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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_bsm_greeks (s30ab)

## Purpose

nag_specfun_opt_bsm_greeks (s30ab) computes the European option price given by the Black–Scholes–Merton formula together with its sensitivities (Greeks).

## Syntax

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30ab(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_bsm_greeks(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)

## Description

nag_specfun_opt_bsm_greeks (s30ab) computes the price of a European call (or put) 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, by the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). The annual volatility, σ$\sigma$, risk-free interest rate, r$r$, and dividend yield, q$q$, must be supplied as input. 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:     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_bsm_greeks_example
put = 'p';
s = 55;
sigma = 0.3;
r = 0.1;
q = 0;
x = [60];
t = [0.7];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, ...
zomma, vomma, ifail] = nag_specfun_opt_bsm_greeks(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(' 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));
```
```

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

Time to Expiry :   0.7000
Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
60.0000   6.0245  -0.4770   0.0289  18.3273  -0.7014 -22.5811 -18.3639

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
60.0000   6.0245   0.2566  -0.2137  -0.0006   0.0215  -0.0972  -0.6816

```
```function s30ab_example
put = 'p';
s = 55;
sigma = 0.3;
r = 0.1;
q = 0;
x = [60];
t = [0.7];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, ...
zomma, vomma, ifail] = s30ab(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(' 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));
```
```

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

Time to Expiry :   0.7000
Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
60.0000   6.0245  -0.4770   0.0289  18.3273  -0.7014 -22.5811 -18.3639

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
60.0000   6.0245   0.2566  -0.2137  -0.0006   0.0215  -0.0972  -0.6816

```