NAG Library Routine Document

1Purpose

s30aaf computes the European option price given by the Black–Scholes–Merton formula.

2Specification

Fortran Interface
 Subroutine s30aaf ( m, n, x, s, t, r, q, p, ldp,
 Integer, Intent (In) :: m, n, ldp Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(m), s, t(n), sigma, r, q Real (Kind=nag_wp), Intent (Inout) :: p(ldp,n) Character (1), Intent (In) :: calput
#include nagmk26.h
 void s30aaf_ (const char *calput, const Integer *m, const Integer *n, const double x[], const double *s, const double t[], const double *sigma, const double *r, const double *q, double p[], const Integer *ldp, Integer *ifail, const Charlen length_calput)

3Description

s30aaf computes the price of a European call (or put) option for constant volatility, $\sigma$, and risk-free interest rate, $r$, with a possible dividend yield, $q$, using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). For a given strike price, $X$, the price of a European call with underlying price, $S$, and time to expiry, $T$, is
 $Pcall = Se-qT Φd1 - Xe-rT Φd2$
and the corresponding European put price is
 $Pput = Xe-rT Φ-d2 - Se-qT Φ-d1$
and where $\Phi$ denotes the cumulative Normal distribution function,
 $Φx = 12π ∫ -∞ x exp -y2/2 dy$
and
 $d1 = ln S/X + r-q+ σ2 / 2 T σ⁢T , d2 = d1 - σ⁢T .$
The option price ${P}_{ij}=P\left(X={X}_{i},T={T}_{j}\right)$ is computed for each strike price in a set ${X}_{i}$, $i=1,2,\dots ,m$, and for each expiry time in a set ${T}_{j}$, $j=1,2,\dots ,n$.

4References

Black F and Scholes M (1973) The pricing of options and corporate liabilities Journal of Political Economy 81 637–654
Merton R C (1973) Theory of rational option pricing Bell Journal of Economics and Management Science 4 141–183

5Arguments

1:     $\mathbf{calput}$ – Character(1)Input
On entry: determines whether the option is a call or a put.
${\mathbf{calput}}=\text{'C'}$
A call; the holder has a right to buy.
${\mathbf{calput}}=\text{'P'}$
A put; the holder has a right to sell.
Constraint: ${\mathbf{calput}}=\text{'C'}$ or $\text{'P'}$.
2:     $\mathbf{m}$ – IntegerInput
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathbf{n}$ – IntegerInput
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{x}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(i\right)$ must contain ${X}_{\mathit{i}}$, the $\mathit{i}$th strike price, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{x}}\left(\mathit{i}\right)\le 1/z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
5:     $\mathbf{s}$ – Real (Kind=nag_wp)Input
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter.
6:     $\mathbf{t}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{t}}\left(i\right)$ must contain ${T}_{\mathit{i}}$, the $\mathit{i}$th time, in years, to expiry, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
7:     $\mathbf{sigma}$ – Real (Kind=nag_wp)Input
On entry: $\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: ${\mathbf{sigma}}>0.0$.
8:     $\mathbf{r}$ – Real (Kind=nag_wp)Input
On entry: $r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
9:     $\mathbf{q}$ – Real (Kind=nag_wp)Input
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: ${\mathbf{q}}\ge 0.0$.
10:   $\mathbf{p}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{p}}\left(i,j\right)$ contains ${P}_{ij}$, the option price evaluated for the strike price ${{\mathbf{x}}}_{i}$ at expiry ${{\mathbf{t}}}_{j}$ for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
11:   $\mathbf{ldp}$ – IntegerInput
On entry: the first dimension of the array p as declared in the (sub)program from which s30aaf is called.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
12:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=〈\mathit{\text{value}}〉$ was an illegal value.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{x}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\left(i\right)\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left(i\right)\le 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{s}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{s}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{s}}\le 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{t}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{t}}\left(i\right)\ge 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{sigma}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=8$
On entry, ${\mathbf{r}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{q}}\ge 0.0$.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{ldp}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, $\Phi$. 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 s15abf and s15adf). An accuracy close to machine precision can generally be expected.

8Parallelism and Performance

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.

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 $0.7$ and $0.8$ years respectively. The stock price is $55$, with strike prices, $58$, $60$ and $62$. The risk-free interest rate is $10%$ per year and the volatility is $30%$ per year.

10.1Program Text

Program Text (s30aafe.f90)

10.2Program Data

Program Data (s30aafe.d)

10.3Program Results

Program Results (s30aafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017