# naginterfaces.library.specfun.opt_​amer_​bs_​price¶

naginterfaces.library.specfun.opt_amer_bs_price(calput, x, s, t, sigma, r, q)[source]

opt_amer_bs_price computes the Bjerksund and Stensland (2002) approximation to the price of an American option.

For full information please refer to the NAG Library document for s30qc

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s30qcf.html

Parameters
calputstr, length 1

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.

xfloat, array-like, shape

must contain , the th strike price, for .

sfloat

, the price of the underlying asset.

tfloat, array-like, shape

must contain , the th time, in years, to expiry, for .

sigmafloat

, the volatility of the underlying asset. Note that a rate of 15% should be entered as .

rfloat

, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as .

qfloat

, the annual continuous yield rate. Note that a rate of 8% should be entered as .

Returns
pfloat, ndarray, shape

contains , the option price evaluated for the strike price at expiry for and .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: and .

(errno )

On entry, .

Constraint: and .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, and .

Constraint: .

Notes

opt_amer_bs_price 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,

The option price is computed for each strike price in a set , , and for each expiry time in a set , .

References

Bjerksund, P and Stensland, G, 2002, Closed form valuation of American options (Discussion Paper 2002/09), NHH Bergen Norway

Genz, A, 2004, Numerical computation of rectangular bivariate and trivariate Normal and probabilities, Statistics and Computing (14), 151–160