nag_amer_bs_price (s30qcc) computes the
Bjerksund and Stensland (2002) approximation to the price of an American option.
nag_amer_bs_price (s30qcc) computes the price of an American option using the closed form approximation of
Bjerksund and Stensland (2002). The time to maturity,
, is divided into two periods, each with a flat early exercise boundary, by choosing a time
, such that
. The two boundary values are defined as
,
with
where
with
, the cost of carry, where
is the risk-free interest rate and
is the annual dividend rate. Here
is the strike price and
is the annual volatility.
The price of an American call option is approximated as
where
,
and
are as defined in
Bjerksund and Stensland (2002).
The price of a put option is obtained by the put-call transformation,
Bjerksund P and Stensland G (2002) Closed form valuation of American options
Discussion Paper 2002/09 NHH Bergen Norway http://www.nhh.no/
Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and probabilities Statistics and Computing 14 151–160
- 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 and
where
is as defined in
Section 3.
- 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:
sigma – doubleInput
On entry: , the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint:
.
- 9:
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:
.
- 10:
q – doubleInput
On entry: , the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint:
.
- 11:
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 .
- 12:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- 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, .
Constraint: and .
On entry, and .
Constraint: .
On entry, .
Constraint: .
- NE_REAL_ARRAY
-
On entry, .
Constraint: .
On entry, .
Constraint: and .
The accuracy of the output will be bounded by the accuracy of the cumulative bivariate Normal distribution function. The algorithm of
Genz (2004) is used, as described in the document for
nag_bivariate_normal_dist (g01hac), giving a maximum absolute error of less than
. The univariate cumulative Normal distribution function also forms part of the evaluation (see
nag_cumul_normal (s15abc) and
nag_erfc (s15adc)).
nag_amer_bs_price (s30qcc) 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.