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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_binary_con_greeks (s30cb)

## Purpose

nag_specfun_opt_binary_con_greeks (s30cb) computes the price of a binary or digital cash-or-nothing option together with its sensitivities (Greeks).

## Syntax

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30cb(calput, x, s, k, 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_con_greeks(calput, x, s, k, t, sigma, r, q, 'm', m, 'n', n)

## Description

nag_specfun_opt_binary_con_greeks (s30cb) computes the price of a binary or digital cash-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 a fixed amount, $K$, at expiration if the option is in-the-money (see Option Pricing Routines in the S Chapter Introduction). For a strike price, $X$, underlying asset price, $S$, and time to expiry, $T$, the payoff is therefore $K$, if $S>X$ for a call or $S for a put. Nothing is paid out when this condition is not met.
The price of a call with volatility, $\sigma$, risk-free interest rate, $r$, and annualised dividend yield, $q$, is
 $Pcall = K e-rT Φd2$
and for a put,
 $Pput = K e-rT Φ-d2$
where $\Phi$ is the cumulative Normal distribution function,
 $Φx = 1 2π ∫ -∞ x exp -y2/2 dy ,$
and
 $d2 = ln S/X + r-q- σ2 / 2 T σ⁢T .$
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

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

## 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:     $\mathrm{k}$ – double scalar
The amount, $K$, to be paid at expiration if the option is in-the-money, i.e., if ${\mathbf{s}}>{\mathbf{x}}\left(\mathit{i}\right)$ when ${\mathbf{calput}}=\text{'C'}$, or if ${\mathbf{s}}<{\mathbf{x}}\left(\mathit{i}\right)$ when ${\mathbf{calput}}=\text{'P'}$, for $\mathit{i}=1,2,\dots ,m$.
Constraint: ${\mathbf{k}}\ge 0.0$.
5:     $\mathrm{t}\left({\mathbf{n}}\right)$ – 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 $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
6:     $\mathrm{sigma}$ – double scalar
$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: ${\mathbf{sigma}}>0.0$.
7:     $\mathrm{r}$ – double scalar
$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
8:     $\mathrm{q}$ – 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:     $\mathrm{delta}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
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.
3:     $\mathrm{gamma}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
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.
4:     $\mathrm{vega}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{vega}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
5:     $\mathrm{theta}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{theta}}\left(i,j\right)$, 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 $b=r-q$.
6:     $\mathrm{rho}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{rho}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
7:     $\mathrm{crho}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{crho}}\left(i,j\right)$, 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 $b=r-q$.
8:     $\mathrm{vanna}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{vanna}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
9:     $\mathrm{charm}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{charm}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
10:   $\mathrm{speed}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{speed}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
11:   $\mathrm{colour}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{colour}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
12:   $\mathrm{zomma}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{zomma}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
13:   $\mathrm{vomma}\left(\mathit{ldp},{\mathbf{n}}\right)$ – double array
$\mathit{ldp}={\mathbf{m}}$.
${\mathbf{vomma}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
14:   $\mathrm{ifail}$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: ${\mathbf{k}}\ge 0.0$.
${\mathbf{ifail}}=7$
Constraint: ${\mathbf{t}}\left(i\right)\ge _$.
${\mathbf{ifail}}=8$
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=9$
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=10$
Constraint: ${\mathbf{q}}\ge 0.0$.
${\mathbf{ifail}}=12$
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 price of a cash-or-nothing call with a time to expiry of $0.75$ years, a stock price of $110$ and a strike price of $87$. The risk-free interest rate is $5%$ per year, there is an annual dividend return of $4%$ and the volatility is $35%$ per year. If the option is in-the-money at expiration, i.e., if $S>X$, the payoff is $5$.
```function s30cb_example

fprintf('s30cb example results\n\n');

put = 'C';
s = 110.0;
k = 5.0;
sigma = 0.35;
r = 0.05;
q = 0.04;
x = [87.0];
t = [0.75];

[p, delta, gamma,  vega, theta,    rho,  crho, ...
vanna, charm, speed, colour, zomma, vomma, ifail] = ...
s30cb(...
put, x, s, k, t, sigma, r, q);

fprintf('\nBinary (Digital): Cash-or-Nothing\n European Call :\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(' Time to Expiry : %8.4f\n', t(1));
fprintf('%8s%9s%9s%9s%9s%9s%9s%9s\n','Strike','Price','Delta','Gamma',...
'Vega','Theta','Rho','CRho');
fprintf('%8.4f%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f\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('%26s%9s%9s%9s%9s%9s\n','Vanna','Charm','Speed','Colour',...
'Zomma','Vomma');
fprintf('%17s%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f\n\n', ' ', vanna(1,1), ...
charm(1,1), speed(1,1), colour(1,1), zomma(1,1), vomma(1,1));

```
```s30cb example results

Binary (Digital): Cash-or-Nothing
European Call :
Spot       =    110.0000
Payout     =      5.0000
Volatility =      0.3500
Rate       =      0.0500
Dividend   =      0.0400

Time to Expiry :   0.7500
Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
87.0000   3.5696   0.0467  -0.0013  -4.2307   1.1142   1.1788   3.8560

Vanna    Charm    Speed   Colour    Zomma    Vomma
-0.0514   0.0153   0.0000  -0.0019   0.0079  12.8874

```