NAG CL Interfaces30nac (opt_​heston_​price)

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1Purpose

s30nac computes the European option price given by Heston's stochastic volatility model.

2Specification

 #include
 void s30nac (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[], NagError *fail)
The function may be called by the names: s30nac, nag_specfun_opt_heston_price or nag_heston_price.

3Description

s30nac 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).
 $Pcall = S e-qT - X e-rT 1π Re[ ∫ 0+i/2 ∞+i/2 e-ikX¯ H^ (k,v,T) k2 - ik dk] ,$ (1)
where $\overline{X}=\mathrm{ln}\left(S/X\right)+\left(r-q\right)T$ and
 $H^ (k,v,T) = exp( 2κη σv2 [tg -ln( 1-he-ξt 1-h )]+vtg[ 1-e-ξt 1-he-ξt ]) ,$
 $g = 12 (b-ξ) , h = b-ξ b+ξ , t = σv2 T/2 ,$
 $ξ = [b2+4 k2-ik σv2 ] 12 ,$
 $b = 2 σv2 [(1-γ+ik)ρσv+ κ2 - γ(1-γ) σv2 ]$
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: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface 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_CallPut Input
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: $\mathbf{m}$Integer Input
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
4: $\mathbf{n}$Integer Input
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
5: $\mathbf{x}\left[{\mathbf{m}}\right]$const double Input
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: $\mathbf{s}$double Input
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter.
7: $\mathbf{t}\left[{\mathbf{n}}\right]$const double Input
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: $\mathbf{sigmav}$double 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: ${\mathbf{sigmav}}>0.0$.
9: $\mathbf{kappa}$double Input
On entry: $\kappa$, the long term mean reversion rate of the volatility.
Constraint: ${\mathbf{kappa}}>0.0$.
10: $\mathbf{corr}$double Input
On entry: the correlation between the two standard Brownian motions for the asset price and the volatility.
Constraint: $-1.0\le {\mathbf{corr}}\le 1.0$.
11: $\mathbf{var0}$double Input
On entry: the initial value of the variance, ${v}_{t}$, of the asset price.
Constraint: ${\mathbf{var0}}\ge 0.0$.
12: $\mathbf{eta}$double Input
On entry: $\eta$, the long term mean of the variance of the asset price.
Constraint: ${\mathbf{eta}}>0.0$.
13: $\mathbf{grisk}$double Input
On entry: the risk aversion parameter, $\gamma$, of the representative agent.
Constraint: $0.0\le {\mathbf{grisk}}\le 1.0$ and ${\mathbf{grisk}}×\left(1.0-{\mathbf{grisk}}\right)×{\mathbf{sigmav}}×{\mathbf{sigmav}}\le {\mathbf{kappa}}×{\mathbf{kappa}}$.
14: $\mathbf{r}$double 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: ${\mathbf{r}}\ge 0.0$.
15: $\mathbf{q}$double Input
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.
Constraint: ${\mathbf{q}}\ge 0.0$.
16: $\mathbf{p}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ACCURACY
Solution cannot be computed accurately. Check values of input arguments.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
Quadrature has not converged to the specified 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{corr}}=⟨\mathit{\text{value}}⟩$.
Constraint: $|{\mathbf{corr}}|\le 1.0$.
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 ${\mathbf{x}}\left[i-1\right]\le ⟨\mathit{\text{value}}⟩$.

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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({10}^{-8},{10}^{-10}×|I|\right)$, where $|I|$ is the absolute value of the integral.

8Parallelism and Performance

s30nac 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.

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 (s30nace.c)

10.2Program Data

Program Data (s30nace.d)

10.3Program Results

Program Results (s30nace.r)