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NAG C Library Manual

# NAG Library Function Documentnag_barrier_std_price (s30fac)

## 1  Purpose

nag_barrier_std_price (s30fac) computes the price of a standard barrier option.

## 2  Specification

 #include #include
 void nag_barrier_std_price (Nag_OrderType order, Nag_CallPut option, Nag_Barrier itype, 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)

## 3  Description

nag_barrier_std_price (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.
 $A = j S e-qT Φ jx1 - j X e-rT Φ j x1 - σ⁢T B = j S e-qT Φ j x2 - j X e-rT Φ j x2 - σ⁢T C = j S e-qT HS 2 μ+1 Φ ky1 - j X e-rT HS 2μ Φ k y1 - σ⁢T D = j S e-qT HS 2μ+1 Φ ky2 - j X e-rT HS 2μ Φ k y2 - σ⁢T E = K e-rT Φ k x2 - σ⁢T - HS 2μ Φ k y2 - σ⁢T F = K HS μ+λ Φ kz + HS μ-λ Φ k z-σ⁢T$
with
 $x1 = ln S/X σ⁢T + 1+μ σ⁢T x2 = ln S/H σ⁢T + 1+μ σ⁢T y1 = ln H2 / SX σ⁢T + 1+μσ⁢T y2 = lnH/S σ⁢T + 1+μσ⁢T z = lnH/S σ⁢T + λσ⁢T μ = r-q-σ 2 / 2 σ2 λ = μ2 + 2r σ2$
and where $\Phi$ denotes the cumulative Normal distribution function,
 $Φx = 12π ∫ -∞ x exp -y2/2 dy .$
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 $j=k=-1$,
 $Pput = A - C + F .$

## 4  References

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

## 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:     itypeNag_BarrierInput
On entry: indicates the barrier type as In or Out and its relation to the price of the underlying asset as Up or Down.
${\mathbf{itype}}=\mathrm{Nag_DownandIn}$
Down-and-In.
${\mathbf{itype}}=\mathrm{Nag_DownandOut}$
Down-and-Out.
${\mathbf{itype}}=\mathrm{Nag_UpandIn}$
Up-and-In.
${\mathbf{itype}}=\mathrm{Nag_UpandOut}$
Up-and-Out.
Constraint: ${\mathbf{itype}}=\mathrm{Nag_DownandIn}$, $\mathrm{Nag_DownandOut}$, $\mathrm{Nag_UpandIn}$ or $\mathrm{Nag_UpandOut}$.
4:     mIntegerInput
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
5:     nIntegerInput
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
6:     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 $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
7:     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.
8:     hdoubleInput
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:     kdoubleInput
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:   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: ${\mathbf{sigma}}>0.0$.
12:   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: ${\mathbf{r}}\ge 0.0$.
13:   qdoubleInput
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: ${\mathbf{q}}\ge 0.0$.
14:   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.
15:   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_ENUM_REAL_2
On entry, s and h are inconsistent with itype: ${\mathbf{s}}=〈\mathit{\text{value}}〉$ 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.
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 ${\mathbf{x}}\left[i\right]\le 〈\mathit{\text{value}}〉$.

## 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 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.

### 9.1  Program Text

Program Text (s30face.c)

### 9.2  Program Data

Program Data (s30face.d)

### 9.3  Program Results

Program Results (s30face.r)