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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_opt_binary_aon_greeks (s30cd)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

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, S, 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 S, 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 = S e-qT Φd1  
and for a put,
Pput = S e-qT Φ-d1  
where Φ is the cumulative Normal distribution function,
Φx = 1 2π - x exp -y2/2 dy ,  
and
d1 = 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:     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
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:     deltaldpn – double array
ldp=m.
The leading m×n part of the array delta contains the sensitivity, PS, of the option price to change in the price of the underlying asset.
3:     gammaldpn – double array
ldp=m.
The leading m×n part of the array gamma contains the sensitivity, 2PS2, of delta to change in the price of the underlying asset.
4:     vegaldpn – double array
ldp=m.
vegaij, contains the first-order Greek measuring the sensitivity of the option price Pij to change in the volatility of the underlying asset, i.e., Pij σ , for i=1,2,,m and j=1,2,,n.
5:     thetaldpn – double array
ldp=m.
thetaij, contains the first-order Greek measuring the sensitivity of the option price Pij to change in time, i.e., - Pij T , for i=1,2,,m and j=1,2,,n, where b=r-q.
6:     rholdpn – double array
ldp=m.
rhoij, contains the first-order Greek measuring the sensitivity of the option price Pij to change in the annual risk-free interest rate, i.e., - Pij r , for i=1,2,,m and j=1,2,,n.
7:     crholdpn – double array
ldp=m.
crhoij, contains the first-order Greek measuring the sensitivity of the option price Pij to change in the annual cost of carry rate, i.e., - Pij b , for i=1,2,,m and j=1,2,,n, where b=r-q.
8:     vannaldpn – double array
ldp=m.
vannaij, contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the volatility of the asset price, i.e., - Δij T = - 2 Pij Sσ , for i=1,2,,m and j=1,2,,n.
9:     charmldpn – double array
ldp=m.
charmij, contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the time, i.e., - Δij T = - 2 Pij ST , for i=1,2,,m and j=1,2,,n.
10:   speedldpn – double array
ldp=m.
speedij, contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the price of the underlying asset, i.e., - Γij S = - 3 Pij S3 , for i=1,2,,m and j=1,2,,n.
11:   colourldpn – double array
ldp=m.
colourij, contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the time, i.e., - Γij T = - 3 Pij ST , for i=1,2,,m and j=1,2,,n.
12:   zommaldpn – double array
ldp=m.
zommaij, contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the volatility of the underlying asset, i.e., - Γij σ = - 3 Pij S2σ , for i=1,2,,m and j=1,2,,n.
13:   vommaldpn – double array
ldp=m.
vommaij, contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the volatility of the underlying asset, i.e., - Δij σ = - 2 Pij σ2 , for i=1,2,,m and j=1,2,,n.
14:   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: ti_.
   ifail=7
Constraint: sigma>0.0.
   ifail=8
Constraint: r0.0.
   ifail=9
Constraint: q0.0.
   ifail=11
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 an asset-or-nothing put with a time to expiry of 292 days, a stock price of 70 and a strike price of 65. The risk-free interest rate is 5% per year, there is an annual dividend return of 3% and the volatility is 15% per year.
function s30cd_example


fprintf('s30cd example results\n\n');

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('%8s%9s%9s%9s%9s%9s%11s%11s\n','Strike','Price','Delta','Gamma',...
        'Vega','Theta','Rho','CRho');
fprintf('%8.4f%9.4f%9.4f%9.4f%9.4f%9.4f%11.4f%11.4f\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('%26s%9s%9s%9s%11s%11s\n','Vanna','Charm','Speed','Colour',...
        'Zomma','Vomma');
fprintf('%17s%9.4f%9.4f%9.4f%9.4f%11.4f%11.4f\n\n', ' ', vanna(1,1), ...
        charm(1,1), speed(1,1), colour(1,1), zomma(1,1), vomma(1,1));


s30cd example results


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

                     Vanna    Charm    Speed   Colour      Zomma      Vomma
                    9.3479  -1.1351   0.0118   0.2316    -2.6319  -989.9610


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Chapter Introduction
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