naginterfaces.library.pde.dim1_blackscholes_closed¶
- naginterfaces.library.pde.dim1_blackscholes_closed(kopt, x, s, t, tmat, tdpar, r, q, sigma)[source]¶
dim1_blackscholes_closed
computes an analytic solution to the Black–Scholes equation for a certain set of option types.For full information please refer to the NAG Library document for d03nd
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d03/d03ndf.html
- Parameters
- koptint
Specifies the kind of option to be valued:
A European call option.
An American call option.
A European put option.
- xfloat
The exercise price .
- sfloat
The stock price at which the option value and the Greeks should be evaluated.
- tfloat
The time at which the option value and the Greeks should be evaluated.
- tmatfloat
The maturity time of the option.
- tdparbool, array-like, shape
Specifies whether or not various arguments are time-dependent. More precisely, is time-dependent if and constant otherwise. Similarly, specifies whether is time-dependent and specifies whether is time-dependent.
- rfloat, array-like, shape
Note: the required length for this argument is determined as follows: if : ; otherwise: .
If then must contain the constant value of . The remaining elements need not be set.
If then must contain the value of at time and must contain its average instantaneous value over the remaining life of the option:
The auxiliary function
dim1_blackscholes_means()
may be used to construct from a set of values of at discrete times.- qfloat, array-like, shape
Note: the required length for this argument is determined as follows: if : ; otherwise: .
If then must contain the constant value of . The remaining elements need not be set.
If then must contain the constant value of and must contain its average instantaneous value over the remaining life of the option:
The auxiliary function
dim1_blackscholes_means()
may be used to construct from a set of values of at discrete times.- sigmafloat, array-like, shape
Note: the required length for this argument is determined as follows: if : ; otherwise: .
If then must contain the constant value of . The remaining elements need not be set.
If then must contain the value of at time , the average instantaneous value , and the second-order average , where:
The auxiliary function
dim1_blackscholes_means()
may be used to compute from a set of values at discrete times.
- Returns
- ffloat
The value of the option at the stock price and time .
- thetafloat
The values of various Greeks at the stock price and time , as follows:
- deltafloat
The values of various Greeks at the stock price and time , as follows:
- gammafloat
The values of various Greeks at the stock price and time , as follows:
- lamdafloat
The values of various Greeks at the stock price and time , as follows:
- rhofloat
The values of various Greeks at the stock price and time , as follows:
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, and .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, is not equal to with American call option. .
- (errno )
On entry, .
Constraint: , or .
- Notes
dim1_blackscholes_closed
computes an analytic solution to the Black–Scholes equation (see Hull (1989) and Wilmott et al. (1995))for the value of a European put or call option, or an American call option with zero dividend . In equation (1) is time, is the stock price, is the exercise price, is the risk free interest rate, is the continuous dividend, and is the stock volatility. The parameter , and may be either constant, or functions of time. In the latter case their average instantaneous values over the remaining life of the option should be provided to
dim1_blackscholes_closed
. An auxiliary functiondim1_blackscholes_means()
is available to compute such averages from values at a set of discrete times. Equation (1) is subject to different boundary conditions depending on the type of option. For a call option the boundary condition iswhere is the maturity time of the option. For a put option the equation (1) is subject to
dim1_blackscholes_closed
also returns values of the Greeksspecfun.opt_bsm_greeks
also computes the European option price given by the Black–Scholes–Merton formula together with a more comprehensive set of sensitivities (Greeks).Further details of the analytic solution returned are given in Algorithmic Details.
- References
Hull, J, 1989, Options, Futures and Other Derivative Securities, Prentice–Hall
Wilmott, P, Howison, S and Dewynne, J, 1995, The Mathematics of Financial Derivatives, Cambridge University Press