s Chapter Contents
s Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_bsm_greeks (s30abc)

## 1  Purpose

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

## 2  Specification

 #include #include
 void nag_bsm_greeks (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)

## 3  Description

nag_bsm_greeks (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
 $Pcall = Se-qT Φd1 - Xe-rT Φd2$
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,
 $Φx = 12π ∫ -∞ x exp -y2/2 dy$
and
 $d1 = ln S/X + r-q+ σ2 / 2 T σ⁢T , d2 = d1 - σ⁢T .$

## 4  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

## 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:   qdoubleInput
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:   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, $b$, where $b=r-q$.
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, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
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 ${\mathbf{x}}\left[i\right]\le 〈\mathit{\text{value}}〉$.

## 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 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.

### 9.1  Program Text

Program Text (s30abce.c)

### 9.2  Program Data

Program Data (s30abce.d)

### 9.3  Program Results

Program Results (s30abce.r)