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NAG Toolbox: nag_specfun_opt_binary_aon_greeks (s30cd)

Purpose

nag_specfun_opt_binary_aon_greeks (s30cd) computes the price of a binary or digital asset-or-nothing option together with its sensitivities (Greeks).

Syntax

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30cd(calput, x, s, 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_aon_greeks(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)

Description

nag_specfun_opt_binary_aon_greeks (s30cd) computes the price of a binary or digital asset-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 the underlying asset itself, SS, 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 SS, 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 = S eqT Φ(d1)
Pcall = S e-qT Φ(d1)
and for a put,
Pput = S eqT Φ(d1)
Pput = S e-qT Φ(-d1)
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
d1 = ( ln (S / X) + (rq + σ2 / 2) T )/(σ×sqrt(T)) .
d1 = 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:     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:     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, 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_binary_aon_greeks_example
put = 'P';
s = 70;
sigma = 0.15;
r = 0.05;
q = 0.03;
x = [65.0];
t = [0.8];


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


fprintf('\nBinary (Digital): Asset-or-Nothing\n European Put :\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(' 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): Asset-or-Nothing
 European Put :
  Spot       =     70.0000
  Volatility =      0.1500
  Rate       =      0.0500
  Dividend   =      0.0300

 Time to Expiry :   0.8000
 Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
 65.0000  15.7211  -1.9852   0.1422  83.6424  -4.2761 -123.7497 -111.1728

 Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
 65.0000  15.7211   9.3479  -1.1351   0.0118   0.2316  -2.6319 -989.9610

function s30cd_example
put = 'P';
s = 70;
sigma = 0.15;
r = 0.05;
q = 0.03;
x = [65.0];
t = [0.8];


[p, delta, gamma, vega, theta, rho, crho, vanna, charm, ...
 speed, colour, zomma, vomma, ifail] = ...
              s30cd(put, x, s, t, sigma, r, q);


fprintf('\nBinary (Digital): Asset-or-Nothing\n European Put :\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(' 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): Asset-or-Nothing
 European Put :
  Spot       =     70.0000
  Volatility =      0.1500
  Rate       =      0.0500
  Dividend   =      0.0300

 Time to Expiry :   0.8000
 Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
 65.0000  15.7211  -1.9852   0.1422  83.6424  -4.2761 -123.7497 -111.1728

 Strike    Price    Vanna    Charm    Speed   Colour    Zomma    Vomma
 65.0000  15.7211   9.3479  -1.1351   0.0118   0.2316  -2.6319 -989.9610


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