s30nbc computes the price and sensitivities 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.
Writing the expression for the price of a call option as
then the sensitivities or Greeks can be obtained in the following manner,
- Delta
-
- Vega
-
- Rho
-
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/
-
1:
– 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
. 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:
or .
-
2:
– Nag_CallPut
Input
-
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:
– Integer
Input
-
On entry: the number of strike prices to be used.
Constraint:
.
-
4:
– Integer
Input
-
On entry: the number of times to expiry to be used.
Constraint:
.
-
5:
– const double
Input
-
On entry: must contain
, the th strike price, for .
Constraint:
, where , the safe range parameter, for .
-
6:
– double
Input
-
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
-
7:
– const double
Input
-
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
-
8:
– double
Input
-
On entry: the volatility, , of the volatility process, . Note that a rate of 20% should be entered as .
Constraint:
.
-
9:
– double
Input
-
On entry: , the long term mean reversion rate of the volatility.
Constraint:
.
-
10:
– double
Input
-
On entry: the correlation between the two standard Brownian motions for the asset price and the volatility.
Constraint:
.
-
11:
– double
Input
-
On entry: the initial value of the variance, , of the asset price.
Constraint:
.
-
12:
– double
Input
-
On entry: , the long term mean of the variance of the asset price.
Constraint:
.
-
13:
– double
Input
-
On entry: the risk aversion parameter, , of the representative agent.
Constraint:
and .
-
14:
– double
Input
-
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as .
Constraint:
.
-
15:
– double
Input
-
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as .
Constraint:
.
-
16:
– double
Output
-
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:
– double
Output
-
Note: the
th element of the matrix is stored in
- when ;
- when .
On exit: the
array
delta contains the sensitivity,
, of the option price to change in the price of the underlying asset.
-
18:
– double
Output
-
Note: the
th element of the matrix is stored in
- when ;
- when .
On exit: the
array
gamma contains the sensitivity,
, of
delta to change in the price of the underlying asset.
-
19:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the first-order Greek measuring the sensitivity of the option price to change in the volatility of the underlying asset, i.e., , for and .
-
20:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the first-order Greek measuring the sensitivity of the option price to change in time, i.e., , for and , where .
-
21:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the first-order Greek measuring the sensitivity of the option price to change in the annual risk-free interest rate, i.e., , for and .
-
22:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the second-order Greek measuring the sensitivity of the first-order Greek to change in the volatility of the asset price, i.e., , for and .
-
23:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the second-order Greek measuring the sensitivity of the first-order Greek to change in the time, i.e., , for and .
-
24:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the third-order Greek measuring the sensitivity of the second-order Greek to change in the price of the underlying asset, i.e., , for and .
-
25:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the third-order Greek measuring the sensitivity of the second-order Greek to change in the volatility of the underlying asset, i.e., , for and .
-
26:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the second-order Greek measuring the sensitivity of the option price to second-order changes in the volatility of the underlying asset, i.e., , for and .
-
27:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
- 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.
- NE_BAD_PARAM
-
On entry, argument 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, .
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.
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, .
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 .
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.
Background information to multithreading can be found in the
Multithreading documentation.
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.
This example computes the price and sensitivities of a European call using Heston's stochastic volatility model. The time to expiry is
year, 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
.