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NAG Toolbox

# NAG Toolbox: nag_specfun_opt_binary_con_price (s30ca)

## Purpose

nag_specfun_opt_binary_con_price (s30ca) computes the price of a binary or digital cash-or-nothing option.

## Syntax

[p, ifail] = s30ca(calput, x, s, k, t, sigma, r, q, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_binary_con_price(calput, x, s, k, t, sigma, r, q, 'm', m, 'n', n)

## Description

nag_specfun_opt_binary_con_price (s30ca) computes the price of a binary or digital cash-or-nothing option which pays a fixed amount, K$K$, at expiration if the option is in-the-money (see Section [Option Pricing s] in the S Chapter Introduction). For a strike price, X$X$, underlying asset price, S$S$, and time to expiry, T$T$, the payoff is therefore K$K$, if S > X$S>X$ for a call or S < X$S for a put. Nothing is paid out when this condition is not met.
The price of a call with volatility, σ$\sigma$, risk-free interest rate, r$r$, and annualised dividend yield, q$q$, is
 Pcall = K e − rT Φ(d2) $Pcall = K e-rT Φ(d2)$
and for a put,
 Pput = K e − rT Φ( − d2) $Pput = K e-rT Φ(-d2)$
where Φ$\Phi$ is the cumulative Normal distribution function,
 x Φ(x) = 1/(sqrt(2π)) ∫ ( − y2 / 2)dy, − ∞
$Φ(x) = 1 2π ∫ -∞ x ( -y2/2 ) dy ,$
and
 d2 = ( ln (S / X) + (r − q − σ2 / 2) T )/(σ×sqrt(T)) . $d2 = ln (S/X) + ( r-q - σ2 / 2 ) T σ⁢T .$

## References

Reiner E and Rubinstein M (1991) Unscrambling the binary code Risk 4

## 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:     k – double scalar
The amount, K$K$, to be paid at expiration if the option is in-the-money, i.e., if s > x(i)${\mathbf{s}}>{\mathbf{x}}\left(\mathit{i}\right)$ when calput = 'C'${\mathbf{calput}}=\text{'C'}$, or if s < x(i)${\mathbf{s}}<{\mathbf{x}}\left(\mathit{i}\right)$ when calput = 'P'${\mathbf{calput}}=\text{'P'}$, for i = 1,2,,m$\mathit{i}=1,2,\dots ,m$.
Constraint: k0.0${\mathbf{k}}\ge 0.0$.
5:     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$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
6:     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${\mathbf{sigma}}>0.0$.
7:     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${\mathbf{r}}\ge 0.0$.
8:     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, k < 0.0${\mathbf{k}}<0.0$.
ifail = 7${\mathbf{ifail}}=7$
On entry, t(i) < z${\mathbf{t}}\left(\mathit{i}\right), where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 8${\mathbf{ifail}}=8$
On entry, sigma0.0${\mathbf{sigma}}\le 0.0$.
ifail = 9${\mathbf{ifail}}=9$
On entry, r < 0.0${\mathbf{r}}<0.0$.
ifail = 10${\mathbf{ifail}}=10$
On entry, q < 0.0${\mathbf{q}}<0.0$.
ifail = 12${\mathbf{ifail}}=12$
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_binary_con_price_example
put = 'P';
s = 100.0;
k = 10.0;
sigma = 0.35;
r = 0.06;
q = 0.0;
x = [80.0];
t = [0.75];
[p, ifail] = nag_specfun_opt_binary_con_price(put, x, s, k, t, sigma, r, q);

fprintf('\nBinary (Digital): Cash-or-Nothing\n European Put :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Payout     =   %9.4f\n', k);
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=1:1
for j=1:1
fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
end
end
```
```

Binary (Digital): Cash-or-Nothing
European Put :
Spot       =    100.0000
Payout     =     10.0000
Volatility =      0.3500
Rate       =      0.0600
Dividend   =      0.0000

Strike    Expiry   Option Price
80.0000    0.7500    2.2155

```
```function s30ca_example
put = 'P';
s = 100.0;
k = 10.0;
sigma = 0.35;
r = 0.06;
q = 0.0;
x = [80.0];
t = [0.75];
[p, ifail] = s30ca(put, x, s, k, t, sigma, r, q);

fprintf('\nBinary (Digital): Cash-or-Nothing\n European Put :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Payout     =   %9.4f\n', k);
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=1:1
for j=1:1
fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
end
end
```
```

Binary (Digital): Cash-or-Nothing
European Put :
Spot       =    100.0000
Payout     =     10.0000
Volatility =      0.3500
Rate       =      0.0600
Dividend   =      0.0000

Strike    Expiry   Option Price
80.0000    0.7500    2.2155

```

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Chapter Contents
Chapter Introduction
NAG Toolbox

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