s30cdc computes the price of a binary or digital asset-or-nothing option, together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters. This option pays the underlying asset itself,
, at expiration if the option is in-the-money (see
Section 2.4 in the
S Chapter Introduction). For a strike price,
, underlying asset price,
, and time to expiry,
, the payoff is, therefore,
, if
for a call or
for a put. Nothing is paid out when this condition is not met.
The price of a call with volatility,
, risk-free interest rate,
, and annualised dividend yield,
, is
and for a put,
where
is the cumulative Normal distribution function,
and
-
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 underlying asset. Note that a rate of 15% should be entered as .
Constraint:
.
-
9:
– double
Input
-
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as .
Constraint:
.
-
10:
– double
Input
-
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as .
Constraint:
.
-
11:
– 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 .
-
12:
– 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.
-
13:
– 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.
-
14:
– 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 .
-
15:
– 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 .
-
16:
– 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 .
-
17:
– 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 cost of carry rate, i.e., , for and , where .
-
18:
– 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 .
-
19:
– 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 .
-
20:
– 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 .
-
21:
– 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 time, 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 third-order Greek measuring the sensitivity of the second-order Greek to change in the volatility of the underlying asset, 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 volatility of the underlying asset, i.e., , for and .
-
24:
– 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 dependent on the accuracy of the cumulative Normal distribution function,
. This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the
machine precision (see
s15abc and
s15adc). An accuracy close to
machine precision can generally be expected.
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 of an asset-or-nothing put with a time to expiry of days, a stock price of and a strike price of . The risk-free interest rate is per year, there is an annual dividend return of and the volatility is per year.