. 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

References

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

Parameters

Compulsory Input Parameters

1: $calput$ – string (length ≥ 1)

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: $type$ – string (length at least 2) (length ≥ 2)

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: $x (m)$ – double array

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 = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, m

4: $s$ – double scalar

S

, the price of the underlying asset.

Constraint:

s \geq z ​ and ​ s \leq 1.0 / z

, where

z = x02am ()

, the safe range parameter.

5: $h$ – double scalar

The barrier price.

Constraint:

h \geq z ​ and ​ h \leq 1 / z

, where

z = x02am ()

, the safe range parameter.

6: $k$ – double scalar

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

7: $t (n)$ – double array

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 = x02am ()

, the safe range parameter, for

i = 1, 2, \dots, n

8: $sigma$ – double scalar

σ

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

Constraint:

sigma > 0.0

9: $r$ – double scalar

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

10: $q$ – double scalar

q

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

Constraint:

q \geq 0.0

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the array x.
The number of strike prices to be used.

Constraint: $m \geq 1$ .
2: $n$ – int64int32nag_int scalar: Default: the dimension of the array t.
The number of times to expiry to be used.

Constraint: $n \geq 1$ .

Output Parameters

1: $p (ldp, n)$ – double array: $ldp = m$ .
$p (i, j)$ 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 $j = 1, 2, \dots, n$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, $calput =_$ was an illegal value.

$ifail = 2$: On entry, $type =_$ was an illegal value.

$ifail = 3$: Constraint: $m \geq 1$ .

$ifail = 4$: Constraint: $n \geq 1$ .

$ifail = 5$: Constraint: $x (i) \geq_$ and $x (i) \leq_$ .

$ifail = 6$: Constraint: $s \geq_$ and $s \leq_$ .

$ifail = 7$: Constraint: $h \geq_$ and $h \leq_$ .

$ifail = 8$: Constraint: $k \geq 0.0$ .

$ifail = 9$: Constraint: $t (i) \geq_$ .

$ifail = 10$: Constraint: $sigma > 0.0$ .

$ifail = 11$: Constraint: $r \geq 0.0$ .

$ifail = 12$: Constraint: $q \geq 0.0$ .

$ifail = 14$: Constraint: $ldp \geq m$ .

$ifail = 15$: On entry, s and h are inconsistent with type.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

Further Comments

None.

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.

Open in the MATLAB editor: s30fa_example

function s30fa_example


fprintf('s30fa example results\n\n');

put = 'P';
type = 'DI';
s = 100.0;
h = 95.0;
k = 3.0;
sigma = 0.3;
r = 0.08;
q = 0.04;
x = [100.0];
t = [0.5];


[p, ifail] = s30fa( ...
                    put, type, x, s, h, k, t, sigma, r, q);


fprintf('\nStandard Barrier Option\n Put :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Barrier    =   %9.4f\n', h);
fprintf('  Rebate     =   %9.4f\n', k);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Dividend   =   %9.4f\n\n', q);

fprintf('   Strike    Expiry   Option Price\n');

for i=1:1
  for j=1:1
    fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
  end
end

s30fa example results


Standard Barrier Option
 Put :
  Spot       =    100.0000
  Barrier    =     95.0000
  Rebate     =      3.0000
  Volatility =      0.3000
  Rate       =      0.0800
  Dividend   =      0.0400

   Strike    Expiry   Option Price
 100.0000    0.5000    7.7988

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

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