# NAG FL Interfaces30abf (opt_​bsm_​greeks)

## 1Purpose

s30abf computes the European option price given by the Black–Scholes–Merton formula together with its sensitivities (Greeks).

## 2Specification

Fortran Interface
 Subroutine s30abf ( m, n, x, s, t, r, q, p, ldp, vega, rho, crho,
 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), delta(ldp,n), gamma(ldp,n), vega(ldp,n), theta(ldp,n), rho(ldp,n), crho(ldp,n), vanna(ldp,n), charm(ldp,n), speed(ldp,n), colour(ldp,n), zomma(ldp,n), vomma(ldp,n) Character (1), Intent (In) :: calput
#include <nag.h>
 void s30abf_ (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, double delta[], double gamma[], double vega[], double theta[], double rho[], double crho[], double vanna[], double charm[], double speed[], double colour[], double zomma[], double vomma[], Integer *ifail, const Charlen length_calput)
The routine may be called by the names s30abf or nagf_specfun_opt_bsm_greeks.

## 3Description

s30abf computes the price of a European call (or put) option together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters, by the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). The annual volatility, $\sigma$, risk-free interest rate, $r$, and dividend yield, $q$, must be supplied as input. 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: $\mathbf{m}$Integer Input
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
3: $\mathbf{n}$Integer Input
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) array Input
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) array Input
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) array Output
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: $\mathbf{ldp}$Integer Input
On entry: the first dimension of the arrays p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma and vomma as declared in the (sub)program from which s30abf is called.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
12: $\mathbf{delta}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the leading ${\mathbf{m}}×{\mathbf{n}}$ part of the array delta contains the sensitivity, $\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.
13: $\mathbf{gamma}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the leading ${\mathbf{m}}×{\mathbf{n}}$ part of the array gamma contains the sensitivity, $\frac{{\partial }^{2}P}{\partial {S}^{2}}$, of delta to change in the price of the underlying asset.
14: $\mathbf{vega}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{vega}}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price ${P}_{ij}$ to change in the volatility of the underlying asset, i.e., $\frac{\partial {P}_{ij}}{\partial \sigma }$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
15: $\mathbf{theta}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{theta}}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price ${P}_{ij}$ to change in time, i.e., $-\frac{\partial {P}_{ij}}{\partial T}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$, where $b=r-q$.
16: $\mathbf{rho}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{rho}}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price ${P}_{ij}$ to change in the annual risk-free interest rate, i.e., $-\frac{\partial {P}_{ij}}{\partial r}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
17: $\mathbf{crho}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{crho}}\left(i,j\right)$, contains the first-order Greek measuring the sensitivity of the option price ${P}_{ij}$ to change in the annual cost of carry rate, i.e., $-\frac{\partial {P}_{ij}}{\partial b}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$, where $b=r-q$.
18: $\mathbf{vanna}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{vanna}}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the asset price, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^{2}{P}_{ij}}{\partial S\partial \sigma }$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
19: $\mathbf{charm}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{charm}}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^{2}{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
20: $\mathbf{speed}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{speed}}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the price of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial S}=-\frac{{\partial }^{3}{P}_{ij}}{\partial {S}^{3}}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
21: $\mathbf{colour}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{colour}}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial T}=-\frac{{\partial }^{3}{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
22: $\mathbf{zomma}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{zomma}}\left(i,j\right)$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial \sigma }=-\frac{{\partial }^{3}{P}_{ij}}{\partial {S}^{2}\partial \sigma }$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
23: $\mathbf{vomma}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{vomma}}\left(i,j\right)$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial \sigma }=-\frac{{\partial }^{2}{P}_{ij}}{\partial {\sigma }^{2}}$, for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
24: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value 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 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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
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, $\Phi$. 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

s30abf 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 price of a European put with a time to expiry of $0.7$ years, a stock price of $55$ and a strike price of $60$. The risk-free interest rate is $10%$ per year and the volatility is $30%$ per year.

### 10.1Program Text

Program Text (s30abfe.f90)

### 10.2Program Data

Program Data (s30abfe.d)

### 10.3Program Results

Program Results (s30abfe.r)