s Chapter Contents
s Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_heston_greeks (s30nbc)

## 1  Purpose

nag_heston_greeks (s30nbc) computes the European option price given by Heston's stochastic volatility model together with its sensitivities (Greeks).

## 2  Specification

 #include #include
 void nag_heston_greeks (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigmav, double kappa, double corr, double var0, double eta, double grisk, double r, double q, double p[], double delta[], double gamma[], double vega[], double theta[], double rho[], double vanna[], double charm[], double speed[], double zomma[], double vomma[], NagError *fail)

## 3  Description

nag_heston_greeks (s30nbc) 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 ${T}_{j}$, $j=1,2,\dots ,n$.

## 4  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

## 5  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{option}$Nag_CallPutInput
On entry: determines whether the option is a call or a put.
${\mathbf{option}}=\mathrm{Nag_Call}$
A call; the holder has a right to buy.
${\mathbf{option}}=\mathrm{Nag_Put}$
A put; the holder has a right to sell.
Constraint: ${\mathbf{option}}=\mathrm{Nag_Call}$ or $\mathrm{Nag_Put}$.
3:    $\mathbf{m}$IntegerInput
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
4:    $\mathbf{n}$IntegerInput
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
5:    $\mathbf{x}\left[{\mathbf{m}}\right]$const doubleInput
On entry: ${\mathbf{x}}\left[i-1\right]$ must contain ${X}_{\mathit{i}}$, the $\mathit{i}$th strike price, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: ${\mathbf{x}}\left[\mathit{i}-1\right]\ge z\text{​ and ​}{\mathbf{x}}\left[\mathit{i}-1\right]\le 1/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
6:    $\mathbf{s}$doubleInput
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter.
7:    $\mathbf{t}\left[{\mathbf{n}}\right]$const doubleInput
On entry: ${\mathbf{t}}\left[i-1\right]$ must contain ${T}_{\mathit{i}}$, the $\mathit{i}$th time, in years, to expiry, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{t}}\left[\mathit{i}-1\right]\ge z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
8:    $\mathbf{sigmav}$doubleInput
On entry: the volatility, ${\sigma }_{v}$, of the volatility process, $\sqrt{{v}_{t}}$. Note that a rate of 20% should be entered as $0.2$.
Constraint: ${\mathbf{sigmav}}>0.0$.
9:    $\mathbf{kappa}$doubleInput
On entry: $\kappa$, the long term mean reversion rate of the volatility.
Constraint: ${\mathbf{kappa}}>0.0$.
10:  $\mathbf{corr}$doubleInput
On entry: the correlation between the two standard Brownian motions for the asset price and the volatility.
Constraint: $-1.0\le {\mathbf{corr}}\le 1.0$.
11:  $\mathbf{var0}$doubleInput
On entry: the initial value of the variance, ${v}_{t}$, of the asset price.
Constraint: ${\mathbf{var0}}\ge 0.0$.
12:  $\mathbf{eta}$doubleInput
On entry: $\eta$, the long term mean of the variance of the asset price.
Constraint: ${\mathbf{eta}}>0.0$.
13:  $\mathbf{grisk}$doubleInput
On entry: the risk aversion parameter, $\gamma$, of the representative agent.
Constraint: $0.0\le {\mathbf{grisk}}\le 1.0$ and ${\mathbf{grisk}}×\left(1-{\mathbf{grisk}}\right)×{\mathbf{sigmav}}×{\mathbf{sigmav}}\le {\mathbf{kappa}}×{\mathbf{kappa}}$.
14:  $\mathbf{r}$doubleInput
On entry: $r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
15:  $\mathbf{q}$doubleInput
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: ${\mathbf{q}}\ge 0.0$.
16:  $\mathbf{p}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{P}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{p}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{p}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{P}}\left(i,j\right)$ 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 $j=1,2,\dots ,{\mathbf{n}}$.
17:  $\mathbf{delta}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{delta}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{delta}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array delta contains the sensitivity, $\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.
18:  $\mathbf{gamma}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{gamma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{gamma}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array gamma contains the sensitivity, $\frac{{\partial }^{2}P}{\partial {S}^{2}}$, of delta to change in the price of the underlying asset.
19:  $\mathbf{vega}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{VEGA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vega}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vega}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{VEGA}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
20:  $\mathbf{theta}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{THETA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{theta}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{theta}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{THETA}}\left(i,j\right)$, 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 $b=r-q$.
21:  $\mathbf{rho}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{RHO}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{rho}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{rho}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{RHO}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
22:  $\mathbf{vanna}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{VANNA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vanna}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vanna}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{VANNA}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
23:  $\mathbf{charm}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{CHARM}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{charm}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{charm}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{CHARM}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
24:  $\mathbf{speed}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{SPEED}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{speed}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{speed}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{SPEED}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
25:  $\mathbf{zomma}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{ZOMMA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{zomma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{zomma}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{ZOMMA}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
26:  $\mathbf{vomma}\left[{\mathbf{m}}×{\mathbf{n}}\right]$doubleOutput
Note: where ${\mathbf{VOMMA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vomma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vomma}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{VOMMA}}\left(i,j\right)$, 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 $j=1,2,\dots ,{\mathbf{n}}$.
27:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ACCURACY
Solution cannot be computed accurately. Check values of input arguments.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
Quadrature has not converged to the required accuracy. However, the result should be a reasonable approximation.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{corr}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{corr}}\right|\le 1.0$.
On entry, ${\mathbf{eta}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{eta}}>0.0$.
On entry, ${\mathbf{grisk}}=〈\mathit{\text{value}}〉$, ${\mathbf{sigmav}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kappa}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0\le {\mathbf{grisk}}\le 1.0$ and ${\mathbf{grisk}}×\left(1.0-{\mathbf{grisk}}\right)×{{\mathbf{sigmav}}}^{2}\le {{\mathbf{kappa}}}^{2}$.
On entry, ${\mathbf{kappa}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kappa}}>0.0$.
On entry, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{q}}\ge 0.0$.
On entry, ${\mathbf{r}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{r}}\ge 0.0$.
On entry, ${\mathbf{s}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{s}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{s}}\le 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{sigmav}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sigmav}}>0.0$.
On entry, ${\mathbf{var0}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{var0}}\ge 0.0$.
NE_REAL_ARRAY
On entry, ${\mathbf{t}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{t}}\left[i-1\right]\ge 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{x}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\left[i-1\right]\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[i-1\right]\le 〈\mathit{\text{value}}〉$.

## 7  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.

## 8  Parallelism and Performance

nag_heston_greeks (s30nbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10  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 $0.0175$.

### 10.1  Program Text

Program Text (s30nbce.c)

### 10.2  Program Data

Program Data (s30nbce.d)

### 10.3  Program Results

Program Results (s30nbce.r)