void	nag_barrier_std_price (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)

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

\max (S - X, 0)

for a call or

\max (X - S, 0)

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,

σ

, risk-free interest rate,

r

, and annualised dividend yield,

q

, are constants. The integer parameters,

j

and

k

, take the values

\pm 1

, depending on the type of barrier.

\begin{array}{l} A = j S e^{- q T} Φ (j x_{1}) - j X e^{- r T} Φ (j [x_{1} - σ \sqrt{T}]) \\ B = j S e^{- q T} Φ (j x_{2}) - j X e^{- r T} Φ (j [x_{2} - σ \sqrt{T}]) \\ C = j S e^{- q T} {(\frac{H}{S})}^{2 (μ + 1)} Φ (k y_{1}) - j X e^{- r T} {(\frac{H}{S})}^{2 μ} Φ (k [y_{1} - σ \sqrt{T}]) \\ D = j S e^{- q T} {(\frac{H}{S})}^{2 (μ + 1)} Φ (k y_{2}) - j X e^{- r T} {(\frac{H}{S})}^{2 μ} Φ (k [y_{2} - σ \sqrt{T}]) \\ E = K e^{- r T} \{Φ (k [x_{2} - σ \sqrt{T}]) - {(\frac{H}{S})}^{2 μ} Φ (k [y_{2} - σ \sqrt{T}])\} \\ F = K \{{(\frac{H}{S})}^{μ + λ} Φ (k z) + {(\frac{H}{S})}^{μ - λ} Φ (k [z - σ \sqrt{T}])\} \end{array}

with

\begin{array}{l} x_{1} = \frac{\ln (S / X)}{σ \sqrt{T}} + (1 + μ) σ \sqrt{T} \\ x_{2} = \frac{\ln (S / H)}{σ \sqrt{T}} + (1 + μ) σ \sqrt{T} \\ y_{1} = \frac{\ln (H^{2} / (S X))}{σ \sqrt{T}} + (1 + μ) σ \sqrt{T} \\ y_{2} = \frac{\ln (H / S)}{σ \sqrt{T}} + (1 + μ) σ \sqrt{T} \\ z = \frac{\ln (H / S)}{σ \sqrt{T}} + λ σ \sqrt{T} \\ μ = \frac{{r - q - σ}^{2} / 2}{σ^{2}} \\ λ = \sqrt{μ^{2} + \frac{2 r}{σ^{2}}} \end{array}

and where

Φ

denotes the cumulative Normal distribution function,

Φ (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x} \exp (- y^{2} / 2) d y .

Down-and-In ( $S > H$ ):

When $X \geq H$ , with $j = k = 1$ ,
$P_{call} = C + E$
and with $j = - 1$ , $k = 1$
$P_{put} = B - C + D + E$

When $X < H$ , with $j = k = 1$
$P_{call} = A - B + D + E$
and with $j = - 1$ , $k = 1$
$P_{put} = A + E .$

Down-and-Out ( $S > H$ ):

When $X \geq H$ , with $j = k = 1$ ,
$P_{call} = A - C + F$
and with $j = - 1$ , $k = 1$
$P_{put} = A - B + C - D + F$

When $X < H$ , with $j = k = 1$ ,
$P_{call} = B - D + F$
and with $j = - 1$ , $k = 1$
$P_{put} = F .$

Up-and-In ( $S < H$ ):

When $X \geq H$ , with $j = 1$ , $k = - 1$ ,
$P_{call} = A + E$
and with $j = k = - 1$ ,
$P_{put} = A - B + D + E$

When $X < H$ , with $j = 1$ , $k = - 1$ ,
$P_{call} = B - C + D + E$
and with $j = k = - 1$ ,
$P_{put} = C + E .$

Up-and-Out ( $S < H$ ):

When $X \geq H$ , with $j = 1$ , $k = - 1$ ,
$P_{call} = F$
and with $j = k = - 1$ ,
$P_{put} = B - D + F$

When $X < H$ , with $j = 1$ , $k = - 1$ ,
$P_{call} = A - B + C - D + F$
and with $j = k = - 1$ ,
$P_{put} = A - C + F .$

The option price

P_{i j} = P (X = X_{i}, T = T_{j})

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

4

References

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

5

Arguments

1: $order$ – Nag_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

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $option$ – Nag_CallPutInput

On entry: determines whether the option is a call or a put.

$option = Nag_Call$: A call; the holder has a right to buy.
$option = Nag_Put$: A put; the holder has a right to sell.

Constraint:

option = Nag_Call

Nag_Put

3: $type$ – Nag_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.

$type = Nag_DownandIn$: Down-and-In.
$type = Nag_DownandOut$: Down-and-Out.
$type = Nag_UpandIn$: Up-and-In.
$type = Nag_UpandOut$: Up-and-Out.

Constraint:

type = Nag_DownandIn

Nag_DownandOut

Nag_UpandIn

Nag_UpandOut

4: $m$ – IntegerInput

On entry: the number of strike prices to be used.

Constraint:

m \geq 1

5: $n$ – IntegerInput

On entry: the number of times to expiry to be used.

Constraint:

n \geq 1

6: $x [m]$ – const doubleInput

On entry:

x [i - 1]

must contain

X_{i}

, the

i

th strike price, for

i = 1, 2, \dots, m

Constraint:

x [i - 1] \geq z ​ and ​ x [i - 1] \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, m

7: $s$ – doubleInput

On entry:

S

, the price of the underlying asset.

Constraint:

s \geq z ​ and ​ s \leq 1.0 / z

, where

z = nag_real_safe_small_number

, the safe range parameter.

8: $h$ – doubleInput

On entry: the barrier price.

Constraint:

h \geq z ​ and ​ h \leq 1 / z

, where

z = nag_real_safe_small_number

, the safe range parameter.

9: $k$ – doubleInput

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:

k \geq 0.0

10: $t [n]$ – const doubleInput

On entry:

t [i - 1]

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, n

Constraint:

t [i - 1] \geq z

, where

z = nag_real_safe_small_number

, the safe range parameter, for

i = 1, 2, \dots, n

11: $sigma$ – doubleInput

On entry:

σ

, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.

Constraint:

sigma > 0.0

12: $r$ – doubleInput

On entry:

r

, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.

Constraint:

r \geq 0.0

13: $q$ – doubleInput

On entry:

q

, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.

Constraint:

q \geq 0.0

14: $p [m \times n]$ – doubleOutput

Note: where

P (i, j)

appears in this document, it refers to the array element

$p [(j - 1) \times m + i - 1]$ when $order = Nag_ColMajor$ ;
$p [(i - 1) \times n + j - 1]$ when $order = Nag_RowMajor$ .

On exit:

P (i, j)

contains

P_{i j}

, the option price evaluated for the strike price

x_{i}

at expiry

t_{j}

for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

15: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_ENUM_REAL_2: On entry, s and h are inconsistent with type: $s = 〈value〉$ and $h = 〈value〉$ .
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $h = 〈value〉$ .
Constraint: $h \geq 〈value〉$ and $h \leq 〈value〉$ .

On entry, $k = 〈value〉$ .
Constraint: $k \geq 0.0$ .

On entry, $q = 〈value〉$ .
Constraint: $q \geq 0.0$ .

On entry, $r = 〈value〉$ .
Constraint: $r \geq 0.0$ .

On entry, $s = 〈value〉$ .
Constraint: $s \geq 〈value〉$ and $s \leq 〈value〉$ .

On entry, $sigma = 〈value〉$ .
Constraint: $sigma > 0.0$ .
NE_REAL_ARRAY: On entry, $t [〈value〉] = 〈value〉$ .
Constraint: $t [i] \geq 〈value〉$ .

On entry, $x [〈value〉] = 〈value〉$ .
Constraint: $x [i] \geq 〈value〉$ and $x [i] \leq 〈value〉$ .

7

Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function,

Φ

. 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.

8

Parallelism and Performance

nag_barrier_std_price (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.

9

Further Comments

None.

10

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.

NAG Library Function Document

nag_barrier_std_price (s30fac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results