The routine may be called by the names s30aaf or nagf_specfun_opt_bsm_price.
3Description
s30aaf computes the price of a European call (or put) option for constant volatility, , and risk-free interest rate, , with possible dividend yield, , using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). For a given strike price, , the price of a European call with underlying price, , and time to expiry, , is
and the corresponding European put price is
and where denotes the cumulative Normal distribution function,
and
The option price is computed for each strike price in a set , , and for each expiry time in a set , .
4References
Black F and Scholes M (1973) The pricing of options and corporate liabilities Journal of Political Economy81 637–654
Merton R C (1973) Theory of rational option pricing Bell Journal of Economics and Management Science4 141–183
5Arguments
1: – Character(1)Input
On entry: determines whether the option is a call or a put.
A call; the holder has a right to buy.
A put; the holder has a right to sell.
Constraint:
or .
2: – IntegerInput
On entry: the number of strike prices to be used.
Constraint:
.
3: – IntegerInput
On entry: the number of times to expiry to be used.
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput
On entry: must contain
, the th strike price, for .
Constraint:
, where , the safe range parameter, for .
5: – Real (Kind=nag_wp)Input
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
6: – Real (Kind=nag_wp) arrayInput
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
7: – Real (Kind=nag_wp)Input
On entry: , the volatility of the underlying asset. Note that a rate of 15% should be entered as .
Constraint:
.
8: – Real (Kind=nag_wp)Input
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as .
Constraint:
.
9: – Real (Kind=nag_wp)Input
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as .
Constraint:
.
10: – Real (Kind=nag_wp) arrayOutput
On exit: contains , the option price evaluated for the strike price at expiry for and .
11: – IntegerInput
On entry: the first dimension of the array p as declared in the (sub)program from which s30aaf is called.
Constraint:
.
12: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, was an illegal value.
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: and .
On entry, .
Constraint: and .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: .
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, . This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the machine precision (see s15abfands15adf). An accuracy close to machine precision can generally be expected.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s30aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
None.
10Example
This example computes the prices for six European call options using two expiry times and three strike prices as input. The times to expiry are taken as and years respectively. The stock price is , with strike prices, , and . The risk-free interest rate is per year and the volatility is per year.