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NAG Toolbox: nag_specfun_opt_binary_con_price (s30ca)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_opt_binary_con_price (s30ca) computes the price of a binary or digital cash-or-nothing option.

Syntax

[p, ifail] = s30ca(calput, x, s, k, t, sigma, r, q, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_binary_con_price(calput, x, s, k, t, sigma, r, q, 'm', m, 'n', n)

Description

nag_specfun_opt_binary_con_price (s30ca) computes the price of a binary or digital cash-or-nothing option which pays a fixed amount, K, at expiration if the option is in-the-money (see Option Pricing Routines in the S Chapter Introduction). For a strike price, X, underlying asset price, S, and time to expiry, T, the payoff is therefore K, if S>X for a call or S<X for a put. Nothing is paid out when this condition is not met.
The price of a call with volatility, σ, risk-free interest rate, r, and annualised dividend yield, q, is
Pcall = K e-rT Φd2  
and for a put,
Pput = K e-rT Φ-d2  
where Φ is the cumulative Normal distribution function,
Φx = 1 2π - x -y2/2 dy ,  
and
d2 = ln S/X + r-q - σ2 / 2 T 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.

References

Reiner E and Rubinstein M (1991) Unscrambling the binary code Risk 4

Parameters

Compulsory Input Parameters

1:     calput – string (length ≥ 1)
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' 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:     k – double scalar
The amount, K, to be paid at expiration if the option is in-the-money, i.e., if s>xi when calput='C', or if s<xi when calput='P', for i=1,2,,m.
Constraint: k0.0.
5:     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.
6:     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.
7:     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.
8:     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
ldp=m.
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:
   ifail=1
On entry, calput=_ was an illegal value.
   ifail=2
Constraint: m1.
   ifail=3
Constraint: n1.
   ifail=4
Constraint: xi_ and xi_.
   ifail=5
Constraint: s_ and s_.
   ifail=6
Constraint: k0.0.
   ifail=7
Constraint: ti_.
   ifail=8
Constraint: sigma>0.0.
   ifail=9
Constraint: r0.0.
   ifail=10
Constraint: q0.0.
   ifail=12
Constraint: ldpm.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

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 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

This example computes the price of a cash-or-nothing put with a time to expiry of 0.75 years, a stock price of 100 and a strike price of 80. The risk-free interest rate is 6% per year and the volatility is 35% per year. If the option is in-the-money at expiration, i.e., if S>X, the payoff is 10.
function s30ca_example


fprintf('s30ca example results\n\n');

put = 'P';
s = 100.0;
k = 10.0;
sigma = 0.35;
r = 0.06;
q = 0.0;
x = [80.0];
t = [0.75];
[p, ifail] = s30ca( ...
                    put, x, s, k, t, sigma, r, q);


fprintf('\nBinary (Digital): Cash-or-Nothing\n European Put :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Payout     =   %9.4f\n', k);
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=1:1
  for j=1:1
    fprintf('%9.4f %9.4f %9.4f\n', x(i), t(j), p(i,j));
  end
end


s30ca example results


Binary (Digital): Cash-or-Nothing
 European Put :
  Spot       =    100.0000
  Payout     =     10.0000
  Volatility =      0.3500
  Rate       =      0.0600
  Dividend   =      0.0000

   Strike    Expiry   Option Price
  80.0000    0.7500    2.2155

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