hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_opt_bsm_price (s30aa)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_opt_bsm_price (s30aa) computes the European option price given by the Black–Scholes–Merton formula.


[p, ifail] = s30aa(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_bsm_price(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)


nag_specfun_opt_bsm_price (s30aa) 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 .  
The option price Pij=PX=Xi,T=Tj is computed for each strike price in a set Xi, i=1,2,,m, and for each expiry time in a set Tj, j=1,2,,n.


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


Compulsory Input Parameters

1:     calput – string (length ≥ 1)
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:     xm – double array
xi must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: xiz ​ and ​ xi 1 / z , where z = x02am , the safe range parameter, for i=1,2,,m.
3:     s – double scalar
S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=x02am, the safe range parameter.
4:     tn – double array
ti must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: tiz, where z = x02am , the safe range parameter, for i=1,2,,n.
5:     sigma – double scalar
σ, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma>0.0.
6:     r – double scalar
r, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0.
7:     q – double scalar
q, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the dimension of the array x.
The number of strike prices to be used.
Constraint: m1.
2:     n int64int32nag_int scalar
Default: the dimension of the array t.
The number of times to expiry to be used.
Constraint: n1.

Output Parameters

1:     pldpn – double array
pij contains Pij, the option price evaluated for the strike price xi at expiry tj for i=1,2,,m and j=1,2,,n.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry, calput=_ was an illegal value.
Constraint: m1.
Constraint: n1.
Constraint: xi_ and xi_.
Constraint: s_ and s_.
Constraint: ti_.
Constraint: sigma>0.0.
Constraint: r0.0.
Constraint: q0.0.
Constraint: ldpm.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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 nag_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

Further Comments



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.
function s30aa_example

fprintf('s30aa example results\n\n');

put = 'c';
s = 55;
sigma = 0.3;
r = 0.1;
q = 0;
x = [58, 60, 62];
t = [0.7, 0.8];

[p, ifail] = s30aa( ...
                    put, x, s, t, sigma, r, q);

fprintf('\nBlack-Scholes-Merton formula\n European Call :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Dividend   =   %9.4f\n\n', q);

fprintf('   Strike    Expiry   Option Price\n');
for i=1:3
  for j=1:2
    fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));

s30aa example results

Black-Scholes-Merton formula
 European Call :
  Spot       =     55.0000
  Volatility =      0.3000
  Rate       =      0.1000
  Dividend   =      0.0000

   Strike    Expiry   Option Price
  58.0000    0.7000    5.9198
  58.0000    0.8000    6.5506
  60.0000    0.7000    5.0809
  60.0000    0.8000    5.6992
  62.0000    0.7000    4.3389
  62.0000    0.8000    4.9379

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015