naginterfaces.library.specfun.opt_​bsm_​price¶

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

opt_bsm_price computes the European option price given by the Black–Scholes–Merton formula.

For full information please refer to the NAG Library document for s30aa

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s30aaf.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: .

Notes

opt_bsm_price computes the price of a European call (or put) option for constant volatility, , and risk-free interest rate, , with possible dividend yield, , using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). 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

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

References

Black, F and Scholes, M, 1973, The pricing of options and corporate liabilities, Journal of Political Economy (81), 637–654

Merton, R C, 1973, Theory of rational option pricing, Bell Journal of Economics and Management Science (4), 141–183