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NAG Toolbox: nag_specfun_opt_bsm_price (s30aa)

Purpose

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

Syntax

[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)

Description

nag_specfun_opt_bsm_price (s30aa) computes the price of a European call (or put) option for constant volatility, σσ, and risk-free interest rate, rr, with a possible dividend yield, qq, using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). For a given strike price, XX, the price of a European call with underlying price, SS, and time to expiry, TT, is
Pcall = SeqT Φ(d1) XerT Φ(d2)
Pcall = Se-qT Φ(d1) - Xe-rT Φ(d2)
and the corresponding European put price is
Pput = XerT Φ(d2) SeqT Φ(d1)
Pput = Xe-rT Φ(-d2) - Se-qT Φ(-d1)
and where ΦΦ denotes the cumulative Normal distribution function,
x
Φ(x) = 1/(sqrt(2π))exp(y2 / 2)dy
Φ(x) = 12π - x exp( -y2/2 ) dy
and
d1 = ( ln (S / X) + (rq + σ2 / 2) T )/(σ×sqrt(T)) ,
d2 = d1 σ×sqrt(T) .
d1 = ln (S/X) + ( r-q+ σ2 / 2 ) T σT , d2 = d1 - σT .

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

Parameters

Compulsory Input Parameters

1:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
calput = 'C'calput='C'
A call. The holder has a right to buy.
calput = 'P'calput='P'
A put. The holder has a right to sell.
Constraint: calput = 'C'calput='C' or 'P''P'.
2:     x(m) – double array
m, the dimension of the array, must satisfy the constraint m1m1.
x(i)xi must contain XiXi, the iith strike price, for i = 1,2,,mi=1,2,,m.
Constraint: x(i)z ​ and ​ x(i) 1 / z xiz ​ and ​ xi 1 / z , where z = x02am () z = x02am () , the safe range parameter, for i = 1,2,,mi=1,2,,m.
3:     s – double scalar
SS, the price of the underlying asset.
Constraint: sz ​ and ​s1.0 / zsz ​ and ​s1.0/z, where z = x02am()z=x02am(), the safe range parameter.
4:     t(n) – double array
n, the dimension of the array, must satisfy the constraint n1n1.
t(i)ti must contain TiTi, the iith time, in years, to expiry, for i = 1,2,,ni=1,2,,n.
Constraint: t(i)ztiz, where z = x02am () z = x02am () , the safe range parameter, for i = 1,2,,ni=1,2,,n.
5:     sigma – double scalar
σσ, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma > 0.0sigma>0.0.
6:     r – double scalar
rr, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0r0.0.
7:     q – double scalar
qq, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0q0.0.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array x.
The number of strike prices to be used.
Constraint: m1m1.
2:     n – int64int32nag_int scalar
Default: The dimension of the array t.
The number of times to expiry to be used.
Constraint: n1n1.

Input Parameters Omitted from the MATLAB Interface

ldp

Output Parameters

1:     p(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array p contains the computed option prices.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, calput = 'C'calput='C' or 'P''P'.
  ifail = 2ifail=2
On entry, m0m0.
  ifail = 3ifail=3
On entry, n0n0.
  ifail = 4ifail=4
On entry, x(i) < zxi<z or x(i) > 1 / zxi>1/z, where z = x02am()z=x02am(), the safe range parameter.
  ifail = 5ifail=5
On entry, s < zs<z or s > 1.0 / zs>1.0/z, where z = x02am()z=x02am(), the safe range parameter.
  ifail = 6ifail=6
On entry, t(i) < zti<z, where z = x02am()z=x02am(), the safe range parameter.
  ifail = 7ifail=7
On entry, sigma0.0sigma0.0.
  ifail = 8ifail=8
On entry, r < 0.0r<0.0.
  ifail = 9ifail=9
On entry, q < 0.0q<0.0.
  ifail = 11ifail=11
On entry, ldp < mldp<m.

Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, ΦΦ. 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

None.

Example

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

[p, ifail] = nag_specfun_opt_bsm_price(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));
  end
end
 

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

function s30aa_example
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));
  end
end
 

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

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