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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_opt_amer_bs_price (s30qc)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_opt_amer_bs_price (s30qc) computes the Bjerksund and Stensland (2002) approximation to the price of an American option.


[p, ifail] = s30qc(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_amer_bs_price(calput, x, s, t, sigma, r, q, 'm', m, 'n', n)


nag_specfun_opt_amer_bs_price (s30qc) computes the price of an American option using the closed form approximation of Bjerksund and Stensland (2002). The time to maturity, T, is divided into two periods, each with a flat early exercise boundary, by choosing a time t 0,T , such that t = 12 5-1 T . The two boundary values are defined as x~=X~t, X~=X~T with
X~τ = B0 + B - B0 1 - exp hτ ,  
hτ = - bτ+2στ X2 B - B0 B0 ,  
B β β-1 X ,  B0 maxX, rr-b X ,  
β = 12 - bσ2 + b σ2 - 12 2 + 2 r σ2 .  
with b=r-q, the cost of carry, where r is the risk-free interest rate and q is the annual dividend rate. Here X is the strike price and σ is the annual volatility.
The price of an American call option is approximated as
Pcall = αX~ Sβ - αX~ ϕ S,t|β,X~,X~+ ϕ S,t|1,X~,X~ - ϕ S,t|1,x~,X~ - X ϕ S,t|0,X~,X~ + X ϕ S,t|0,x~,X~ + α x~ ϕ S,t|β,x~,X~ - αx~ Ψ S,T|β,x~,X~,x~,t + Ψ S,T|1,x~,X~,x~,t - Ψ S,T|1,X,X~,x~,t - X Ψ S,T|0,x~,X~,x~,t + X Ψ S,T|0,X,X~,x~,t ,  
where α, ϕ and Ψ are as defined in Bjerksund and Stensland (2002).
The price of a put option is obtained by the put-call transformation,
Pput X,S,T,σ,r,q = Pcall S,X,T,σ,q,r .  
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.


Bjerksund P and Stensland G (2002) Closed form valuation of American options Discussion Paper 2002/09 NHH Bergen Norway http://www.nhh.no/
Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and t probabilities Statistics and Computing 14 151–160


Compulsory Input Parameters

1:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
A call; the holder has a right to buy.
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 ​s1z, where z=x02am, the safe range parameter and sβ<1z where β is as defined in Description.
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
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:
On entry, calput=_ was an illegal value.
Constraint: m1.
Constraint: n1.
Constraint: xi_ and xi_.
Constraint: s_ and s_.
Constraint: ti_.
Constraint: sigma>0.0.
Constraint: r0.0.
Constraint: q0.0.
Constraint: ldpm.
Constraint: sβ<_.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The accuracy of the output will be bounded by the accuracy of the cumulative bivariate Normal distribution function. The algorithm of Genz (2004) is used, as described in the document for nag_stat_prob_bivariate_normal (g01ha), giving a maximum absolute error of less than 5×10-16. The univariate cumulative Normal distribution function also forms part of the evaluation (see nag_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)).

Further Comments



This example computes the price of an American call with a time to expiry of 3 months, a stock price of 110 and a strike price of 100. The risk-free interest rate is 8% per year, there is an annual dividend return of 12% and the volatility is 20% per year.
function s30qc_example

fprintf('s30qc example results\n\n');

put = 'c';
s = 110.0;
sigma = 0.2;
r = 0.08;
q = 0.12;
x = [100.0];
t = [0.25];

[p, ifail] = s30qc( ...
                    put, x, s, t, sigma, r, q);

fprintf('\nAmerican Call :\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('   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));

s30qc example results

American Call :
  Spot       =    110.0000
  Volatility =      0.2000
  Rate       =      0.0800
  Dividend   =      0.1200

   Strike    Expiry   Option Price
 100.0000    0.2500   10.3340

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