Integer, Intent (In)	::	n, mode
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	p(n), k(n), s0(n), t(n), r(n)
Real (Kind=nag_wp), Intent (Out)	::	sigma(n)
Character (1), Intent (In)	::	calput

C Header Interface

#include <nag.h>

void	s30acf_ (const char calput, const Integer n, const double p[], const double k[], const double s0[], const double t[], const double r[], double sigma[], const Integer mode, Integer ivalid[], Integer ifail, const Charlen length_calput)

The routine may be called by the names s30acf or nagf_specfun_opt_imp_vol.

3 Description

The Black–Scholes formula for the price of a European option is

P_{call} = S_{0} Φ (\frac{\ln (\frac{S_{0}}{K}) + [r + \frac{σ^{2}}{2}] T}{σ \sqrt{T}}) - K e^{- r T} Φ (\frac{\ln (\frac{S_{0}}{K}) + [r - \frac{σ^{2}}{2}] T}{σ \sqrt{T}}),

for a call option, and

P_{put} = K e^{- r T} Φ (\frac{- \ln (\frac{S_{0}}{K}) - [r - \frac{σ^{2}}{2}] T}{σ \sqrt{T}}) - S_{0} Φ (\frac{- \ln (\frac{S_{0}}{K}) - [r + \frac{σ^{2}}{2}] T}{σ \sqrt{T}}),

for a put option, where

Φ

is the cumulative Normal distribution function,

T

is the time to maturity,

S_{0}

is the spot price of the underlying asset,

K

is the strike price,

r

is the interest rate and

σ

is the volatility.

Given arrays of values

P_{i}

K_{i}

S_{0 i}

T_{i}

and

r_{i}

, for

i = 1, 2, \dots n

, s30acf computes the implied volatilities

σ_{i}

s30acf offers the choice of two algorithms. The algorithm of Glau et al. (2018) uses Chebyshev interpolation to compute the implied volatilities, and performs best for long arrays of input data. The algorithm of Jäckel (2015) uses a third order Householder iteration and performs better for short arrays of input data.

4 References

Glau K, Herold P, Madan D B and Pötz C (2018) The Chebyshev method for the implied volatility Accepted for publication in the Journal of Computational Finance

Jäckel P (2015) Let's be Rational Wilmott Magazine 2015(75) 40–53

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: $n$ – Integer Input

On entry:

n

, the number of implied volatilities to be computed.

Constraint:

n \geq 0

3: $p (n)$ – Real (Kind=nag_wp) array Input

On entry:

p (i)

must contain

P_{i}

, the

i

th option price, for

i = 1, 2, \dots, n

Constraint:

p (i) \geq 0.0

, for

i = 1, 2, \dots, n

4: $k (n)$ – Real (Kind=nag_wp) array Input

On entry:

k (i)

must contain

K_{i}

, the

i

th strike price, for

i = 1, 2, \dots, n

Constraint:

k (i) > 0.0

, for

i = 1, 2, \dots, n

5: $s0 (n)$ – Real (Kind=nag_wp) array Input

On entry:

s0 (i)

must contain

S_{0 i}

, the

i

th spot price, for

i = 1, 2, \dots, n

Constraint:

s0 (i) > 0.0

, for

i = 1, 2, \dots, n

6: $t (n)$ – Real (Kind=nag_wp) array Input

On entry:

t (i)

must contain

T_{i}

, the

i

th time, in years, to maturity, for

i = 1, 2, \dots, n

Constraint:

t (i) > 0.0

, for

i = 1, 2, \dots, n

7: $r (n)$ – Real (Kind=nag_wp) array Input

On entry:

r (i)

must contain

r_{i}

, the

i

th interest rate, for

i = 1, 2, \dots, n

. Note that a rate of 5% should be entered as 0.05.

8: $sigma (n)$ – Real (Kind=nag_wp) array Output

On exit:

sigma (i)

contains

σ_{i}

, the

i

th implied volatility, for

i = 1, 2, \dots, n

9: $mode$ – Integer Input

On entry: specifies which algorithm will be used to compute the implied volatilities. See Sections 7 and 8 for further guidance on the choice of mode.

$mode = 0$: The Glau et al. (2018) algorithm will be used. The nodes used in the Chebyshev interpolation will be chosen to achieve relative accuracy to approximately seven decimal places;
$mode = 1$: The Glau et al. (2018) algorithm will be used. The nodes used in the Chebyshev interpolation will be chosen to achieve relative accuracy to approximately $15$ decimal places, but limited by the machine precision;
$mode = 2$: The Jäckel (2015) algorithm will be used, aiming for accuracy to approximately $15$ – $16$ decimal places, but limited by machine precision.

Constraint:

mode = 0

1

2

10: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

indicates any errors with the input arguments that prevented

σ_{i}

from being computed. If

ivalid (i) \neq 0

sigma (i)

contains

0.0

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $P_{i} < 0.0$ .
$ivalid (i) = 2$: $K_{i} \leq 0.0$ .
$ivalid (i) = 3$: $S_{0 i} \leq 0.0$ .
$ivalid (i) = 4$: $T_{i} \leq 0.0$ .
$ivalid (i) = 5$: The combination of $P_{i}$ , $K_{i}$ , $S_{0 i}$ , $T_{i}$ and $r_{i}$ is out of the domain in which $σ_{i}$ can be computed. See Section 9 for further details.

11: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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 at least one input argument was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = 3$: On entry, $mode = ⟨ value ⟩$ .
Constraint: $mode = 0$ , $1$ or $2$ .

$ifail = 4$: On entry, $calput = ⟨ value ⟩$ was an illegal value.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

mode = 0

1

then s30acf uses Chebyshev interpolation. For

mode = 0

it aims for relative accuracy to roughly single precision (approximately seven decimal places). For

mode = 1

it aims for relative accuracy to roughly double precision (approximately sixteen decimal places).

mode = 2

, a Householder iteration is used to achieve relative accuracy to roughly double precision (approximately sixteen decimal places). In practice there is very little difference in accuracy between

mode = 1

and

2

, though for more extreme input values,

mode = 2

is likely to be more accurate.

8 Parallelism and Performance

s30acf 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.

When

mode = 0

1

, s30acf uses an algorithm optimized for performance on large vectors of input data (

n ≳ 1000

) by exploiting vector instructions when available.

When

mode = 2

, s30acf uses an algorithm optimized for performance on smaller vectors of input data.

Numerical experiments suggest that for

n ≲ 1000

, the best performance is always achieved by choosing

mode = 2

, whereas for

n ≳ 1000

mode = 0

1

should be used, according to the desired accuracy.

9 Further Comments

The domain of inputs,

P_{i}

K_{i}

S_{0 i}

T_{i}

and

r_{i}

, for which s30acf is able to accurately compute

σ_{i}

is very large and should cover all practical applications of this routine. Thus, encountering arguments for which

ivalid (i) = 5

is returned is highly unlikely. Note that it is not possible to give a closed-form expression for the allowed range of arguments because this range is based on a transformation to the normalized call price.

Note that some formulations of the Black–Scholes equation also include the annual yield rate,

q

. If you wish to incorporate

q

here, then you must first compute the quantities

{\tilde{S}}_{0} = S_{0} e^{- q T}

\tilde{K} = K e^{- q T}

and

\tilde{r} = r - q

. s30acf should then be called with

{\tilde{S}}_{0}

\tilde{K}

and

\tilde{r}

in place of

S_{0}

K

and

r

respectively.

s30acf can also be used with Black's model for European futures options. In this case, the forward price,

F

, and the discount factor,

D

, are used, which are related to

S_{0}

via

F = S_{0} / D

. In addition, the option price is scaled by a factor

e^{- r T}

Approximately

7 \times n

of real allocatable memory and

3 \times n

of integer allocatable memory is required by the routine.

10 Example

This example reads in values of

P_{i}

K_{i}

S_{0 i}

T_{i}

and

r

from a file, evaluates the implied volatilities,

σ_{i}

, and prints the results.

s30ac: FL CL CPP AD

NAG FL Interfaces30acf (opt_​imp_​vol)

▸▿ 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 FL Interface
s30acf (opt_imp_vol)