naginterfaces.library.specfun.opt_​bsm_​greeks

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

opt_bsm_greeks computes the European option price given by the Black–Scholes–Merton formula together with its sensitivities (Greeks).

For full information please refer to the NAG Library document for s30ab

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s30abf.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 $j=1,2,\dots ,n$.

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

The leading $m\times n$ part of the array $delta$ contains the sensitivity, $\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.

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

The leading $m\times n$ part of the array $gamma$ contains the sensitivity, $\frac{{\partial }^{2}P}{\partial S^2}$, of $delta$ to change in the price of the underlying asset.

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

$vega[i-1,j-1]$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in the volatility of the underlying asset, i.e., $\frac{\partial P_{ij}}{\partial \sigma }$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

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

$theta[i-1,j-1]$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in time, i.e., $-\frac{\partial P_{ij}}{\partial T}$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$, where $b=r-q$.

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

$rho[i-1,j-1]$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in the annual risk-free interest rate, i.e., $-\frac{\partial P_{ij}}{\partial r}$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

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

$crho[i-1,j-1]$, contains the first-order Greek measuring the sensitivity of the option price $P_{ij}$ to change in the annual cost of carry rate, i.e., $-\frac{\partial P_{ij}}{\partial b}$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$, where $b=r-q$.

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

$vanna[i-1,j-1]$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the asset price, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^2{P}_{ij}}{\partial S\partial \sigma }$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

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

$charm[i-1,j-1]$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial T}=-\frac{{\partial }^2{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

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

$speed[i-1,j-1]$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the price of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial S}=-\frac{{\partial }^3{P}_{ij}}{\partial S^3}$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

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

$colour[i-1,j-1]$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the time, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial T}=-\frac{{\partial }^3{P}_{ij}}{\partial S\partial T}$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

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

$zomma[i-1,j-1]$, contains the third-order Greek measuring the sensitivity of the second-order Greek ${\Gamma }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Gamma }_{ij}}{\partial \sigma }=-\frac{{\partial }^3{P}_{ij}}{\partial S^2\partial \sigma }$, for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

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

$vomma[i-1,j-1]$, contains the second-order Greek measuring the sensitivity of the first-order Greek ${\Delta }_{ij}$ to change in the volatility of the underlying asset, i.e., $-\frac{\partial {\Delta }_{ij}}{\partial \sigma }=-\frac{{\partial }^2{P}_{ij}}{\partial {\sigma }^2}$, 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_greeks 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, $\sigma$, risk-free interest rate, $r$, and dividend yield, $q$, must be supplied as input. 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