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# 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, T$T$, is divided into two periods, each with a flat early exercise boundary, by choosing a time t [0,T] $t\in \left[0,T\right]$, such that t = (1/2) (sqrt(5)1) T $t=\frac{1}{2}\left(\sqrt{5}-1\right)T$. The two boundary values are defined as = (t)$\stackrel{~}{x}=\stackrel{~}{X}\left(t\right)$, = (T)$\stackrel{~}{X}=\stackrel{~}{X}\left(T\right)$ with
 X̃(τ) = B0 + (B∞-B0) (1 − exp{h(τ)}) , $X~(τ) = B0 + ( B∞ - B0 ) ( 1 - exp{ h(τ) } ) ,$
where
 h(τ) = − (bτ + 2σ×sqrt(τ)) ((X2)/( (B∞-B0) B0 )) , $h(τ) = - ( bτ+2σ⁢τ ) ( X2 ( B∞ - B0 ) B0 ) ,$
 B∞ ≡ β/(β − 1) X ,  B0 ≡ max {X, (r/(r − b)) 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 = rq$b=r-q$, the cost of carry, where r$r$ is the risk-free interest rate and q$q$ is the annual dividend rate. Here X$X$ is the strike price and σ$\sigma$ is the annual volatility.
The price of an American call option is approximated as
 Pcall = α(X̃) Sβ − α(X̃) φ (S,t|β,X̃,X̃) + φ (S,t|1,X̃,X̃) − φ (S,t|1,x̃,X̃) − X φ (S,t|0,X̃,X̃) + X φ (S,t|0,x̃,X̃) + α (x̃) φ (S,t|β,x̃,X̃) − α(x̃) Ψ (S,T|β,x̃,X̃,x̃,t) + Ψ (S,T|1,x̃,X̃,x̃,t) − Ψ (S,T|1,X,X̃,x̃,t) − X Ψ (S,T|0,x̃,X̃,x̃,t) + X Ψ (S,T|0,X,X̃,x̃,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 α$\alpha$, φ$\varphi$ and Ψ$\Psi$ 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 t$t$ 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'${\mathbf{calput}}=\text{'C'}$
A call. The holder has a right to buy.
calput = 'P'${\mathbf{calput}}=\text{'P'}$
A put. The holder has a right to sell.
Constraint: calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
2:     x(m) – double array
m, the dimension of the array, must satisfy the constraint m1${\mathbf{m}}\ge 1$.
x(i)${\mathbf{x}}\left(i\right)$ must contain Xi${X}_{\mathit{i}}$, the i$\mathit{i}$th strike price, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: x(i)z ​ and ​ x(i) 1 / z ${\mathbf{x}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{x}}\left(\mathit{i}\right)\le 1/z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     s – double scalar
S$S$, the price of the underlying asset.
Constraint: sz ​ and ​s1/z${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le \frac{1}{z}$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter and sβ < 1/z${{\mathbf{s}}}^{\beta }<\frac{1}{z}$ where β$\beta$ is as defined in Section [Description].
4:     t(n) – double array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
t(i)${\mathbf{t}}\left(i\right)$ must contain Ti${T}_{\mathit{i}}$, the i$\mathit{i}$th time, in years, to expiry, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: t(i)z${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where z = x02am () $z=\mathbf{x02am}\left(\right)$, the safe range parameter, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5:     sigma – double scalar
σ$\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma > 0.0${\mathbf{sigma}}>0.0$.
6:     r – double scalar
r$r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0${\mathbf{r}}\ge 0.0$.
7:     q – double scalar
q$q$, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0${\mathbf{q}}\ge 0.0$.

### 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${\mathbf{m}}\ge 1$.
2:     n – int64int32nag_int scalar
Default: The dimension of the array t.
The number of times to expiry to be used.
Constraint: n1${\mathbf{n}}\ge 1$.

ldp

### Output Parameters

1:     p(ldp,n) – double array
ldpm$\mathit{ldp}\ge {\mathbf{m}}$.
The leading m × n${\mathbf{m}}×{\mathbf{n}}$ part of the array p contains the computed option prices.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, calput = 'C'${\mathbf{calput}}=\text{'C'}$ or 'P'$\text{'P'}$.
ifail = 2${\mathbf{ifail}}=2$
On entry, m0${\mathbf{m}}\le 0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, n0${\mathbf{n}}\le 0$.
ifail = 4${\mathbf{ifail}}=4$
On entry, x(i) < z${\mathbf{x}}\left(\mathit{i}\right) or x(i) > 1 / z${\mathbf{x}}\left(\mathit{i}\right)>1/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 5${\mathbf{ifail}}=5$
On entry, s < z${\mathbf{s}} or s > 1.0 / z${\mathbf{s}}>1.0/z$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 6${\mathbf{ifail}}=6$
On entry, t(i) < z${\mathbf{t}}\left(\mathit{i}\right), where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter.
ifail = 7${\mathbf{ifail}}=7$
On entry, sigma0.0${\mathbf{sigma}}\le 0.0$.
ifail = 8${\mathbf{ifail}}=8$
On entry, r < 0.0${\mathbf{r}}<0.0$.
ifail = 9${\mathbf{ifail}}=9$
On entry, q < 0.0${\mathbf{q}}<0.0$.
ifail = 11${\mathbf{ifail}}=11$
On entry, ldp < m$\mathit{ldp}<{\mathbf{m}}$.
ifail = 14${\mathbf{ifail}}=14$
On entry, β1/z$\beta \ge \frac{1}{z}$, where z = x02am()$z=\mathbf{x02am}\left(\right)$, the safe range parameter (see Section [Description]).
ifail = 15${\mathbf{ifail}}=15$
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 × 1016$5×{10}^{-16}$. The univariate cumulative Normal distribution function also forms part of the evaluation (see nag_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)).

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

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