# NAG Library Routine Document

## 1Purpose

s30nbf computes the European option price given by Heston's stochastic volatility model together with its sensitivities (Greeks).

## 2Specification

Fortran Interface
 Subroutine s30nbf ( m, n, x, s, t, corr, var0, eta, r, q, p, ldp, vega, rho,
 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), delta(ldp,n), gamma(ldp,n), vega(ldp,n), theta(ldp,n), rho(ldp,n), vanna(ldp,n), charm(ldp,n), speed(ldp,n), zomma(ldp,n), vomma(ldp,n) Character (1), Intent (In) :: calput
#include nagmk26.h
 void s30nbf_ (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, double delta[], double gamma[], double vega[], double theta[], double rho[], double vanna[], double charm[], double speed[], double zomma[], double vomma[], Integer *ifail, const Charlen length_calput)

## 3Description

s30nbf 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 ${T}_{j}$, $j=1,2,\dots ,n$.

## 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 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

## 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:     $\mathbf{m}$ – IntegerInput
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathbf{n}$ – IntegerInput
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) arrayInput
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) arrayInput
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 ${\mathbf{grisk}}×\left(1-{\mathbf{grisk}}\right)×{\mathbf{sigmav}}×{\mathbf{sigmav}}\le {\mathbf{kappa}}×{\mathbf{kappa}}$.
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) arrayOutput
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:   $\mathbf{ldp}$ – IntegerInput
On entry: the first dimension of the arrays p, delta, gamma, vega, theta, rho, vanna, charm, speed, zomma and vomma as declared in the (sub)program from which s30nbf is called.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
17:   $\mathbf{delta}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: 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.
18:   $\mathbf{gamma}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: 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.
19:   $\mathbf{vega}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{theta}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{rho}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{vanna}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{charm}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{speed}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{zomma}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{vomma}\left({\mathbf{ldp}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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 required accuracy. However, the result should be a reasonable approximation.
${\mathbf{ifail}}=18$
Solution cannot be computed accurately. Check values of input arguments.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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

s30nbf 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 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 $0.0175$.

### 10.1Program Text

Program Text (s30nbfe.f90)

### 10.2Program Data

Program Data (s30nbfe.d)

### 10.3Program Results

Program Results (s30nbfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017