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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_opt_asian_geom_greeks (s30sb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_opt_asian_geom_greeks (s30sb) computes the Asian geometric continuous average-rate option price together with its sensitivities (Greeks).

Syntax

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = s30sb(calput, x, s, t, sigma, r, b, 'm', m, 'n', n)
[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, zomma, vomma, ifail] = nag_specfun_opt_asian_geom_greeks(calput, x, s, t, sigma, r, b, 'm', m, 'n', n)

Description

nag_specfun_opt_asian_geom_greeks (s30sb) computes the price of an Asian geometric continuous average-rate 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. The annual volatility, σ, risk-free rate, r, and cost of carry, b, are constants (see Kemna and Vorst (1990)). For a given strike price, X, the price of a call option with underlying price, S, and time to expiry, T, is
Pcall = S e b--r T Φ d- 1 - X e-rT Φ d- 2 ,  
and the corresponding put option price is
Pput = X e-rT Φ -d-2 - S e b--r T Φ - d-1 ,  
where
d-1 = lnS/X + b- + σ-2 / 2 T σ- T  
and
d-2 = d-1 - σ- T ,  
with
σ- = σ 3 ,  b- = 1 2 b- σ2 6 .  
Φ is the cumulative Normal distribution function,
Φx = 1 2π - x exp -y2/2 dy .  
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

Kemna A and Vorst A (1990) A pricing method for options based on average asset values Journal of Banking and Finance 14 113–129

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:     b – double scalar
b, the annual cost of carry rate. Note that a rate of 8% should be entered as 0.08.

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.
deltaij, contains the first-order Greek measuring the sensitivity of the option price Pij to change in the price of the underlying asset, i.e., - Pij S , for i=1,2,,m and j=1,2,,n.
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=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 Asian geometric continuous average-rate call with a time to expiry of 3 months, a stock price of 80 and a strike price of 97. The risk-free interest rate is 5% per year, the cost of carry is 8% and the volatility is 20% per year.
function s30sb_example


fprintf('s30sb example results\n\n');

put = 'C';
s = 80.0;
sigma = 0.2;
r = 0.05;
b = 0.08;
x = [97.0];
t = [0.25];

[p, delta, gamma, vega, theta, rho, crho, vanna, charm, speed, colour, ...
  zomma, vomma, ifail] = s30sb( ...
                                put, x, s, t, sigma, r, b);


fprintf('\nAsian Option: Geometric Continuous Average-Rate\nAsian Call :\n');
fprintf('  Spot          =   %9.4f\n', s);
fprintf('  Volatility    =   %9.4f\n', sigma);
fprintf('  Rate          =   %9.4f\n', r);
fprintf('  Cost of carry =   %9.4f\n\n', b);

fprintf(' Time to Expiry : %8.4f\n', t(1));
fprintf('%8s%9s%9s%9s%9s%9s%9s%9s\n','Strike','Price','Delta','Gamma',...
        'Vega','Theta','Rho','CRho');
fprintf('%8.4f%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f\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%9s%9s\n','Vanna','Charm','Speed','Colour',...
        'Zomma','Vomma');
fprintf('%17s%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f\n\n', ' ', vanna(1,1), ...
        charm(1,1), speed(1,1), colour(1,1), zomma(1,1), vomma(1,1));


s30sb example results


Asian Option: Geometric Continuous Average-Rate
Asian Call :
  Spot          =     80.0000
  Volatility    =      0.2000
  Rate          =      0.0500
  Cost of carry =      0.0800

 Time to Expiry :   0.2500
  Strike    Price    Delta    Gamma     Vega    Theta      Rho     CRho
 97.0000   0.0010   0.0008   0.0006   0.0638  -0.0281   0.0079   0.0081

                     Vanna    Charm    Speed   Colour    Zomma    Vomma
                    0.0443  -0.0196   0.0004  -0.0122   0.0272   3.1893


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