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NAG Toolbox: nag_specfun_opt_binary_con_greeks (s30cb)

Purpose

nag_specfun_opt_binary_con_greeks (s30cb) computes the price of a binary or digital cash-or-nothing option together with its sensitivities (Greeks).

Syntax

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30cb(calput, x, s, k, t, sigma, r, q, 'm', m, 'n', n)
[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = nag_specfun_opt_binary_con_greeks(calput, x, s, k, t, sigma, r, q, 'm', m, 'n', n)

Description

nag_specfun_opt_binary_con_greeks (s30cb) computes the price of a binary or digital cash-or-nothing 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. This option pays a fixed amount, KK, at expiration if the option is in-the-money (see Section [Option Pricing s] in the S Chapter Introduction). For a strike price, XX, underlying asset price, SS, and time to expiry, TT, the payoff is therefore KK, if S > XS>X for a call or S < XS<X for a put. Nothing is paid out when this condition is not met.
The price of a call with volatility, σσ, risk-free interest rate, rr, and annualised dividend yield, qq, is
Pcall = K erT Φ(d2)
Pcall = K e-rT Φ(d2)
and for a put,
Pput = K erT Φ(d2)
Pput = K e-rT Φ(-d2)
where ΦΦ is the cumulative Normal distribution function,
x
Φ(x) = 1/(sqrt(2π))exp(y2 / 2)dy,
Φ(x) = 1 2π - x exp( -y2/2 ) dy ,
and
d2 = ( ln (S / X) + (rqσ2 / 2) T )/(σ×sqrt(T)) .
d2 = ln (S/X) + ( r-q- σ2 / 2 ) T σT .

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'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:     k – double scalar
The amount, KK, to be paid at expiration if the option is in-the-money, i.e., if s > x(i)s>xi when calput = 'C'calput='C', or if s < x(i)s<xi when calput = 'P'calput='P', for i = 1,2,,mi=1,2,,m.
Constraint: k0.0k0.0.
5:     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.
6:     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.
7:     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.
8:     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:     delta(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array delta contains the sensitivity, (P)/(S)PS, of the option price to change in the price of the underlying asset.
3:     gamma(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array gamma contains the sensitivity, (2P)/(S2)2PS2, of delta to change in the price of the underlying asset.
4:     vega(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array vega contains the sensitivity, (P)/(σ)Pσ, of the option price to change in the volatility of the underlying asset.
5:     theta(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array theta contains the sensitivity, (P)/(T)-PT, of the option price to change in the time to expiry of the option.
6:     rho(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array rho contains the sensitivity, (P)/(r)Pr, of the option price to change in the annual risk-free interest rate.
7:     crho(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array crho containing the sensitivity, (P)/(b)Pb, of the option price to change in the annual cost of carry rate, bb, where b = rqb=r-q.
8:     vanna(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array vanna contains the sensitivity, (2P)/(Sσ)2PSσ, 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.
9:     charm(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array charm contains the sensitivity, (2P)/(S T)-2PS T, of delta to change in the time to expiry of the option.
10:   speed(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array speed contains the sensitivity, (3P)/(S3)3PS3, of gamma to change in the price of the underlying asset.
11:   colour(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array colour contains the sensitivity, (3P)/(S2 T)-3PS2 T, of gamma to change in the time to expiry of the option.
12:   zomma(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array zomma contains the sensitivity, (3P)/(S2σ)3PS2σ, of gamma to change in the volatility of the underlying asset.
13:   vomma(ldp,n) – double array
ldpmldpm.
The leading m × nm×n part of the array vomma contains the sensitivity, (2P)/(σ2)2Pσ2, of vega to change in the volatility of the underlying asset.
14:   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, k < 0.0k<0.0.
  ifail = 7ifail=7
On entry, t(i) < zti<z, where z = x02am()z=x02am(), the safe range parameter.
  ifail = 8ifail=8
On entry, sigma0.0sigma0.0.
  ifail = 9ifail=9
On entry, r < 0.0r<0.0.
  ifail = 10ifail=10
On entry, q < 0.0q<0.0.
  ifail = 12ifail=12
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_binary_con_greeks_example
put = 'C';
s = 110.0;
k = 5.0;
sigma = 0.35;
r = 0.05;
q = 0.04;
x = [87.0];
t = [0.75];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, ...
 zomma, vomma, ifail] = nag_specfun_opt_binary_con_greeks(put, x, s, k, t, sigma, r, q);


fprintf('\nBinary (Digital): Cash-or-Nothing\n European Call :\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(' Time to Expiry : %8.4f\n', t(1));
fprintf(' Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n\n', x(1), ...
         p(1,1), delta(1,1), gamma(1,1), vega(1,1), theta(1,1), rho(1,1), ...
         crho(1,1));

fprintf(' Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
         p(1,1), vanna(1,1), charm(1,1), speed(1,1), colour(1,1), ...
         zomma(1,1), vomma(1,1));
 

Binary (Digital): Cash-or-Nothing
 European Call :
  Spot       =    110.0000
  Payout     =      5.0000
  Volatility =      0.3500
  Rate       =      0.0500
  Dividend   =      0.0400

 Time to Expiry :   0.7500
 Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
 87.0000   3.5696   0.0467  -0.0013  -4.2307   1.1142   1.1788   3.8560

 Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
 87.0000   3.5696  -0.0514   0.0153   0.0000  -0.0019   0.0079  12.8874

function s30cb_example
put = 'C';
s = 110.0;
k = 5.0;
sigma = 0.35;
r = 0.05;
q = 0.04;
x = [87.0];
t = [0.75];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, ...
 zomma, vomma, ifail] = s30cb(put, x, s, k, t, sigma, r, q);


fprintf('\nBinary (Digital): Cash-or-Nothing\n European Call :\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(' Time to Expiry : %8.4f\n', t(1));
fprintf(' Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n\n', x(1), ...
         p(1,1), delta(1,1), gamma(1,1), vega(1,1), theta(1,1), rho(1,1), ...
         crho(1,1));

fprintf(' Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma\n');
fprintf('%8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f %8.4f\n', x(1), ...
         p(1,1), vanna(1,1), charm(1,1), speed(1,1), colour(1,1), ...
         zomma(1,1), vomma(1,1));
 

Binary (Digital): Cash-or-Nothing
 European Call :
  Spot       =    110.0000
  Payout     =      5.0000
  Volatility =      0.3500
  Rate       =      0.0500
  Dividend   =      0.0400

 Time to Expiry :   0.7500
 Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
 87.0000   3.5696   0.0467  -0.0013  -4.2307   1.1142   1.1788   3.8560

 Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
 87.0000   3.5696  -0.0514   0.0153   0.0000  -0.0019   0.0079  12.8874


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