NAG Toolbox: nag_specfun_opt_bsm_greeks (s30ab)

Purpose

Syntax

Description

References

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:     $\mathrm{t}\left(\mathbf{n}\right)$ – 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 $\mathit{i}=1,2,\dots ,\mathbf{n}$.

5:     $\mathrm{sigma}$ – double scalar

$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.

Constraint: $\mathbf{sigma}>0.0$.

6:     $\mathrm{r}$ – double scalar

$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.

Constraint: $\mathbf{r}\ge 0.0$.

7:     $\mathrm{q}$ – 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 $\mathbf{m}\times \mathbf{n}$ 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 $\mathbf{m}\times \mathbf{n}$ 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

   $\mathbf{ifail}=1$

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: $\mathbf{t}\left(i\right)\ge \_$.

   $\mathbf{ifail}=7$

Constraint: $\mathbf{sigma}>0.0$.

   $\mathbf{ifail}=8$

Constraint: $\mathbf{r}\ge 0.0$.

   $\mathbf{ifail}=9$

Constraint: $\mathbf{q}\ge 0.0$.

   $\mathbf{ifail}=11$

Constraint: $\mathit{ldp}\ge \mathbf{m}$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: s30ab_example

function s30ab_example


fprintf('s30ab example results\n\n');

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('%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));

s30ab example results


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

                     Vanna    Charm    Speed   Colour    Zomma    Vomma
                    0.2566  -0.2137  -0.0006   0.0215  -0.0972  -0.6816