s30ndc 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)).
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
-
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:
– const double
Input
-
On entry: must contain
, the th annual risk-free interest rate, continuously compounded, for . Note that a rate of 5% should be entered as .
-
15:
– const double
Input
-
On entry: must contain
, the th annual continuous yield rate, for . Note that a rate of 8% should be entered as .
-
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 .
-
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:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the derivative measuring the sensitivity of the option price to change in the strick price, , i.e., , for and .
-
28:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the derivative measuring the sensitivity of the option price to change in the annual continuous yield rate, , i.e., , for and .
-
29:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the derivative measuring the sensitivity of the option price to change in the long term mean of the variance of the asset price, i.e., , for and .
-
30:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the derivative measuring the sensitivity of the option price to change in the long term mean reversion rate of the volatility, i.e., , for and .
-
31:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the derivative measuring the sensitivity of the option price to change in the volatility of the volatility process, , i.e., , for and .
-
32:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the derivative measuring the sensitivity of the option price to change in the correlation between the two standard Brownian motions for the asset price and the volatility, i.e., , for and .
-
33:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: , contains the derivative measuring the sensitivity of the option price to change in the risk aversion parameter of the representative agent,, i.e., , for and .
-
34:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
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.
None.
This example computes the price and sensitivities of four European calls using Heston's stochastic volatility model. In each case, the time to expiry is
year, the stock price is
, the strike price is
, the volatility of the variance (
) is
per year, the mean reversion parameter (
) is
, the long term mean of the variance (
) is
, 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
. The risk-free interest rate values for each call are
,
,
,
per year. The annual continuous yield rate values are
,
,
,
per year.