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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_binary_aon_greeks (s30cd)

## Purpose

nag_specfun_opt_binary_aon_greeks (s30cd) computes the price of a binary or digital asset-or-nothing option together with its sensitivities (Greeks).

## Syntax

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30cd(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)
[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = nag_specfun_opt_binary_aon_greeks(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)

## Description

nag_specfun_opt_binary_aon_greeks (s30cd) computes the price of a binary or digital asset-or-nothing 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. This option pays the underlying asset itself, S$S$, 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 S$S$, 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 = S e − qT Φ(d1) $Pcall = S e-qT Φ(d1)$
and for a put,
 Pput = S e − qT Φ( − d1) $Pput = S e-qT Φ(-d1)$
where Φ$\Phi$ is the cumulative Normal distribution function,
 x Φ(x) = 1/(sqrt(2π)) ∫ exp( − y2 / 2)dy, − ∞
$Φ(x) = 1 2π ∫ -∞ x exp( -y2/2 ) dy ,$
and
 d1 = ( ln (S / X) + (r − q + σ2 / 2) T )/(σ×sqrt(T)) . $d1 = ln (S/X) + ( r-q + σ2 / 2 ) T σ⁢T .$

## 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:     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:     delta(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array delta contains the sensitivity, (P)/(S)$\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.
3:     gamma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array gamma contains the sensitivity, (2P)/(S2)$\frac{{\partial }^{2}P}{\partial {S}^{2}}$, of delta to change in the price of the underlying asset.
4:     vega(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vega contains the sensitivity, (P)/(σ)$\frac{\partial P}{\partial \sigma }$, of the option price to change in the volatility of the underlying asset.
5:     theta(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array theta contains the sensitivity, (P)/(T)$-\frac{\partial P}{\partial T}$, of the option price to change in the time to expiry of the option.
6:     rho(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array rho contains the sensitivity, (P)/(r)$\frac{\partial P}{\partial r}$, of the option price to change in the annual risk-free interest rate.
7:     crho(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array crho containing the sensitivity, (P)/(b)$\frac{\partial P}{\partial b}$, of the option price to change in the annual cost of carry rate, b$b$, where b = rq$b=r-q$.
8:     vanna(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vanna contains the sensitivity, (2P)/(Sσ)$\frac{{\partial }^{2}P}{\partial S\partial \sigma }$, of vega to change in the price of the underlying asset or, equivalently, the sensitivity of delta to change in the volatility of the asset price.
9:     charm(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array charm contains the sensitivity, (2P)/(S T)$-\frac{{\partial }^{2}P}{\partial S\partial T}$, of delta to change in the time to expiry of the option.
10:   speed(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array speed contains the sensitivity, (3P)/(S3)$\frac{{\partial }^{3}P}{\partial {S}^{3}}$, of gamma to change in the price of the underlying asset.
11:   colour(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array colour contains the sensitivity, (3P)/(S2 T)$-\frac{{\partial }^{3}P}{\partial {S}^{2}\partial T}$, of gamma to change in the time to expiry of the option.
12:   zomma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array zomma contains the sensitivity, (3P)/(S2σ)$\frac{{\partial }^{3}P}{\partial {S}^{2}\partial \sigma }$, of gamma to change in the volatility of the underlying asset.
13:   vomma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vomma contains the sensitivity, (2P)/(σ2)$\frac{{\partial }^{2}P}{\partial {\sigma }^{2}}$, of vega to change in the volatility of the underlying asset.
14:   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_binary_aon_greeks_example
put = 'P';
s = 70;
sigma = 0.15;
r = 0.05;
q = 0.03;
x = [65.0];
t = [0.8];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, ...
speed, colour, zomma, vomma, ifail] = ...
nag_specfun_opt_binary_aon_greeks(put, x, s, t, sigma, r, q);

fprintf('\nBinary (Digital): Asset-or-Nothing\n European Put :\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(' Time to Expiry : %8.4f\n', t(1));
fprintf(' Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n\n', x(1), ...
p(1,1), delta(1,1), gamma(1,1), vega(1,1), theta(1,1), rho(1,1), ...
crho(1,1));

fprintf(' Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
p(1,1), vanna(1,1), charm(1,1), speed(1,1), colour(1,1), ...
zomma(1,1), vomma(1,1));
```
```

Binary (Digital): Asset-or-Nothing
European Put :
Spot       =     70.0000
Volatility =      0.1500
Rate       =      0.0500
Dividend   =      0.0300

Time to Expiry :   0.8000
Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
65.0000  15.7211  -1.9852   0.1422  83.6424  -4.2761 -123.7497 -111.1728

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
65.0000  15.7211   9.3479  -1.1351   0.0118   0.2316  -2.6319 -989.9610

```
```function s30cd_example
put = 'P';
s = 70;
sigma = 0.15;
r = 0.05;
q = 0.03;
x = [65.0];
t = [0.8];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, ...
speed, colour, zomma, vomma, ifail] = ...
s30cd(put, x, s, t, sigma, r, q);

fprintf('\nBinary (Digital): Asset-or-Nothing\n European Put :\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(' Time to Expiry : %8.4f\n', t(1));
fprintf(' Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n\n', x(1), ...
p(1,1), delta(1,1), gamma(1,1), vega(1,1), theta(1,1), rho(1,1), ...
crho(1,1));

fprintf(' Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
p(1,1), vanna(1,1), charm(1,1), speed(1,1), colour(1,1), ...
zomma(1,1), vomma(1,1));
```
```

Binary (Digital): Asset-or-Nothing
European Put :
Spot       =     70.0000
Volatility =      0.1500
Rate       =      0.0500
Dividend   =      0.0300

Time to Expiry :   0.8000
Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
65.0000  15.7211  -1.9852   0.1422  83.6424  -4.2761 -123.7497 -111.1728

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
65.0000  15.7211   9.3479  -1.1351   0.0118   0.2316  -2.6319 -989.9610

```