naginterfaces.library.specfun.opt_​asian_​geom_​greeks

naginterfaces.library.specfun.opt_asian_geom_greeks(calput, x, s, t, sigma, r, b)

opt_asian_geom_greeks computes the Asian geometric continuous average-rate option price together with its sensitivities (Greeks).

For full information please refer to the NAG Library document for s30sb

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s30sbf.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 $0.05$.

bfloat

$b$, the annual cost of carry 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 $(m,n)$

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

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: $r\ge 0.0$.

Notes

opt_asian_geom_greeks computes the price of an Asian geometric continuous average-rate 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. The annual volatility, $\sigma$, risk-free rate, $r$, and cost of carry, $b$, are constants (see Kemna and Vorst (1990)). For a given strike price, $X$, the price of a call option with underlying price, $S$, and time to expiry, $T$, is

$$P_{call}=S{e}^{(\overline{b}-r)T}\Phi \left({\overline{d}}_1\right)-X{e}^{-rT}\Phi \left({\overline{d}}_2\right)\text{,}$$

and the corresponding put option price is

$$P_{put}=X{e}^{-rT}\Phi (-{\overline{d}}_2)-S{e}^{(\overline{b}-r)T}\Phi (-{\overline{d}}_1)\text{,}$$

where

$${\overline{d}}_1=\frac{ln\left(S/X\right)+(\overline{b}+{\overline{\sigma }}^2/2)T}{\overline{\sigma }\sqrt{T}}$$

and

$${\overline{d}}_2={\overline{d}}_1-\overline{\sigma }\sqrt{T}\text{,}$$

with

$$\overline{\sigma }=\frac{\sigma }{\sqrt{3}}\text{,}\overline{b}=\frac{1}{2}(b-\frac{{\sigma }^2}{6})\text{.}$$

$\Phi$ is the cumulative Normal distribution function,

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

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

Kemna, A and Vorst, A, 1990, A pricing method for options based on average asset values, Journal of Banking and Finance (14), 113–129