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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_heston_greeks (s30nb)

## Purpose

nag_specfun_opt_heston_greeks (s30nb) computes the European option price given by Heston's stochastic volatility model together with its sensitivities (Greeks).

## Syntax

[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, 'm', m, 'n', n)
[p, delta, gamma, vega, theta, rho, vanna, charm, speed, zomma, vomma, ifail] = nag_specfun_opt_heston_greeks(calput, x, s, t, sigmav, kappa, corr, var0, eta, grisk, r, q, 'm', m, 'n', n)

## Description

nag_specfun_opt_heston_greeks (s30nb) computes the price and sensitivities of a European option using Heston's stochastic volatility model. The return on the asset price, $S$, is
 $dS S = r-q dt + vt d W t 1$
and the instantaneous variance, ${v}_{t}$, is defined by a mean-reverting square root stochastic process,
 $dvt = κ η-vt dt + σv vt d W t 2 ,$
where $r$ is the risk free annual interest rate; $q$ is the annual dividend rate; ${v}_{t}$ is the variance of the asset price; ${\sigma }_{v}$ is the volatility of the volatility, $\sqrt{{v}_{t}}$; $\kappa$ is the mean reversion rate; $\eta$ is the long term variance. $d{W}_{t}^{\left(\mathit{i}\right)}$, for $\mathit{i}=1,2$, denotes two correlated standard Brownian motions with
 $ℂov d W t 1 , d W t 2 = ρ d t .$
The option price is computed by evaluating the integral transform given by Lewis (2000) using the form of the characteristic function discussed by Albrecher et al. (2007), see also Kilin (2006).
 $Pcall = S e-qT - X e-rT 1π Re ∫ 0+i/2 ∞+i/2 e-ikX- H^ k,v,T k2 - ik d k ,$ (1)
where $\stackrel{-}{X}=\mathrm{ln}\left(S/X\right)+\left(r-q\right)T$ and
 $H^ k,v,T = exp 2κη σv2 tg - ln 1-he-ξt 1-h + vt g 1-e-ξt 1-he-ξt ,$
 $g = 12 b-ξ , h = b-ξ b+ξ , t = σv2 T/2 ,$
 $ξ = b2 + 4 k2-ik σv2 12 ,$
 $b = 2 σv2 1-γ+ik ρσv + κ2 - γ1-γ σv2$
with $t={\sigma }_{v}^{2}T/2$. Here $\gamma$ is the risk aversion parameter of the representative agent with $0\le \gamma \le 1$ and $\gamma \left(1-\gamma \right){\sigma }_{v}^{2}\le {\kappa }^{2}$. The value $\gamma =1$ corresponds to $\lambda =0$, where $\lambda$ is the market price of risk in Heston (1993) (see Lewis (2000) and Rouah and Vainberg (2007)).
The price of a put option is obtained by put-call parity.
 $Pput = Pcall + Xe-rT - S e-qT .$
Writing the expression for the price of a call option as
 $Pcall = Se-qT - Xe-rT 1π Re ∫ 0+i/2 ∞+i/2 I k,r,S,T,v d k$
then the sensitivities or Greeks can be obtained in the following manner,
Delta
 $∂ Pcall ∂S = e-qT + Xe-rT S 1π Re ∫ 0+i/2 ∞+i/2 ik I k,r,S,T,v dk ,$
Vega
 $∂P ∂v = - X e-rT 1π Re ∫ 0-i/2 0+i/2 f2 I k,r,j,S,T,v dk , where ​ f2 = g 1 - e-ξt 1 - h e-ξt ,$
Rho
 $∂Pcall ∂r = T X e-rT 1π Re ∫ 0+i/2 ∞+i/2 1+ik I k,r,S,T,v dk .$
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

Albrecher H, Mayer P, Schoutens W and Tistaert J (2007) The little Heston trap Wilmott Magazine January 2007 83–92
Heston S (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options Review of Financial Studies 6 327–343
Kilin F (2006) Accelerating the calibration of stochastic volatility models MPRA Paper No. 2975 http://mpra.ub.uni-muenchen.de/2975/
Lewis A L (2000) Option valuation under stochastic volatility Finance Press, USA
Rouah F D and Vainberg G (2007) Option Pricing Models and Volatility using Excel-VBA John Wiley and Sons, Inc

## 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 ${\mathbf{grisk}}×\left(1-{\mathbf{grisk}}\right)×{\mathbf{sigmav}}×{\mathbf{sigmav}}\le {\mathbf{kappa}}×{\mathbf{kappa}}$.
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 ${\mathbf{m}}×{\mathbf{n}}$ 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:     $\mathrm{vanna}\left(\mathit{ldp},{\mathbf{n}}\right)$ – 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

Errors or warnings detected by the function:

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 ${\mathbf{grisk}}×\left(1.0-{\mathbf{grisk}}\right)×{{\mathbf{sigmav}}}^{2}\le {{\mathbf{kappa}}}^{2}$.
${\mathbf{ifail}}=13$
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.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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 determined by the accuracy of the numerical quadrature used to evaluate the integral in (1). An adaptive method is used which evaluates the integral to within a tolerance of $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({10}^{-8},{10}^{-10}×\left|I\right|\right)$, where $\left|I\right|$ is the absolute value of the integral.

None.

## Example

This example computes the price and sensitivities of a European call using Heston's stochastic volatility model. The time to expiry is $1$ year, the stock price is $100$ and the strike price is $100$. The risk-free interest rate is $2.5%$ per year, the volatility of the variance, ${\sigma }_{v}$, is $57.51%$ per year, the mean reversion parameter, $\kappa$, is $1.5768$, the long term mean of the variance, $\eta$, is $0.0398$ and the correlation between the volatility process and the stock price process, $\rho$, is $-0.5711$. The risk aversion parameter, $\gamma$, is $1.0$ and the initial value of the variance, var0, is $0.0175$.
```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
```