NAG Library Routine Document

S30ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S30ABF computes the European option price given by the Black–Scholes–Merton formula together with its sensitivities (Greeks).

2 Specification

SUBROUTINE S30ABF (

CALPUT, M, N, X, S, T, SIGMA, R, Q, P, LDP, DELTA, GAMMA, VEGA, THETA, RHO, CRHO, VANNA, CHARM, SPEED, COLOUR, ZOMMA, VOMMA, IFAIL)

INTEGER	M, N, LDP, IFAIL
REAL (KIND=nag_wp)	X(M), S, T(N), SIGMA, R, Q, P(LDP,N), DELTA(LDP,N), GAMMA(LDP,N), VEGA(LDP,N), THETA(LDP,N), RHO(LDP,N), CRHO(LDP,N), VANNA(LDP,N), CHARM(LDP,N), SPEED(LDP,N), COLOUR(LDP,N), ZOMMA(LDP,N), VOMMA(LDP,N)
CHARACTER(1)	CALPUT

3 Description

S30ABF 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}

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 Parameters

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: the leading

M \times N

part of the array P contains the computed option prices.

11: LDP – INTEGERInput

On entry: the first dimension of the arrays P, DELTA, GAMMA, VEGA, THETA, RHO, CRHO, VANNA, CHARM, SPEED, COLOUR, ZOMMA and VOMMA as declared in the (sub)program from which S30ABF is called.

Constraint:

LDP \geq M

12: DELTA(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the 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(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the 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(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array VEGA contains the sensitivity,

\frac{\partial P}{\partial σ}

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

15: THETA(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array THETA contains the sensitivity,

- \frac{\partial P}{\partial T}

, of the option price to change in the time to expiry of the option.

16: RHO(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array RHO contains the sensitivity,

\frac{\partial P}{\partial r}

, of the option price to change in the annual risk-free interest rate.

17: CRHO(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array CRHO containing the sensitivity,

\frac{\partial P}{\partial b}

, of the option price to change in the annual cost of carry rate,

b

, where

b = r - q

18: VANNA(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array VANNA contains the sensitivity,

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

, of VEGA to change in the price of the underlying asset or, equivalently, the sensitivity of DELTA to change in the volatility of the asset price.

19: CHARM(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array CHARM contains the sensitivity,

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

, of DELTA to change in the time to expiry of the option.

20: SPEED(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array SPEED contains the sensitivity,

\frac{\partial^{3} P}{\partial S^{3}}

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

21: COLOUR(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array COLOUR contains the sensitivity,

- \frac{\partial^{3} P}{\partial S^{2} \partial T}

, of GAMMA to change in the time to expiry of the option.

22: ZOMMA(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array ZOMMA contains the sensitivity,

\frac{\partial^{3} P}{\partial S^{2} \partial σ}

, of GAMMA to change in the volatility of the underlying asset.

23: VOMMA(LDP,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leading

M \times N

part of the array VOMMA contains the sensitivity,

\frac{\partial^{2} P}{\partial σ^{2}}

, of VEGA to change in the volatility of the underlying asset.

24: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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, $CALPUT \neq'C'$ or $'P'$ .

$IFAIL = 2$: On entry, $M \leq 0$ .

$IFAIL = 3$: On entry, $N \leq 0$ .

$IFAIL = 4$: On entry, $X (i) < z$ or $X (i) > 1 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 5$: On entry, $S < z$ or $S > 1.0 / z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 6$: On entry, $T (i) < z$ , where $z = X02AMF ()$ , the safe range parameter.

$IFAIL = 7$: On entry, $SIGMA \leq 0.0$ .

$IFAIL = 8$: On entry, $R < 0.0$ .

$IFAIL = 9$: On entry, $Q < 0.0$ .

$IFAIL = 11$: On entry, $LDP < M$ .

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 Further Comments

None.

9 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.

NAG Library Routine DocumentS30ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S30ABF