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=∂∂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.
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');
−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, sigma, r, d);
BlackScholesRho(100, t -> max(t-100, 0), 1, 0.3, 0.05, 0.03);
44.40274728
Rho := BlackScholesRho(100, K, 1, sigma, 0.05, 0.03, 'call');
Ρ≔38.71517541⁢ⅇ−0.4999999997⁢4.625170186+ln⁡1K+0.5⁢σ22σ2+0.4756147122⁢K⁢σ+0.4756147122⁢K⁢σ⁢erf⁡3.270489202+0.707106781⁢ln⁡1K−0.3535533905⁢σ2σ−0.3794856357⁢K⁢ⅇ−1.⁢3.270489202+0.707106781⁢ln⁡1K−0.3535533905⁢σ22σ2σ
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');
25⁢−2⁢ⅇ−r⁢erf⁡σ2+2⁢ln⁡2+2⁢d−2⁢r⁢24⁢σ⁢π⁢σ−2⁢ⅇ−r⁢π⁢σ+ⅇ−σ4−4⁢ln⁡2⁢σ2+4⁢d⁢σ2+4⁢r⁢σ2+4⁢ln⁡22+8⁢ln⁡2⁢d−8⁢ln⁡2⁢r+4⁢d2−8⁢d⁢r+4⁢r28⁢σ2⁢2−2⁢2⁢ⅇ−σ4+4⁢ln⁡2⁢σ2+4⁢d⁢σ2+4⁢r⁢σ2+4⁢ln⁡22+8⁢ln⁡2⁢d−8⁢ln⁡2⁢r+4⁢d2−8⁢d⁢r+4⁢r28⁢σ2π⁢σ
BlackScholesRho(50, 100, 1, 0.3, 0.05, 0.03, 'put');
−94.32991431
BlackScholesRho(50, t -> max(100-t, 0), 1, sigma, r, d);
−25⁢ⅇ−r⁢2⁢erf⁡σ2+2⁢ln⁡2+2⁢d−2⁢r⁢24⁢σ⁢π⁢σ+2⁢ⅇ−σ2+2⁢ln⁡2+2⁢d−2⁢r28⁢σ2⁢2−ⅇ−σ4−4⁢ln⁡2⁢σ2+4⁢d⁢σ2−4⁢r⁢σ2+4⁢ln⁡22+8⁢ln⁡2⁢d−8⁢ln⁡2⁢r+4⁢d2−8⁢d⁢r+4⁢r28⁢σ2⁢2+2⁢π⁢σπ⁢σ
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, sigma, r, d);
S≔25⁢ⅇ−r⁢2⁢2⁢ⅇ−σ4+4⁢ln⁡2⁢σ2+4⁢d⁢σ2−4⁢r⁢σ2+4⁢ln⁡22−8⁢ln⁡2⁢d+8⁢ln⁡2⁢r+4⁢d2−8⁢d⁢r+4⁢r28⁢σ2+erf⁡2⁢−σ2+2⁢ln⁡2−2⁢d+2⁢r4⁢σ⁢π⁢σ−2⁢π⁢erf⁡σ2+2⁢d−2⁢r⁢24⁢σ⁢σ−ⅇ−−σ2+2⁢ln⁡2−2⁢d+2⁢r28⁢σ2⁢2+π⁢σπ⁢σ
C := BlackScholesRho(100, 100, 1, sigma, r, d, 'call');
C≔−50⁢ⅇ−r⁢erf⁡σ2+2⁢d−2⁢r⁢24⁢σ−1
P := BlackScholesRho(100, 50, 1, sigma, r, d, 'put');
P≔−25⁢−ⅇ−r⁢erf⁡2⁢−σ2+2⁢ln⁡2−2⁢d+2⁢r4⁢σ⁢π⁢σ+ⅇ−r⁢π⁢σ+2⁢ⅇ−σ4−4⁢ln⁡2⁢σ2+4⁢d⁢σ2+4⁢r⁢σ2+4⁢ln⁡22−8⁢ln⁡2⁢d+8⁢ln⁡2⁢r+4⁢d2−8⁢d⁢r+4⁢r28⁢σ2−2⁢2⁢ⅇ−σ4+4⁢ln⁡2⁢σ2+4⁢d⁢σ2+4⁢r⁢σ2+4⁢ln⁡22−8⁢ln⁡2⁢d+8⁢ln⁡2⁢r+4⁢d2−8⁢d⁢r+4⁢r28⁢σ2π⁢σ
Check:
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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