NAG Library Function Document

nag_binary_aon_greeks (s30cdc)

nag_binary_aon_greeks (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigma, double r, double q, double p[], double delta[], double gamma[], double vega[], double theta[], double rho[], double crho[], double vanna[], double charm[], double speed[], double colour[], double zomma[], double vomma[], NagError *fail)

3 Description

nag_binary_aon_greeks (s30cdc) computes the price of a binary or digital asset-or-nothing option, together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters. This option pays the underlying asset itself,

S

, 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

S

, 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} = S e^{- q T} Φ (d_{1})

and for a put,

P_{put} = S e^{- q T} Φ (- d_{1})

where

Φ

is the cumulative Normal distribution function,

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

and

d_{1} = \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 3.2.1.3 in the Essential Introduction 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: 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

8: 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

9: 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

10: 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

11: 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

12: delta[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

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

On exit: the

m \times n

array delta contains the sensitivity,

\frac{\partial P}{\partial S}

, of the option price to change in the price of the underlying asset.

13: gamma[ $m \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

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

On exit: the

m \times n

array gamma contains the sensitivity,

\frac{\partial^{2} P}{\partial S^{2}}

, of delta to change in the price of the underlying asset.

14: vega[ $m \times n$ ] – doubleOutput

Note: where

VEGA (i, j)

appears in this document, it refers to the array element

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

On exit:

VEGA (i, j)

, contains the first-order Greek measuring the sensitivity of the option price

P_{i j}

to change in the volatility of the underlying asset, i.e.,

\frac{\partial P_{i j}}{\partial σ}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

15: theta[ $m \times n$ ] – doubleOutput

Note: where

THETA (i, j)

appears in this document, it refers to the array element

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

On exit:

THETA (i, j)

, contains the first-order Greek measuring the sensitivity of the option price

P_{i j}

to change in time, i.e.,

- \frac{\partial P_{i j}}{\partial T}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

, where

b = r - q

16: rho[ $m \times n$ ] – doubleOutput

Note: where

RHO (i, j)

appears in this document, it refers to the array element

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

On exit:

RHO (i, j)

, contains the first-order Greek measuring the sensitivity of the option price

P_{i j}

to change in the annual risk-free interest rate, i.e.,

- \frac{\partial P_{i j}}{\partial r}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

17: crho[ $m \times n$ ] – doubleOutput

Note: where

CRHO (i, j)

appears in this document, it refers to the array element

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

On exit:

CRHO (i, j)

, contains the first-order Greek measuring the sensitivity of the option price

P_{i j}

to change in the annual cost of carry rate, i.e.,

- \frac{\partial P_{i j}}{\partial b}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

, where

b = r - q

18: vanna[ $m \times n$ ] – doubleOutput

Note: where

VANNA (i, j)

appears in this document, it refers to the array element

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

On exit:

VANNA (i, j)

, contains the second-order Greek measuring the sensitivity of the first-order Greek

Δ_{i j}

to change in the volatility of the asset price, i.e.,

- \frac{\partial Δ_{i j}}{\partial T} = - \frac{\partial^{2} P_{i j}}{\partial S \partial σ}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

19: charm[ $m \times n$ ] – doubleOutput

Note: where

CHARM (i, j)

appears in this document, it refers to the array element

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

On exit:

CHARM (i, j)

, contains the second-order Greek measuring the sensitivity of the first-order Greek

Δ_{i j}

to change in the time, i.e.,

- \frac{\partial Δ_{i j}}{\partial T} = - \frac{\partial^{2} P_{i j}}{\partial S \partial T}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

20: speed[ $m \times n$ ] – doubleOutput

Note: where

SPEED (i, j)

appears in this document, it refers to the array element

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

On exit:

SPEED (i, j)

, contains the third-order Greek measuring the sensitivity of the second-order Greek

Γ_{i j}

to change in the price of the underlying asset, i.e.,

- \frac{\partial Γ_{i j}}{\partial S} = - \frac{\partial^{3} P_{i j}}{\partial S^{3}}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

21: colour[ $m \times n$ ] – doubleOutput

Note: where

COLOUR (i, j)

appears in this document, it refers to the array element

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

On exit:

COLOUR (i, j)

, contains the third-order Greek measuring the sensitivity of the second-order Greek

Γ_{i j}

to change in the time, i.e.,

- \frac{\partial Γ_{i j}}{\partial T} = - \frac{\partial^{3} P_{i j}}{\partial S \partial T}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

22: zomma[ $m \times n$ ] – doubleOutput

Note: where

ZOMMA (i, j)

appears in this document, it refers to the array element

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

On exit:

ZOMMA (i, j)

, contains the third-order Greek measuring the sensitivity of the second-order Greek

Γ_{i j}

to change in the volatility of the underlying asset, i.e.,

- \frac{\partial Γ_{i j}}{\partial σ} = - \frac{\partial^{3} P_{i j}}{\partial S^{2} \partial σ}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

23: vomma[ $m \times n$ ] – doubleOutput

Note: where

VOMMA (i, j)

appears in this document, it refers to the array element

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

On exit:

VOMMA (i, j)

, contains the second-order Greek measuring the sensitivity of the first-order Greek

Δ_{i j}

to change in the volatility of the underlying asset, i.e.,

- \frac{\partial Δ_{i j}}{\partial σ} = - \frac{\partial^{2} P_{i j}}{\partial σ^{2}}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

24: fail – NagError *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.
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.
NE_REAL: 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_aon_greeks (s30cdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please 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 an asset-or-nothing put with a time to expiry of

292

days, a stock price of

70

and a strike price of

65

. The risk-free interest rate is

5 %

per year, there is an annual dividend return of

3 %

and the volatility is

15 %

per year.

NAG Library Function Documentnag_binary_aon_greeks (s30cdc)

+− 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

NAG Library Function Document

nag_binary_aon_greeks (s30cdc)