# NAG CL Interfaces30fac (opt_​barrier_​std_​price)

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## 1Purpose

s30fac computes the price of a standard barrier option.

## 2Specification

 #include
 void s30fac (Nag_OrderType order, Nag_CallPut option, Nag_Barrier type, Integer m, Integer n, const double x[], double s, double h, double k, const double t[], double sigma, double r, double q, double p[], NagError *fail)
The function may be called by the names: s30fac, nag_specfun_opt_barrier_std_price or nag_barrier_std_price.

## 3Description

s30fac computes the price of a standard barrier option, where the exercise, for a given strike price, $X$, depends on the underlying asset price, $S$, reaching or crossing a specified barrier level, $H$. Barrier options of type In only become active (are knocked in) if the underlying asset price attains the pre-determined barrier level during the lifetime of the contract. Those of type Out start active and are knocked out if the underlying asset price attains the barrier level during the lifetime of the contract. A cash rebate, $K$, may be paid if the option is inactive at expiration. The option may also be described as Up (the underlying price starts below the barrier level) or Down (the underlying price starts above the barrier level). This gives the following options which can be specified as put or call contracts.
Down-and-In: the option starts inactive with the underlying asset price above the barrier level. It is knocked in if the underlying price moves down to hit the barrier level before expiration.
Down-and-Out: the option starts active with the underlying asset price above the barrier level. It is knocked out if the underlying price moves down to hit the barrier level before expiration.
Up-and-In: the option starts inactive with the underlying asset price below the barrier level. It is knocked in if the underlying price moves up to hit the barrier level before expiration.
Up-and-Out: the option starts active with the underlying asset price below the barrier level. It is knocked out if the underlying price moves up to hit the barrier level before expiration.
The payoff is $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(S-X,0\right)$ for a call or $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(X-S,0\right)$ for a put, if the option is active at expiration, otherwise it may pay a pre-specified cash rebate, $K$. Following Haug (2007), the prices of the various standard barrier options can be written as shown below. The volatility, $\sigma$, risk-free interest rate, $r$, and annualised dividend yield, $q$, are constants. The integer parameters, $j$ and $k$, take the values $±1$, depending on the type of barrier.
 $A = j S e-qT Φ (jx1) - j X e-rT Φ (j[x1-σ⁢T]) B = j S e-qT Φ (jx2) - j X e-rT Φ (j[x2-σ⁢T]) C = j S e-qT (HS) 2 ⁢ (μ+1) Φ (ky1) - j X e-rT (HS) 2⁢μ Φ (k[y1-σ⁢T]) D = j S e-qT (HS) 2⁢(μ+1) Φ (ky2) - j X e-rT (HS) 2⁢μ Φ (k[y2-σ⁢T]) E = K e-rT {Φ(k[x2-σ⁢T])- (HS) 2μ Φ(k[y2-σ⁢T])} F = K { (HS) μ+λ Φ(kz)+ (HS) μ-λ Φ(k[z-σ⁢T])}$
with
 $x1 = ln (S/X) σ⁢T + (1+μ) σ⁢T x2 = ln (S/H) σ⁢T + (1+μ) σ⁢T y1 = ln (H2/(SX)) σ⁢T + (1+μ)σ⁢T y2 = ln(H/S) σ⁢T + (1+μ)σ⁢T z = ln(H/S) σ⁢T + λσ⁢T μ = r-q-σ 2 / 2 σ2 λ = μ2 + 2r σ2$
and where $\Phi$ denotes the cumulative Normal distribution function,
 $Φ(x) = 12π ∫ -∞ x exp(-y2/2) dy .$
Down-and-In ($S>H$):
• When $X\ge H$, with $j=k=1$,
 $Pcall = C + E$
and with $j=-1$, $k=1$
 $Pput = B - C + D + E$
When $X, with $j=k=1$
 $Pcall = A - B + D + E$
and with $j=-1$, $k=1$
 $Pput = A + E .$
Down-and-Out ($S>H$):
• When $X\ge H$, with $j=k=1$,
 $Pcall = A-C + F$
and with $j=-1$, $k=1$
 $Pput = A - B + C - D + F$
When $X, with $j=k=1$,
 $Pcall = B - D + F$
and with $j=-1$, $k=1$
 $Pput = F .$
Up-and-In ($S):
• When $X\ge H$, with $j=1$, $k=-1$,
 $Pcall = A + E$
and with $j=k=-1$,
 $Pput = A - B + D + E$
When $X, with $j=1$, $k=-1$,
 $Pcall = B - C + D + E$
and with $j=k=-1$,
 $Pput = C + E .$
Up-and-Out ($S):
• When $X\ge H$, with $j=1$, $k=-1$,
 $Pcall = F$
and with $j=k=-1$,
 $Pput = B - D + F$
When $X, with $j=1$, $k=-1$,
 $Pcall = A - B + C - D + F$
and with $j=k=-1$,
 $Pput = A - C + F .$
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

Haug E G (2007) The Complete Guide to Option Pricing Formulas (2nd Edition) McGraw-Hill

## 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: $\mathbf{type}$Nag_Barrier Input
On entry: indicates the barrier type as In or Out and its relation to the price of the underlying asset as Up or Down.
${\mathbf{type}}=\mathrm{Nag_DownandIn}$
Down-and-In.
${\mathbf{type}}=\mathrm{Nag_DownandOut}$
Down-and-Out.
${\mathbf{type}}=\mathrm{Nag_UpandIn}$
Up-and-In.
${\mathbf{type}}=\mathrm{Nag_UpandOut}$
Up-and-Out.
Constraint: ${\mathbf{type}}=\mathrm{Nag_DownandIn}$, $\mathrm{Nag_DownandOut}$, $\mathrm{Nag_UpandIn}$ or $\mathrm{Nag_UpandOut}$.
4: $\mathbf{m}$Integer Input
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
5: $\mathbf{n}$Integer Input
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
6: $\mathbf{x}\left[{\mathbf{m}}\right]$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 $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
7: $\mathbf{s}$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.
8: $\mathbf{h}$double Input
On entry: the barrier price.
Constraint: ${\mathbf{h}}\ge z\text{​ and ​}{\mathbf{h}}\le 1/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter.
9: $\mathbf{k}$double Input
On entry: the value of a possible cash rebate to be paid if the option has not been knocked in (or out) before expiration.
Constraint: ${\mathbf{k}}\ge 0.0$.
10: $\mathbf{t}\left[{\mathbf{n}}\right]$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 $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
11: $\mathbf{sigma}$double Input
On entry: $\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as $0.15$.
Constraint: ${\mathbf{sigma}}>0.0$.
12: $\mathbf{r}$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: ${\mathbf{r}}\ge 0.0$.
13: $\mathbf{q}$double Input
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.
Constraint: ${\mathbf{q}}\ge 0.0$.
14: $\mathbf{p}\left[{\mathbf{m}}×{\mathbf{n}}\right]$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 $j=1,2,\dots ,{\mathbf{n}}$.
15: $\mathbf{fail}$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_ENUM_REAL_2
On entry, s and h are inconsistent with type: ${\mathbf{s}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{h}}=⟨\mathit{\text{value}}⟩$.
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, ${\mathbf{h}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{h}}\ge ⟨\mathit{\text{value}}⟩$ and ${\mathbf{h}}\le ⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{k}}\ge 0.0$.
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

s30fac 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 Down-and-In put with a time to expiry of $6$ months, a stock price of $100$ and a strike price of $100$. The barrier value is $95$ and there is a cash rebate of $3$, payable on expiry if the option has not been knocked in. The risk-free interest rate is $8%$ per year, there is an annual dividend return of $4%$ and the volatility is $30%$ per year.

### 10.1Program Text

Program Text (s30face.c)

### 10.2Program Data

Program Data (s30face.d)

### 10.3Program Results

Program Results (s30face.r)