s Chapter Contents
s Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_asian_geom_greeks (s30sbc)

## 1  Purpose

nag_asian_geom_greeks (s30sbc) computes the Asian geometric continuous average-rate option price together with its sensitivities (Greeks).

## 2  Specification

 #include #include
 void nag_asian_geom_greeks (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigma, double r, double b, double p[], double delta[], double gamma[], double vega[], double theta[], double rho[], double crho[], double vanna[], double charm[], double speed[], double colour[], double zomma[], double vomma[], NagError *fail)

## 3  Description

nag_asian_geom_greeks (s30sbc) 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
 $Pcall = S e b--r T Φ d- 1 - X e-rT Φ d- 2 ,$
and the corresponding put option price is
 $Pput = X e-rT Φ -d-2 - S e b--r T Φ - d-1 ,$
where
 $d-1 = lnS/X + b- + σ-2 / 2 T σ- T$
and
 $d-2 = d-1 - σ- T ,$
with
 $σ- = σ 3 , b- = 1 2 b- σ2 6 .$
$\Phi$ is the cumulative Normal distribution function,
 $Φx = 1 2π ∫ -∞ x exp -y2/2 dy .$

## 4  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

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     optionNag_CallPutInput
On entry: determines whether the option is a call or a put.
${\mathbf{option}}=\mathrm{Nag_Call}$
A call. The holder has a right to buy.
${\mathbf{option}}=\mathrm{Nag_Put}$
A put. The holder has a right to sell.
Constraint: ${\mathbf{option}}=\mathrm{Nag_Call}$ or $\mathrm{Nag_Put}$.
3:     mIntegerInput
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
4:     nIntegerInput
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
5:     x[m]const doubleInput
On entry: ${\mathbf{x}}\left[i-1\right]$ must contain ${X}_{\mathit{i}}$, the $\mathit{i}$th strike price, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: ${\mathbf{x}}\left[\mathit{i}-1\right]\ge z\text{​ and ​}{\mathbf{x}}\left[\mathit{i}-1\right]\le 1/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
6:     sdoubleInput
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le 1.0/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter.
7:     t[n]const doubleInput
On entry: ${\mathbf{t}}\left[i-1\right]$ must contain ${T}_{\mathit{i}}$, the $\mathit{i}$th time, in years, to expiry, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{t}}\left[\mathit{i}-1\right]\ge z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
On entry: $\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: ${\mathbf{sigma}}>0.0$.
9:     rdoubleInput
On entry: $r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: ${\mathbf{r}}\ge 0.0$.
10:   bdoubleInput
On entry: $b$, the annual cost of carry rate. Note that a rate of 8% should be entered as $0.08$.
11:   p[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix $P$ is stored in
• ${\mathbf{p}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{p}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array p contains the computed option prices.
12:   delta[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{delta}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{delta}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array delta contains the sensitivity, $\frac{\partial P}{\partial S}$, of the option price to change in the price of the underlying asset.
13:   gamma[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{gamma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{gamma}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array gamma contains the sensitivity, $\frac{{\partial }^{2}P}{\partial {S}^{2}}$, of delta to change in the price of the underlying asset.
14:   vega[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vega}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vega}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array vega contains the sensitivity, $\frac{\partial P}{\partial \sigma }$, of the option price to change in the volatility of the underlying asset.
15:   theta[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{theta}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{theta}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array theta contains the sensitivity, $-\frac{\partial P}{\partial T}$, of the option price to change in the time to expiry of the option.
16:   rho[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{rho}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{rho}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array rho contains the sensitivity, $\frac{\partial P}{\partial r}$, of the option price to change in the annual risk-free interest rate.
17:   crho[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{crho}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{crho}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array crho containing the sensitivity, $\frac{\partial P}{\partial b}$, of the option price to change in the annual cost of carry rate, $b$.
18:   vanna[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vanna}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vanna}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array vanna contains the sensitivity, $\frac{{\partial }^{2}P}{\partial S\partial \sigma }$, 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.
19:   charm[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{charm}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{charm}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array charm contains the sensitivity, $-\frac{{\partial }^{2}P}{\partial S\partial T}$, of delta to change in the time to expiry of the option.
20:   speed[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{speed}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{speed}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array speed contains the sensitivity, $\frac{{\partial }^{3}P}{\partial {S}^{3}}$, of gamma to change in the price of the underlying asset.
21:   colour[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{colour}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{colour}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array colour contains the sensitivity, $-\frac{{\partial }^{3}P}{\partial {S}^{2}\partial T}$, of gamma to change in the time to expiry of the option.
22:   zomma[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{zomma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{zomma}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array zomma contains the sensitivity, $\frac{{\partial }^{3}P}{\partial {S}^{2}\partial \sigma }$, of gamma to change in the volatility of the underlying asset.
23:   vomma[${\mathbf{m}}×{\mathbf{n}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vomma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vomma}}\left[\left(i-1\right)×{\mathbf{n}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $m×n$ array vomma contains the sensitivity, $\frac{{\partial }^{2}P}{\partial {\sigma }^{2}}$, of vega to change in the volatility of the underlying asset.
24:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{r}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{r}}\ge 0.0$.
On entry, ${\mathbf{s}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{s}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{s}}\le 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{sigma}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sigma}}>0.0$.
NE_REAL_ARRAY
On entry, ${\mathbf{t}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{t}}\left[i\right]\ge 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{x}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\left[i\right]\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[i\right]\le 〈\mathit{\text{value}}〉$.

## 7  Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, $\Phi$. 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_cumul_normal (s15abc) and nag_erfc (s15adc)). An accuracy close to machine precision can generally be expected.

None.

## 9  Example

This example computes the price of an Asian geometric continuous average-rate call with a time to expiry of $3$ months, a stock price of $80$ and a strike price of $97$. The risk-free interest rate is $5%$ per year, the cost of carry is $8%$ and the volatility is $20%$ per year.

### 9.1  Program Text

Program Text (s30sbce.c)

### 9.2  Program Data

Program Data (s30sbce.d)

### 9.3  Program Results

Program Results (s30sbce.r)