hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_opt_amer_bs_price (s30qc)

Purpose

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

Syntax

[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)

Description

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, TT, is divided into two periods, each with a flat early exercise boundary, by choosing a time t [0,T] t [0,T] , such that t = (1/2) (sqrt(5)1) T t = 12 (5-1) T . The two boundary values are defined as = (t)x~=X~(t), = (T)X~=X~(T) with
(τ) = B0 + (B-B0) (1exp{h(τ)}) ,
X~(τ) = B0 + ( B - B0 ) ( 1 - exp{ h(τ) } ) ,
where
h(τ) = (bτ + 2σ×sqrt(τ)) ((X2)/( (B-B0) B0 )) ,
h(τ) = - ( bτ+2στ ) ( X2 ( B - B0 ) B0 ) ,
B β/(β1) X ,  B0 max {X, (r/(rb)) X } ,
B β β-1 X ,  B0 max{X, ( rr-b ) X } ,
β = ((1/2)b/(σ2)) + sqrt( (b/(σ2)(1/2))2 + 2 r/(σ2) ) .
β = ( 12 - bσ2 ) + ( b σ2 - 12 ) 2 + 2 r σ2 .
with b = rqb=r-q, the cost of carry, where rr is the risk-free interest rate and qq is the annual dividend rate. Here XX is the strike price and σσ is the annual volatility.
The price of an American call option is approximated as
Pcall = α() Sβ α() φ (S,t|β,,) +
φ (S,t|1,,) φ (S,t|1,,)
X φ (S,t|0,,) + X φ (S,t|0,,) +
α () φ (S,t|β,,) α() Ψ (S,T|β,,,,t) +
Ψ (S,T|1,,,,t) Ψ (S,T|1,X,,,t)
X Ψ (S,T|0,,,,t) + X Ψ (S,T|0,X,,,t) ,
Pcall = α(X~) Sβ - α(X~) 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~) - α(x~) 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 αα, φϕ and Ψ 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) .
Pput (X,S,T,σ,r,q) = Pcall (S,X,T,σ,q,r) .

References

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 tt probabilities Statistics and Computing 14 151–160

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/zsz ​ and ​s1z, where z = x02am()z=x02am(), the safe range parameter and sβ < 1/zsβ<1z where ββ is as defined in Section [Description].
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:     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.
  ifail = 14ifail=14
On entry, β1/zβ1z, where z = x02am()z=x02am(), the safe range parameter (see Section [Description]).
  ifail = 15ifail=15
Internal memory allocation failed.

Accuracy

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 × 10165×10-16. 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

None.

Example

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

[p, ifail] = nag_specfun_opt_amer_bs_price(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));
  end
end
 

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

   Strike    Expiry   Option Price
 100.0000    0.2500   10.3340

function s30qc_example
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));
  end
end
 

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

   Strike    Expiry   Option Price
 100.0000    0.2500   10.3340


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013