nag_bsm_greeks (s30abc) computes the price of a European call (or put) 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, by the Black–Scholes–Merton formula (see
Black and Scholes (1973) and
Merton (1973)). The annual volatility,
, risk-free interest rate,
, and dividend yield,
, must be supplied as input. For a given strike price,
, the price of a European call with underlying price,
, and time to expiry,
, is
and the corresponding European put price is
and where
denotes the cumulative Normal distribution function,
and
- 1:
– 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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– 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:
– IntegerInput
-
On entry: the number of strike prices to be used.
Constraint:
.
- 4:
– IntegerInput
-
On entry: the number of times to expiry to be used.
Constraint:
.
- 5:
– const doubleInput
-
On entry: must contain
, the th strike price, for .
Constraint:
, where , the safe range parameter, for .
- 6:
– doubleInput
-
On entry: , the price of the underlying asset.
Constraint:
, where , the safe range parameter.
- 7:
– const doubleInput
-
On entry: must contain
, the th time, in years, to expiry, for .
Constraint:
, where , the safe range parameter, for .
- 8:
– doubleInput
-
On entry: , the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint:
.
- 9:
– doubleInput
-
On entry: , the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint:
.
- 10:
– doubleInput
-
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint:
.
- 11:
– 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 .
- 12:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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:
– doubleOutput
-
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 3.7 in How to Use the NAG Library and its Documentation).
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
nag_cumul_normal (s15abc) and
nag_erfc (s15adc)). An accuracy close to
machine precision can generally be expected.
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.