void	nag_binary_con_price (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, double k, const double t[], double sigma, double r, double q, double p[], NagError *fail)

3

Description

nag_binary_con_price (s30cac) computes the price of a binary or digital cash-or-nothing option which pays a fixed amount,

K

, at expiration if the option is in-the-money (see Section 2.4 in the s Chapter Introduction). For a strike price,

X

, underlying asset price,

S

, and time to expiry,

T

, the payoff is therefore

K

, if

S > X

for a call or

S < X

for a put. Nothing is paid out when this condition is not met.

The price of a call with volatility,

σ

, risk-free interest rate,

r

, and annualised dividend yield,

q

, is

P_{call} = K e^{- r T} Φ (d_{2})

and for a put,

P_{put} = K e^{- r T} Φ (- d_{2})

where

Φ

is the cumulative Normal distribution function,

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

and

d_{2} = \frac{\ln (S / X) + (r - q - σ^{2} / 2) T}{σ \sqrt{T}} .

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

Reiner E and Rubinstein M (1991) Unscrambling the binary code Risk 4

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: $m$ – 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: $x [m]$ – 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

6: $s$ – 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.

7: $k$ – doubleInput

On entry: the amount,

K

, to be paid at expiration if the option is in-the-money, i.e., if

s > x [i - 1]

when

option = Nag_Call

, or if

s < x [i - 1]

when

option = Nag_Put

, for

i = 1, 2, \dots, m

Constraint:

k \geq 0.0

8: $t [n]$ – 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

9: $sigma$ – 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

10: $r$ – 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

11: $q$ – 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

12: $p [m \times n]$ – 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

13: $fail$ – 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_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, $k = 〈value〉$ .
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_binary_con_price (s30cac) 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 cash-or-nothing put with a time to expiry of

0.75

years, a stock price of

100

and a strike price of

80

. The risk-free interest rate is

6 %

per year and the volatility is

35 %

per year. If the option is in-the-money at expiration, i.e., if

S > X

, the payoff is

10

NAG Library Function Document

nag_binary_con_price (s30cac)

▸▿ 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