s30abc (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)

The function may be called by the names: s30abc, nag_specfun_opt_bsm_greeks or nag_bsm_greeks.

3 Description

s30abc computes the price of a European call (or put) 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, by the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). The annual volatility,

σ

, risk-free interest rate,

r

, and dividend yield,

q

, must be supplied as input. For a given strike price,

X

, the price of a European call with underlying price,

S

, and time to expiry,

T

, is

P_{call} = S e^{- q T} Φ (d_{1}) - X e^{- r T} Φ (d_{2})

and the corresponding European put price is

P_{put} = X e^{- r T} Φ (- d_{2}) - S e^{- q T} Φ (- d_{1})

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

and

\begin{array}{l} d_{1} = \frac{\ln (S / X) + (r - q + σ^{2} / 2) T}{σ \sqrt{T}}, \\ d_{2} = d_{1} - σ \sqrt{T} . \end{array}

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

Black F and Scholes M (1973) The pricing of options and corporate liabilities Journal of Political Economy 81 637–654

Merton R C (1973) Theory of rational option pricing Bell Journal of Economics and Management Science 4 141–183

5 Arguments

1: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $option$ – Nag_CallPut Input

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: $m$ – Integer Input

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

Constraint:

m \geq 1

4: $n$ – Integer Input

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

Constraint:

n \geq 1

5: $x [m]$ – const double Input

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: $s$ – double Input

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 double Input

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: $sigma$ – double 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

9: $r$ – double 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

10: $q$ – double 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

11: $p [m \times n]$ – double Output

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: $delta [m \times n]$ – double Output

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: $gamma [m \times n]$ – double Output

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: $vega [m \times n]$ – double Output

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: $theta [m \times n]$ – double Output

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: $rho [m \times n]$ – double Output

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: $crho [m \times n]$ – double Output

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: $vanna [m \times n]$ – double Output

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: $charm [m \times n]$ – double Output

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: $speed [m \times n]$ – double Output

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: $colour [m \times n]$ – double Output

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: $zomma [m \times n]$ – double Output

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: $vomma [m \times n]$ – double Output

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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 s15abc and s15adc). An accuracy close to machine precision can generally be expected.

8 Parallelism and Performance

s30abc 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 European put with a time to expiry of

0.7

years, a stock price of

55

and a strike price of

60

. The risk-free interest rate is

10 %

per year and the volatility is

30 %

per year.

s30ab: FL CL CPP AD

NAG CL Interfaces30abc (opt_​bsm_​greeks)

▸▿ 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 CL Interface
s30abc (opt_bsm_greeks)