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# 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, σ$\sigma$, and risk-free interest rate, r$r$, with a possible dividend yield, q$q$, using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). For a given strike price, X$X$, the price of a European call with underlying price, S$S$, and time to expiry, T$T$, is
 Pcall = Se − qT Φ(d1) − Xe − rT Φ(d2) $Pcall = Se-qT Φ(d1) - Xe-rT Φ(d2)$
and the corresponding European put price is
 Pput = Xe − rT Φ( − d2) − Se − qT Φ( − d1) $Pput = Xe-rT Φ(-d2) - Se-qT Φ(-d1)$
and where Φ$\Phi$ denotes the cumulative Normal distribution function,
 x Φ(x) = 1/(sqrt(2π)) ∫ exp( − y2 / 2)dy − ∞
$Φ(x) = 12π ∫ -∞ x exp( -y2/2 ) dy$
and
 d1 = ( ln (S / X) + (r − q + σ2 / 2) T )/(σ×sqrt(T)) , d2 = d1 − σ×sqrt(T) .
$d1 = ln (S/X) + ( r-q+ σ2 / 2 ) T σ⁢T , d2 = d1 - σ⁢T .$

## 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:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
calput = 'C'${\mathbf{calput}}=\text{'C'}$
A call. The holder has a right to buy.
calput = 'P'${\mathbf{calput}}=\text{'P'}$
A put. The holder has a right to sell.
Constraint: calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
2:     x(m) – double array
m, the dimension of the array, must satisfy the constraint m1${\mathbf{m}}\ge 1$.
x(i)${\mathbf{x}}\left(i\right)$ must contain Xi${X}_{\mathit{i}}$, the i$\mathit{i}$th strike price, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: x(i)z ​ and ​ x(i) 1 / z ${\mathbf{x}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{x}}\left(\mathit{i}\right)\le 1/z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     s – double scalar
S$S$, the price of the underlying asset.
Constraint: sz ​ and ​s1.0 / z${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
4:     t(n) – double array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
t(i)${\mathbf{t}}\left(i\right)$ must contain Ti${T}_{\mathit{i}}$, the i$\mathit{i}$th time, in years, to expiry, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: t(i)z${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     sigma – double scalar
σ$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma > 0.0${\mathbf{sigma}}>0.0$.
6:     r – double scalar
r$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0${\mathbf{r}}\ge 0.0$.
7:     q – double scalar
q$q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0${\mathbf{q}}\ge 0.0$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array x.
The number of strike prices to be used.
Constraint: m1${\mathbf{m}}\ge 1$.
2:     n – int64int32nag_int scalar
Default: The dimension of the array t.
The number of times to expiry to be used.
Constraint: n1${\mathbf{n}}\ge 1$.

ldp

### Output Parameters

1:     p(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array p contains the computed option prices.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
ifail = 2${\mathbf{ifail}}=2$
On entry, m0${\mathbf{m}}\le 0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, n0${\mathbf{n}}\le 0$.
ifail = 4${\mathbf{ifail}}=4$
On entry, x(i) < z${\mathbf{x}}\left(\mathit{i}\right) or x(i) > 1 / z${\mathbf{x}}\left(\mathit{i}\right)>1/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 5${\mathbf{ifail}}=5$
On entry, s < z${\mathbf{s}} or s > 1.0 / z${\mathbf{s}}>1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 6${\mathbf{ifail}}=6$
On entry, t(i) < z${\mathbf{t}}\left(\mathit{i}\right), where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 7${\mathbf{ifail}}=7$
On entry, sigma0.0${\mathbf{sigma}}\le 0.0$.
ifail = 8${\mathbf{ifail}}=8$
On entry, r < 0.0${\mathbf{r}}<0.0$.
ifail = 9${\mathbf{ifail}}=9$
On entry, q < 0.0${\mathbf{q}}<0.0$.
ifail = 11${\mathbf{ifail}}=11$
On entry, ldp < m$\mathit{ldp}<{\mathbf{m}}$.

## Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, Φ$\Phi$. 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

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

[p, ifail] = nag_specfun_opt_bsm_price(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
```
```

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

```
```function s30aa_example
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
```
```

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

```

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