NAG Library Routine Document

Integer, Intent (In)	::	m, n, ldp
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(m), s, t(n), sigma, r, q
Real (Kind=nag_wp), Intent (Inout)	::	p(ldp,n)
Character (1), Intent (In)	::	calput

C Header Interface

#include <nagmk26.h>

void	s30aaf_ (const char calput, const Integer m, const Integer n, const double x[], const double s, const double t[], const double sigma, const double r, const double q, double p[], const Integer ldp, Integer *ifail, const Charlen length_calput)

3

Description

s30aaf computes the price of a European call (or put) option for constant volatility,

σ

, and risk-free interest rate,

r

, with a possible dividend yield,

q

, using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). 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: $calput$ – 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: $m$ – IntegerInput

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

Constraint:

m \geq 1

3: $n$ – IntegerInput

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

Constraint:

n \geq 1

4: $x (m)$ – 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

5: $s$ – 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.

6: $t (n)$ – 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

7: $sigma$ – 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

8: $r$ – 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

9: $q$ – 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

10: $p (ldp, n)$ – Real (Kind=nag_wp) arrayOutput

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

11: $ldp$ – IntegerInput

On entry: the first dimension of the array p as declared in the (sub)program from which s30aaf is called.

Constraint:

ldp \geq m

12: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $calput = 〈value〉$ was an illegal value.

$ifail = 2$: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

$ifail = 3$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

$ifail = 4$: On entry, $x (〈value〉) = 〈value〉$ .
Constraint: $x (i) \geq 〈value〉$ and $x (i) \leq 〈value〉$ .

$ifail = 5$: On entry, $s = 〈value〉$ .
Constraint: $s \geq 〈value〉$ and $s \leq 〈value〉$ .

$ifail = 6$: On entry, $t (〈value〉) = 〈value〉$ .
Constraint: $t (i) \geq 〈value〉$ .

$ifail = 7$: On entry, $sigma = 〈value〉$ .
Constraint: $sigma > 0.0$ .

$ifail = 8$: On entry, $r = 〈value〉$ .
Constraint: $r \geq 0.0$ .

$ifail = 9$: On entry, $q = 〈value〉$ .
Constraint: $q \geq 0.0$ .

$ifail = 11$: On entry, $ldp = 〈value〉$ and $m = 〈value〉$ .
Constraint: $ldp \geq m$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

Parallelism and Performance

s30aaf 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 routine. 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 prices for six European call options using two expiry times and three strike prices as input. The times to expiry are taken as

0.7

and

0.8

years respectively. The stock price is

55

, with strike prices,

58

60

and

62

. The risk-free interest rate is

10 %

per year and the volatility is

30 %

per year.

NAG Library Routine Document

s30aaf (opt_bsm_price)

▸▿ 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