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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_bsm_price (s30aa)

## Purpose

nag_specfun_opt_bsm_price (s30aa) computes the European option price given by the Black–Scholes–Merton formula.

## Syntax

[p, ifail] = s30aa(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_bsm_price(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)

## Description

nag_specfun_opt_bsm_price (s30aa) 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 ${T}_{j}$, $j=1,2,\dots ,n$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{calput}$ – string (length ≥ 1)
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:     $\mathrm{x}\left({\mathbf{m}}\right)$ – double array
${\mathbf{x}}\left(i\right)$ 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 $z={\mathbf{x02am}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     $\mathrm{s}$ – double scalar
$S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{x02am}}\left(\right)$, the safe range parameter.
4:     $\mathrm{t}\left({\mathbf{n}}\right)$ – double array
${\mathbf{t}}\left(i\right)$ 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 $z={\mathbf{x02am}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     $\mathrm{sigma}$ – double scalar
$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: ${\mathbf{sigma}}>0.0$.
6:     $\mathrm{r}$ – double scalar
$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
7:     $\mathrm{q}$ – double scalar
$q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: ${\mathbf{q}}\ge 0.0$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the array x.
The number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array t.
The number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.

### Output Parameters

1:     $\mathrm{p}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{p}}\left(i,j\right)$ 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 $j=1,2,\dots ,{\mathbf{n}}$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=_$ was an illegal value.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
Constraint: ${\mathbf{x}}\left(i\right)\ge _$ and ${\mathbf{x}}\left(i\right)\le _$.
${\mathbf{ifail}}=5$
Constraint: ${\mathbf{s}}\ge _$ and ${\mathbf{s}}\le _$.
${\mathbf{ifail}}=6$
Constraint: ${\mathbf{t}}\left(i\right)\ge _$.
${\mathbf{ifail}}=7$
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=8$
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=9$
Constraint: ${\mathbf{q}}\ge 0.0$.
${\mathbf{ifail}}=11$
Constraint: $\mathit{ldp}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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 nag_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

None.

## Example

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.
```function s30aa_example

fprintf('s30aa example results\n\n');

put = 'c';
s = 55;
sigma = 0.3;
r = 0.1;
q = 0;
x = [58, 60, 62];
t = [0.7, 0.8];

[p, ifail] = s30aa( ...
put, x, s, t, sigma, r, q);

fprintf('\nBlack-Scholes-Merton formula\n European Call :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Dividend   =   %9.4f\n\n', q);

fprintf('   Strike    Expiry   Option Price\n');
for i=1:3
for j=1:2
fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
end
end

```
```s30aa example results

Black-Scholes-Merton formula
European Call :
Spot       =     55.0000
Volatility =      0.3000
Rate       =      0.1000
Dividend   =      0.0000

Strike    Expiry   Option Price
58.0000    0.7000    5.9198
58.0000    0.8000    6.5506
60.0000    0.7000    5.0809
60.0000    0.8000    5.6992
62.0000    0.7000    4.3389
62.0000    0.8000    4.9379
```