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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_opt_asian_geom_greeks (s30sb)

## Purpose

nag_specfun_opt_asian_geom_greeks (s30sb) computes the Asian geometric continuous average-rate option price together with its sensitivities (Greeks).

## Syntax

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30sb(calput, x, s, t, sigma, r, b, 'm', m, 'n', n)
[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = nag_specfun_opt_asian_geom_greeks(calput, x, s, t, sigma, r, b, 'm', m, 'n', n)

## Description

nag_specfun_opt_asian_geom_greeks (s30sb) 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, σ$\sigma$, risk-free rate, r$r$, and cost of carry, b$b$, are constants (see Kemna and Vorst (1990)). For a given strike price, X$X$, the price of a call option with underlying price, S$S$, and time to expiry, T$T$, is
 Pcall = S e (b − r) T Φ (d1) − X e − rT Φ (d2) , $Pcall = S e ( b--r ) T Φ ( d- 1 ) - X e-rT Φ ( d- 2 ) ,$
and the corresponding put option price is
 Pput = X e − rT Φ ( − d2) − S e (b − r) T Φ ( − d1) , $Pput = X e-rT Φ ( -d-2 ) - S e ( b--r ) T Φ ( - d-1 ) ,$
where
 d1 = ( ln(S / X) + (b + σ2 / 2) T )/( σ sqrt(T) ) $d-1 = ln(S/X) + ( b- + σ-2 / 2 ) T σ- T$
and
 d2 = d1 − σ sqrt(T) , $d-2 = d-1 - σ- T ,$
with
 σ = σ/(sqrt(3)) ,  b = (1/2) (b − (σ2)/6) . $σ- = σ 3 , b- = 1 2 ( b- σ2 6 ) .$
Φ$\Phi$ is the cumulative Normal distribution function,
 Φ(x) = 1/(sqrt(2π)) ∫ − ∞x exp( − y2 / 2) dy . $Φ(x) = 1 2π ∫ -∞ x exp( -y2/2 ) dy .$

## 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

## Parameters

### Compulsory Input Parameters

1:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
calput = 'C'${\mathbf{calput}}=\text{'C'}$
A call. The holder has a right to buy.
calput = 'P'${\mathbf{calput}}=\text{'P'}$
A put. The holder has a right to sell.
Constraint: calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
2:     x(m) – double array
m, the dimension of the array, must satisfy the constraint m1${\mathbf{m}}\ge 1$.
x(i)${\mathbf{x}}\left(i\right)$ must contain Xi${X}_{\mathit{i}}$, the i$\mathit{i}$th strike price, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: x(i)z ​ and ​ x(i) 1 / z ${\mathbf{x}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{x}}\left(\mathit{i}\right)\le 1/z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     s – double scalar
S$S$, the price of the underlying asset.
Constraint: sz ​ and ​s1.0 / z${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
4:     t(n) – double array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
t(i)${\mathbf{t}}\left(i\right)$ must contain Ti${T}_{\mathit{i}}$, the i$\mathit{i}$th time, in years, to expiry, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: t(i)z${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     sigma – double scalar
σ$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma > 0.0${\mathbf{sigma}}>0.0$.
6:     r – double scalar
r$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0${\mathbf{r}}\ge 0.0$.
7:     b – double scalar
b$b$, the annual cost of carry rate. Note that a rate of 8% should be entered as 0.08$0.08$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array x.
The number of strike prices to be used.
Constraint: m1${\mathbf{m}}\ge 1$.
2:     n – int64int32nag_int scalar
Default: The dimension of the array t.
The number of times to expiry to be used.
Constraint: n1${\mathbf{n}}\ge 1$.

ldp

### Output Parameters

1:     p(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array p contains the computed option prices.
2:     delta(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array delta contains the sensitivity, (P)/(S)$\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.
3:     gamma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array gamma contains the sensitivity, (2P)/(S2)$\frac{{\partial }^{2}P}{\partial {S}^{2}}$, of delta to change in the price of the underlying asset.
4:     vega(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vega contains the sensitivity, (P)/(σ)$\frac{\partial P}{\partial \sigma }$, of the option price to change in the volatility of the underlying asset.
5:     theta(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array theta contains the sensitivity, (P)/(T)$-\frac{\partial P}{\partial T}$, of the option price to change in the time to expiry of the option.
6:     rho(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array rho contains the sensitivity, (P)/(r)$\frac{\partial P}{\partial r}$, of the option price to change in the annual risk-free interest rate.
7:     crho(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The m × n$m×n$ array crho containing the sensitivity, (P)/(b)$\frac{\partial P}{\partial b}$, of the option price to change in the annual cost of carry rate, b$b$.
8:     vanna(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vanna contains the sensitivity, (2P)/(Sσ)$\frac{{\partial }^{2}P}{\partial S\partial \sigma }$, of vega to change in the price of the underlying asset or, equivalently, the sensitivity of delta to change in the volatility of the asset price.
9:     charm(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array charm contains the sensitivity, (2P)/(S T)$-\frac{{\partial }^{2}P}{\partial S\partial T}$, of delta to change in the time to expiry of the option.
10:   speed(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array speed contains the sensitivity, (3P)/(S3)$\frac{{\partial }^{3}P}{\partial {S}^{3}}$, of gamma to change in the price of the underlying asset.
11:   colour(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array colour contains the sensitivity, (3P)/(S2 T)$-\frac{{\partial }^{3}P}{\partial {S}^{2}\partial T}$, of gamma to change in the time to expiry of the option.
12:   zomma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array zomma contains the sensitivity, (3P)/(S2σ)$\frac{{\partial }^{3}P}{\partial {S}^{2}\partial \sigma }$, of gamma to change in the volatility of the underlying asset.
13:   vomma(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array vomma contains the sensitivity, (2P)/(σ2)$\frac{{\partial }^{2}P}{\partial {\sigma }^{2}}$, of vega to change in the volatility of the underlying asset.
14:   ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
ifail = 2${\mathbf{ifail}}=2$
On entry, m0${\mathbf{m}}\le 0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, n0${\mathbf{n}}\le 0$.
ifail = 4${\mathbf{ifail}}=4$
On entry, x(i) < z${\mathbf{x}}\left(\mathit{i}\right) or x(i) > 1 / z${\mathbf{x}}\left(\mathit{i}\right)>1/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 5${\mathbf{ifail}}=5$
On entry, s < z${\mathbf{s}} or s > 1.0 / z${\mathbf{s}}>1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 6${\mathbf{ifail}}=6$
On entry, t(i) < z${\mathbf{t}}\left(\mathit{i}\right), where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 7${\mathbf{ifail}}=7$
On entry, sigma0.0${\mathbf{sigma}}\le 0.0$.
ifail = 8${\mathbf{ifail}}=8$
On entry, r < 0.0${\mathbf{r}}<0.0$.
ifail = 11${\mathbf{ifail}}=11$
On entry, ldp < m$\mathit{ldp}<{\mathbf{m}}$.

## Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, Φ$\Phi$. This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the machine precision (see nag_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

None.

## Example

function nag_specfun_opt_asian_geom_greeks_example
put = 'C';
s = 80.0;
sigma = 0.2;
r = 0.05;
b = 0.08;
x = [97.0];
t = [0.25];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, ...
zomma, vomma, ifail] = nag_specfun_opt_asian_geom_greeks(put, x, s, t, sigma, r, b);

fprintf('\nAsian Option: Geometric Continuous Average-Rate\nAsian Call :\n');
fprintf('  Spot          =   %9.4f\n', s);
fprintf('  Volatility    =   %9.4f\n', sigma);
fprintf('  Rate          =   %9.4f\n', r);
fprintf('  Cost of carry =   %9.4f\n\n', b);

fprintf(' Time to Expiry : %8.4f\n', t(1));
fprintf(' Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
p(1,1), delta(1,1), gamma(1,1), vega(1,1), theta(1,1), rho(1,1), ...
crho(1,1));

fprintf('\n Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
p(1,1), vanna(1,1), charm(1,1), speed(1,1), colour(1,1), ...
zomma(1,1), vomma(1,1));

Asian Option: Geometric Continuous Average-Rate
Asian Call :
Spot          =     80.0000
Volatility    =      0.2000
Rate          =      0.0500
Cost of carry =      0.0800

Time to Expiry :   0.2500
Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
97.0000   0.0010   0.0008   0.0006   0.0638  -0.0281   0.0079   0.0081

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
97.0000   0.0010   0.0443  -0.0196   0.0004  -0.0122   0.0272   3.1893

function s30sb_example
put = 'C';
s = 80.0;
sigma = 0.2;
r = 0.05;
b = 0.08;
x = [97.0];
t = [0.25];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, ...
zomma, vomma, ifail] = s30sb(put, x, s, t, sigma, r, b);

fprintf('\nAsian Option: Geometric Continuous Average-Rate\nAsian Call :\n');
fprintf('  Spot          =   %9.4f\n', s);
fprintf('  Volatility    =   %9.4f\n', sigma);
fprintf('  Rate          =   %9.4f\n', r);
fprintf('  Cost of carry =   %9.4f\n\n', b);

fprintf(' Time to Expiry : %8.4f\n', t(1));
fprintf(' Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
p(1,1), delta(1,1), gamma(1,1), vega(1,1), theta(1,1), rho(1,1), ...
crho(1,1));

fprintf('\n Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
p(1,1), vanna(1,1), charm(1,1), speed(1,1), colour(1,1), ...
zomma(1,1), vomma(1,1));

Asian Option: Geometric Continuous Average-Rate
Asian Call :
Spot          =     80.0000
Volatility    =      0.2000
Rate          =      0.0500
Cost of carry =      0.0800

Time to Expiry :   0.2500
Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
97.0000   0.0010   0.0008   0.0006   0.0638  -0.0281   0.0079   0.0081

Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
97.0000   0.0010   0.0443  -0.0196   0.0004  -0.0122   0.0272   3.1893