NAG Library Function Document
nag_heston_price (s30nac)
1 Purpose
nag_heston_price (s30nac) computes the European option price given by Heston's stochastic volatility model.
2 Specification
#include <nag.h> |
#include <nags.h> |
void |
nag_heston_price (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) |
|
3 Description
nag_heston_price (s30nac) computes the price of a European option using Heston's stochastic volatility model. The return on the asset price,
, is
and the instantaneous variance,
, is defined by a mean-reverting square root stochastic process,
where
is the risk free annual interest rate;
is the annual dividend rate;
is the variance of the asset price;
is the volatility of the volatility,
;
is the mean reversion rate;
is the long term variance.
, for
, denotes two correlated standard Brownian motions with
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).
where
and
with
.
Here
is the risk aversion parameter of the representative agent with
and
. The value
corresponds to
, where
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 is computed for each strike price in a set , , and for each expiry time in a set , .
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:
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
. See
Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
option – Nag_CallPutInput
On entry: determines whether the option is a call or a put.
- A call; the holder has a right to buy.
- A put; the holder has a right to sell.
Constraint:
or .
- 3:
m – IntegerInput
On entry: the number of strike prices to be used.
Constraint:
.
- 4:
n – IntegerInput
On entry: the number of times to expiry to be used.
Constraint:
.
- 5:
x[m] – const doubleInput
On entry: must contain
, the th strike price, for .
Constraint:
, where , the safe range parameter, for .
- 6:
s – doubleInput
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
- 7:
t[n] – const doubleInput
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
- 8:
sigmav – doubleInput
On entry: the volatility, , of the volatility process, . Note that a rate of 20% should be entered as .
Constraint:
.
- 9:
kappa – doubleInput
On entry: , the long term mean reversion rate of the volatility.
Constraint:
.
- 10:
corr – doubleInput
On entry: the correlation between the two standard Brownian motions for the asset price and the volatility.
Constraint:
.
- 11:
var0 – doubleInput
On entry: the initial value of the variance, , of the asset price.
Constraint:
.
- 12:
eta – doubleInput
On entry: , the long term mean of the variance of the asset price.
Constraint:
.
- 13:
grisk – doubleInput
On entry: the risk aversion parameter, , of the representative agent.
Constraint:
and .
- 14:
r – doubleInput
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint:
.
- 15:
q – doubleInput
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint:
.
- 16:
p[] – doubleOutput
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: contains , the option price evaluated for the strike price at expiry for and .
- 17:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument 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, .
Constraint: .
On entry, .
Constraint: .
- 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.
- NE_REAL
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, , and .
Constraint: and .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: and .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_REAL_ARRAY
-
On entry, .
Constraint: .
On entry, .
Constraint: and .
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
, where
is the absolute value of the integral.
8 Parallelism and Performance
nag_heston_price (s30nac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
Users' Note for your implementation for any additional implementation-specific information.
None.
10 Example
This example computes the price of a European call using Heston's stochastic volatility model. The time to expiry is
months, the stock price is
and the strike price is
. The risk-free interest rate is
per year, the volatility of the variance,
, is
per year, the mean reversion parameter,
, is
, the long term mean of the variance,
, is
and the correlation between the volatility process and the stock price process,
, is
. The risk aversion parameter,
, is
and the initial value of the variance,
var0, is
.
10.1 Program Text
Program Text (s30nace.c)
10.2 Program Data
Program Data (s30nace.d)
10.3 Program Results
Program Results (s30nace.r)