naginterfaces.library.specfun.opt_​bsm_​price

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

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://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s30aaf.html

Parameters

calputstr, length 1

Determines whether the option is a call or a put.

$calput=\text{‘C'}$

A call; the holder has a right to buy.

$calput=\text{‘P'}$

A put; the holder has a right to sell.

xfloat, array-like, shape $\left(m\right)$

$x[i-1]$ must contain $X_{\text{i}}$, the $\text{i}$th strike price, for $\text{i}=1,2,\dots ,m$.

sfloat

$S$, the price of the underlying asset.

tfloat, array-like, shape $\left(n\right)$

$t[i-1]$ must contain $T_{\text{i}}$, the $\text{i}$th time, in years, to expiry, for $\text{i}=1,2,\dots ,n$.

sigmafloat

$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as $0.15$.

rfloat

$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as $0.05$.

qfloat

$q$, the annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.

Returns

pfloat, ndarray, shape $(m,n)$

$p[i-1,j-1]$ contains $P_{ij}$, the option price evaluated for the strike price $x_i$ at expiry $t_j$ for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

Raises

NagValueError

(errno $1$)

On entry, $calput=⟨value⟩$.

Constraint: $calput=\text{‘C'}\text{or}\text{‘P'}$.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $3$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $4$)

On entry, $x[⟨value⟩]=⟨value⟩$.

Constraint: $x\left[i\right]\ge ⟨value⟩$ and $x\left[i\right]\le ⟨value⟩$.

(errno $5$)

On entry, $s=⟨value⟩$.

Constraint: $s\ge ⟨value⟩$ and $s\le ⟨value⟩$.

(errno $6$)

On entry, $t[⟨value⟩]=⟨value⟩$.

Constraint: $t\left[i\right]\ge ⟨value⟩$.

(errno $7$)

On entry, $sigma=⟨value⟩$.

Constraint: $sigma>0.0$.

(errno $8$)

On entry, $r=⟨value⟩$.

Constraint: $r\ge 0.0$.

(errno $9$)

On entry, $q=⟨value⟩$.

Constraint: $q\ge 0.0$.

Notes

opt_bsm_price computes the price of a European call (or put) option for constant volatility, $\sigma$, and risk-free interest rate, $r$, with possible dividend yield, $q$, using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). For a given strike price, $X$, the price of a European call with underlying price, $S$, and time to expiry, $T$, is

$$P_{call}=S{e}^{-qT}\Phi \left(d_1\right)-X{e}^{-rT}\Phi \left(d_2\right)$$

and the corresponding European put price is

$$P_{put}=X{e}^{-rT}\Phi \left(-d_2\right)-S{e}^{-qT}\Phi \left(-d_1\right)$$

and where $\Phi$ denotes the cumulative Normal distribution function,

$$\Phi \left(x\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{x}exp(-y^2/2)dy$$

and

$$\begin{array}{c}\begin{array}{c}{d}_1=\frac{ln\left(S/X\right)+(r-q+{\sigma }^2/2)T}{\sigma \sqrt{T}}\text{,}\\ \\ d_2=d_1-\sigma \sqrt{T}\text{.}\end{array}\end{array}$$

The option price $P_{ij}=P(X=X_i,T=T_j)$ is computed for each strike price in a set $X_i$, $i=1,2,\dots ,m$, and for each expiry time in a set $T_j$, $j=1,2,\dots ,n$.

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

See also

naginterfaces.library.examples.roots.contfn_brent_interval_ex.main()

naginterfaces.library.examples.specfun.opt_bsm_price_ex.main()