# NAG FL Interfaces30naf (opt_​heston_​price)

## 1Purpose

s30naf computes the European option price given by Heston's stochastic volatility model.

## 2Specification

Fortran Interface
 Subroutine s30naf ( m, n, x, s, t, corr, var0, eta, r, q, p, ldp,
 Integer, Intent (In) :: m, n, ldp Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(m), s, t(n), sigmav, kappa, corr, var0, eta, grisk, r, q Real (Kind=nag_wp), Intent (Inout) :: p(ldp,n) Character (1), Intent (In) :: calput
#include <nag.h>
 void s30naf_ (const char *calput, const Integer *m, const Integer *n, const double x[], const double *s, const double t[], const double *sigmav, const double *kappa, const double *corr, const double *var0, const double *eta, const double *grisk, const double *r, const double *q, double p[], const Integer *ldp, Integer *ifail, const Charlen length_calput)
The routine may be called by the names s30naf or nagf_specfun_opt_heston_price.

## 3Description

s30naf computes the price 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 $\overline{X}=\mathrm{ln}\left(S/X\right)+\left(r-q\right)T$ 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.
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 ${T}_{j}$, $j=1,2,\dots ,n$.

## 4References

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 https://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

## 5Arguments

1: $\mathbf{calput}$Character(1) Input
On entry: 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: $\mathbf{m}$Integer Input
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
3: $\mathbf{n}$Integer Input
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
4: $\mathbf{x}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{x}}\left(i\right)$ 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 $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
5: $\mathbf{s}$Real (Kind=nag_wp) Input
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter.
6: $\mathbf{t}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{t}}\left(i\right)$ 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 $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
7: $\mathbf{sigmav}$Real (Kind=nag_wp) Input
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: ${\mathbf{sigmav}}>0.0$.
8: $\mathbf{kappa}$Real (Kind=nag_wp) Input
On entry: $\kappa$, the long term mean reversion rate of the volatility.
Constraint: ${\mathbf{kappa}}>0.0$.
9: $\mathbf{corr}$Real (Kind=nag_wp) Input
On entry: the correlation between the two standard Brownian motions for the asset price and the volatility.
Constraint: $-1.0\le {\mathbf{corr}}\le 1.0$.
10: $\mathbf{var0}$Real (Kind=nag_wp) Input
On entry: the initial value of the variance, ${v}_{t}$, of the asset price.
Constraint: ${\mathbf{var0}}\ge 0.0$.
11: $\mathbf{eta}$Real (Kind=nag_wp) Input
On entry: $\eta$, the long term mean of the variance of the asset price.
Constraint: ${\mathbf{eta}}>0.0$.
12: $\mathbf{grisk}$Real (Kind=nag_wp) Input
On entry: the risk aversion parameter, $\gamma$, of the representative agent.
Constraint: $0.0\le {\mathbf{grisk}}\le 1.0$ and ${\mathbf{grisk}}×\left(1.0-{\mathbf{grisk}}\right)×{\mathbf{sigmav}}×{\mathbf{sigmav}}\le {\mathbf{kappa}}×{\mathbf{kappa}}$.
13: $\mathbf{r}$Real (Kind=nag_wp) Input
On entry: $r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as $0.05$.
Constraint: ${\mathbf{r}}\ge 0.0$.
14: $\mathbf{q}$Real (Kind=nag_wp) Input
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.
Constraint: ${\mathbf{q}}\ge 0.0$.
15: $\mathbf{p}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
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 $j=1,2,\dots ,{\mathbf{n}}$.
16: $\mathbf{ldp}$Integer Input
On entry: the first dimension of the array p as declared in the (sub)program from which s30naf is called.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
17: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=〈\mathit{\text{value}}〉$ was an illegal value.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{x}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\left(i\right)\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left(i\right)\le 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{s}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{s}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{s}}\le 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{t}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{t}}\left(i\right)\ge 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{sigmav}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sigmav}}>0.0$.
${\mathbf{ifail}}=8$
On entry, ${\mathbf{kappa}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kappa}}>0.0$.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{corr}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{corr}}\right|\le 1.0$.
${\mathbf{ifail}}=10$
On entry, ${\mathbf{var0}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{var0}}\ge 0.0$.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{eta}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{eta}}>0.0$.
${\mathbf{ifail}}=12$
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 ${\mathbf{grisk}}×\left(1.0-{\mathbf{grisk}}\right)×{{\mathbf{sigmav}}}^{2}\le {{\mathbf{kappa}}}^{2}$.
${\mathbf{ifail}}=13$
On entry, ${\mathbf{r}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=14$
On entry, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{q}}\ge 0.0$.
${\mathbf{ifail}}=16$
On entry, ${\mathbf{ldp}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=17$
Quadrature has not converged to the specified accuracy. However, the result should be a reasonable approximation.
${\mathbf{ifail}}=18$
Solution cannot be computed accurately. Check values of input arguments.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

s30naf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example computes the price of a European call using Heston's stochastic volatility model. The time to expiry is $6$ months, the stock price is $100$ and the strike price is $100$. The risk-free interest rate is $5%$ per year, the volatility of the variance, ${\sigma }_{v}$, is $22.5%$ per year, the mean reversion parameter, $\kappa$, is $2.0$, the long term mean of the variance, $\eta$, is $0.01$ and the correlation between the volatility process and the stock price process, $\rho$, is $0.0$. The risk aversion parameter, $\gamma$, is $1.0$ and the initial value of the variance, var0, is $0.01$.

### 10.1Program Text

Program Text (s30nafe.f90)

### 10.2Program Data

Program Data (s30nafe.d)

### 10.3Program Results

Program Results (s30nafe.r)