NAG Library Routine Document

S30FAF

. 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 $j = k = - 1$ ,
$P_{put} = A - C + F .$

4 References

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

5 Parameters

1: CALPUT – CHARACTER(1)Input

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

$CALPUT ='C'$: A call. The holder has a right to buy.
$CALPUT ='P'$: A put. The holder has a right to sell.

Constraint:

CALPUT ='C'

'P'

2: TYPE – CHARACTER(2)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.

$TYPE ='DI'$: Down-and-In.
$TYPE ='DO'$: Down-and-Out.
$TYPE ='UI'$: Up-and-In.
$TYPE ='UO'$: Up-and-Out.

Constraint:

TYPE ='DI'

'DO'

'UI'

'UO'

3: M – INTEGERInput

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

Constraint:

M \geq 1

4: N – INTEGERInput

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

Constraint:

N \geq 1

5: X(M) – REAL (KIND=nag_wp) arrayInput

On entry:

X (i)

must contain

X_{i}

, the

i

th strike price, for

i = 1, 2, \dots, M

Constraint:

X (i) \geq z ​ and ​ X (i) \leq 1 / z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, M

6: S – REAL (KIND=nag_wp)Input

On entry:

S

, the price of the underlying asset.

Constraint:

S \geq z ​ and ​ S \leq 1.0 / z

, where

z = X02AMF ()

, the safe range parameter.

7: H – REAL (KIND=nag_wp)Input

On entry: the barrier price.

Constraint:

H \geq z ​ and ​ H \leq 1 / z

, where

z = X02AMF ()

, the safe range parameter.

8: K – REAL (KIND=nag_wp)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:

K \geq 0.0

9: T(N) – REAL (KIND=nag_wp) arrayInput

On entry:

T (i)

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, N

Constraint:

T (i) \geq z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, N

10: SIGMA – REAL (KIND=nag_wp)Input

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

11: R – REAL (KIND=nag_wp)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:

R \geq 0.0

12: Q – REAL (KIND=nag_wp)Input

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

13: P(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array P contains the computed option prices.

14: LDP – INTEGERInput

On entry: the first dimension of the array P as declared in the (sub)program from which S30FAF is called.

Constraint:

LDP \geq M

15: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $CALPUT \neq'C'$ or $'P'$ .

$IFAIL = 2$: On entry, $TYPE \neq'DI'$ , $'DO'$ , $'UI'$ or $'UO'$ .

$IFAIL = 3$: On entry, $M \leq 0$ .

$IFAIL = 4$: On entry, $N \leq 0$ .

$IFAIL = 5$: On entry, $X (i) < z$ or $X (i) > 1 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 6$: On entry, $S < z$ or $S > 1.0 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 7$: On entry, $H < z$ or $H > 1.0 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 8$: On entry, $K < 0.0$ .

$IFAIL = 9$: On entry, $T (i) < z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 10$: On entry, $SIGMA \leq 0.0$ .

$IFAIL = 11$: On entry, $R < 0.0$ .

$IFAIL = 12$: On entry, $Q < 0.0$ .

$IFAIL = 15$: S and H are not consistent with TYPE.

$IFAIL = 14$: On entry, $LDP < M$ .

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 S15ABF and S15ADF). An accuracy close to machine precision can generally be expected.

8 Further Comments

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.

NAG Library Routine DocumentS30FAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S30FAF