# NAG CL Interfaces30abc (opt_​bsm_​greeks)

Settings help

CL Name Style:

## 1Purpose

s30abc computes the European option price given by the Black–Scholes–Merton formula together with its sensitivities (Greeks).

## 2Specification

 #include
 void s30abc (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigma, double r, double q, 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)
The function may be called by the names: s30abc, nag_specfun_opt_bsm_greeks or nag_bsm_greeks.

## 3Description

s30abc 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
 $Pcall = Se-qT Φ(d1) - Xe-rT Φ(d2)$
and the corresponding European put price is
 $Pput = Xe-rT Φ(-d2) - Se-qT Φ(-d1)$
and where $\Phi$ denotes the cumulative Normal distribution function,
 $Φ(x) = 12π ∫ -∞ x exp(-y2/2) dy$
and
 $d1 = ln (S/X) + (r-q+σ2/2) T σ⁢T , d2 = d1 - σ⁢T .$
The option price ${P}_{ij}=P\left(X={X}_{i},T={T}_{j}\right)$ is computed for each strike price in a set ${X}_{i}$, $i=1,2,\dots ,m$, and for each expiry time in a set ${T}_{j}$, $j=1,2,\dots ,n$.

## 4References

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

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{option}$Nag_CallPut Input
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: $\mathbf{m}$Integer Input
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
4: $\mathbf{n}$Integer Input
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
5: $\mathbf{x}\left[{\mathbf{m}}\right]$const double Input
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: $\mathbf{s}$double Input
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: $\mathbf{t}\left[{\mathbf{n}}\right]$const double Input
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 $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
8: $\mathbf{sigma}$double Input
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: $\mathbf{r}$double Input
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: $\mathbf{q}$double Input
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.
Constraint: ${\mathbf{q}}\ge 0.0$.
11: $\mathbf{p}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{P}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{p}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{P}}\left(i,j\right)$ contains ${P}_{ij}$, the option price evaluated for the strike price ${{\mathbf{x}}}_{i}$ at expiry ${{\mathbf{t}}}_{j}$ for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
12: $\mathbf{delta}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
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: $\mathbf{gamma}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
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: $\mathbf{vega}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{VEGA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vega}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{VEGA}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
15: $\mathbf{theta}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{THETA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{theta}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{THETA}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$, where $b=r-q$.
16: $\mathbf{rho}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{RHO}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{rho}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{RHO}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
17: $\mathbf{crho}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{CRHO}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{crho}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{CRHO}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$, where $b=r-q$.
18: $\mathbf{vanna}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{VANNA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vanna}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{VANNA}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
19: $\mathbf{charm}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{CHARM}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{charm}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{CHARM}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
20: $\mathbf{speed}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{SPEED}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{speed}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{SPEED}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
21: $\mathbf{colour}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{COLOUR}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{colour}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{COLOUR}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
22: $\mathbf{zomma}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{ZOMMA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{zomma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{ZOMMA}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
23: $\mathbf{vomma}\left[{\mathbf{m}}×{\mathbf{n}}\right]$double Output
Note: where ${\mathbf{VOMMA}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vomma}}\left[\left(j-1\right)×{\mathbf{m}}+i-1\right]$ 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: ${\mathbf{VOMMA}}\left(i,j\right)$, 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 $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
24: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{q}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{q}}\ge 0.0$.
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 ${\mathbf{x}}\left[i\right]\le ⟨\mathit{\text{value}}⟩$.

## 7Accuracy

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 s15abc and s15adc). An accuracy close to machine precision can generally be expected.

## 8Parallelism and Performance

s30abc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example computes the price of a European put with a time to expiry of $0.7$ years, a stock price of $55$ and a strike price of $60$. The risk-free interest rate is $10%$ per year and the volatility is $30%$ per year.

### 10.1Program Text

Program Text (s30abce.c)

### 10.2Program Data

Program Data (s30abce.d)

### 10.3Program Results

Program Results (s30abce.r)