NAG Toolbox: nag_specfun_opt_heston_greeks (s30nb)

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:     $\mathrm{sigmav}$ – double scalar

The volatility, ${\sigma }_v$, of the volatility process, $\sqrt{v_t}$. Note that a rate of 20% should be entered as $0.2$.

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

6:     $\mathrm{kappa}$ – double scalar

$\kappa$, the long term mean reversion rate of the volatility.

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

7:     $\mathrm{corr}$ – double scalar

The correlation between the two standard Brownian motions for the asset price and the volatility.

Constraint: $-1.0\le \mathbf{corr}\le 1.0$.

8:     $\mathrm{var0}$ – double scalar

The initial value of the variance, $v_t$, of the asset price.

Constraint: $\mathbf{var0}\ge 0.0$.

9:     $\mathrm{eta}$ – double scalar

$\eta$, the long term mean of the variance of the asset price.

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

10:   $\mathrm{grisk}$ – double scalar

The risk aversion parameter, $\gamma$, of the representative agent.

Constraint: $0.0\le \mathbf{grisk}\le 1.0$ and $\mathbf{grisk}\times \left(1-\mathbf{grisk}\right)\times \mathbf{sigmav}\times \mathbf{sigmav}\le \mathbf{kappa}\times \mathbf{kappa}$.

11:   $\mathrm{r}$ – double scalar

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

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

12:   $\mathrm{q}$ – 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:     $\mathrm{vanna}\left(\mathit{ldp},\mathbf{n}\right)$ – 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 $j=1,2,\dots ,\mathbf{n}$.

8:     $\mathrm{charm}\left(\mathit{ldp},\mathbf{n}\right)$ – 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 $j=1,2,\dots ,\mathbf{n}$.

9:     $\mathrm{speed}\left(\mathit{ldp},\mathbf{n}\right)$ – 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 $j=1,2,\dots ,\mathbf{n}$.

10:   $\mathrm{zomma}\left(\mathit{ldp},\mathbf{n}\right)$ – 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 $j=1,2,\dots ,\mathbf{n}$.

11:   $\mathrm{vomma}\left(\mathit{ldp},\mathbf{n}\right)$ – 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 $j=1,2,\dots ,\mathbf{n}$.

12:   $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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: $\mathbf{sigmav}>0.0$.

   $\mathbf{ifail}=8$

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

   $\mathbf{ifail}=9$

Constraint: $\left|\mathbf{corr}\right|\le 1.0$.

   $\mathbf{ifail}=10$

Constraint: $\mathbf{var0}\ge 0.0$.

   $\mathbf{ifail}=11$

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

   $\mathbf{ifail}=12$

Constraint: $0.0\le \mathbf{grisk}\le 1.0$ and $\mathbf{grisk}\times \left(1.0-\mathbf{grisk}\right)\times {\mathbf{sigmav}}^2\le {\mathbf{kappa}}^2$.

   $\mathbf{ifail}=13$

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

   $\mathbf{ifail}=14$

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

   $\mathbf{ifail}=16$

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

W  $\mathbf{ifail}=17$

Quadrature has not converged to the required accuracy. However, the result should be a reasonable approximation.

W  $\mathbf{ifail}=18$

Solution cannot be computed accurately. Check values of input arguments.

   $\mathbf{ifail}=-99$

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: s30nb_example

function s30nb_example


fprintf('s30nb example results\n\n');

calput = 'C';
s      = 100.0;
r      = 0.025;
q      = 0.0;
kappa  = 1.5768;
eta    = 0.0398;
var0   = 0.0175;
sigmav = 0.5751;
corr   = -0.5711;
grisk  = 1;
x      = [100.0];
t      = [1];


[p, delta, gamma, vega,  theta, rho, ...
    vanna, charm, speed, zomma, vomma, ifail] = ...
    s30nb(...
          calput, x, s, t, sigmav, kappa, corr, var0, eta, grisk, r, q);

fprintf('\nHeston''s Stochastic Volatility Model\n');
if calput == 'C' || calput == 'c'
  fprintf('European Call :\n');
else
  fprintf('European Put :\n');
end
fprintf(' Spot                   =  %9.4f\n', s);
fprintf(' Volatility of vol      =  %9.4f\n', sigmav);
fprintf(' Mean reversion         =  %9.4f\n', kappa);
fprintf(' Correlation            =  %9.4f\n', corr);
fprintf(' Variance               =  %9.4f\n', var0);
fprintf(' Mean of variance       =  %9.4f\n', eta);
fprintf(' Risk aversion          =  %9.4f\n', grisk);
fprintf(' Rate                   =  %9.4f\n', r);
fprintf(' Dividend               =  %9.4f\n\n', q);

for j=1:1
  fprintf('%8s%9s%9s%9s%9s%9s%9s\n','Strike','Price','Delta','Gamma',...
          'Vega','Theta','Rho');
  for i=1:1
    fprintf('%8.4f%9.4f%9.4f%9.4f%9.4f%9.4f%9.2f\n', x(i), p(i,j), ...
          delta(i,j), gamma(i,j), vega(i,j), theta(i,j), rho(i,j));
  end
  fprintf('\n%26s%9s%9s%9s%9s\n','Vanna','Charm','Speed','Zomma','Vomma');
  for i=1:1
    fprintf('%17s%9.4f%9.4f%9.4f%9.4f%9.2f\n', ' ', vanna(i,j), ...
            charm(i,j), speed(i,j), zomma(i,j), vomma(i,j));
  end
end

s30nb example results


Heston's Stochastic Volatility Model
European Call :
 Spot                   =   100.0000
 Volatility of vol      =     0.5751
 Mean reversion         =     1.5768
 Correlation            =    -0.5711
 Variance               =     0.0175
 Mean of variance       =     0.0398
 Risk aversion          =     1.0000
 Rate                   =     0.0250
 Dividend               =     0.0000

  Strike    Price    Delta    Gamma     Vega    Theta      Rho
100.0000   7.2743   0.6945   0.0251  52.5461  -4.9969    62.17

                     Vanna    Charm    Speed    Zomma    Vomma
                   -0.5643  -0.0321  -0.0023  -0.1976  -321.08