S30AAF (PDF version)
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NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

S30AAF computes the European option price given by the Black–Scholes–Merton formula.

2  Specification

REAL (KIND=nag_wp)  X(M), S, T(N), SIGMA, R, Q, P(LDP,N)

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
Pcall = Se-qT Φd1 - Xe-rT Φd2
and the corresponding European put price is
Pput = Xe-rT Φ-d2 - Se-qT Φ-d1
and where Φ denotes the cumulative Normal distribution function,
Φx = 12π - x exp -y2/2 dy
d1 = ln S/X + r-q+ σ2 / 2 T σT , d2 = d1 - σT .

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.
A call. The holder has a right to buy.
A put. The holder has a right to sell.
Constraint: CALPUT='C' or 'P'.
2:     M – INTEGERInput
On entry: the number of strike prices to be used.
Constraint: M1.
3:     N – INTEGERInput
On entry: the number of times to expiry to be used.
Constraint: N1.
4:     X(M) – REAL (KIND=nag_wp) arrayInput
On entry: Xi must contain Xi, the ith strike price, for i=1,2,,M.
Constraint: Xiz ​ and ​ Xi 1 / z , where z = X02AMF , the safe range parameter, for i=1,2,,M.
5:     S – REAL (KIND=nag_wp)Input
On entry: S, the price of the underlying asset.
Constraint: Sz ​ and ​S1.0/z, where z=X02AMF, the safe range parameter.
6:     T(N) – REAL (KIND=nag_wp) arrayInput
On entry: Ti must contain Ti, the ith time, in years, to expiry, for i=1,2,,N.
Constraint: Tiz, where z = X02AMF , the safe range parameter, for i=1,2,,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: R0.0.
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: Q0.0.
10:   P(LDP,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the leading M×N part of the array P contains the computed option prices.
11:   LDP – INTEGERInput
On entry: the first dimension of the array P as declared in the (sub)program from which S30AAF is called.
Constraint: LDPM.
12:   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 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry, CALPUT'C' or 'P'.
On entry, M0.
On entry, N0.
On entry, Xi<z or Xi>1/z, where z=X02AMF, the safe range parameter.
On entry, S<z or S>1.0/z, where z=X02AMF, the safe range parameter.
On entry, Ti<z, where z=X02AMF, the safe range parameter.
On entry, SIGMA0.0.
On entry, R<0.0.
On entry, Q<0.0.
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


9  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.

9.1  Program Text

Program Text (s30aafe.f90)

9.2  Program Data

Program Data (s30aafe.d)

9.3  Program Results

Program Results (s30aafe.r)

S30AAF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012