r⁢erf⁡σ2+2⁢d−2⁢r⁢24⁢σ−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
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,σ,r,d
BlackScholesRho⁡100,t↦max⁡t−100,0,1,0.3,0.05,0.03
Ρ≔BlackScholesRho⁡100,K,1,σ,0.05,0.03,call
Ρ≔3.832995302⁢1K−1.⁢0.4999999997⁢σ2+4.625170183σ2⁢ⅇ−10.69609962−0.4999999997⁢ln⁡1K2−0.1249999999⁢σ4σ2+0.4756147122⁢K⁢σ+0.4756147122⁢K⁢σ⁢erf⁡3.270489202+0.707106781⁢ln⁡1K−0.3535533905⁢σ2σ−3.832995293⁢K⁢1K0.4999999998⁢σ2−4.625170184σ2⁢ⅇ−10.69609962−0.4999999997⁢ln⁡1K2−0.1249999999⁢σ4σ2σ
plot3d⁡Ρ,σ=0..1,K=70..120,axes=BOXED
Here are similar examples for the European put option.
BlackScholesRho⁡50,100,1,σ,r,d,put
−50⁢ⅇ−r⁢1+erf⁡σ2+2⁢ln⁡2+2⁢d−2⁢r⁢24⁢σ
BlackScholesRho⁡50,100,1,0.3,0.05,0.03,put
−94.32991431
BlackScholesRho⁡50,t↦max⁡100−t,0,1,σ,r,d
BlackScholesRho⁡50,t↦max⁡100−t,0,1,0.3,0.05,0.03,d
−94.32991433
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,σ,r,d
S≔−25⁢ⅇ−r⁢2⁢erf⁡σ2+2⁢d−2⁢r⁢24⁢σ−erf⁡2⁢−σ2+2⁢ln⁡2−2⁢d+2⁢r4⁢σ−1
C≔BlackScholesRho⁡100,100,1,σ,r,d,call
C≔−50⁢ⅇ−r⁢erf⁡σ2+2⁢d−2⁢r⁢24⁢σ−1
P≔BlackScholesRho⁡100,50,1,σ,r,d,put
P≔25⁢ⅇ−r⁢−1+erf⁡2⁢−σ2+2⁢ln⁡2−2⁢d+2⁢r4⁢σ
Check:
expand⁡simplify⁡S−C−P
0
Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.
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]
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