# NAG FL Interfaces30qcf (opt_​amer_​bs_​price)

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## 1Purpose

s30qcf computes the Bjerksund and Stensland (2002) approximation to the price of an American option.

## 2Specification

Fortran Interface
 Subroutine s30qcf ( m, n, x, s, t, r, q, p, ldp,
 Integer, Intent (In) :: m, n, ldp Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(m), s, t(n), sigma, r, q Real (Kind=nag_wp), Intent (Inout) :: p(ldp,n) Character (1), Intent (In) :: calput
#include <nag.h>
 void s30qcf_ (const char *calput, const Integer *m, const Integer *n, const double x[], const double *s, const double t[], const double *sigma, const double *r, const double *q, double p[], const Integer *ldp, Integer *ifail, const Charlen length_calput)
The routine may be called by the names s30qcf or nagf_specfun_opt_amer_bs_price.

## 3Description

s30qcf 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 $t\in \left[0,T\right]$, such that $t=\frac{1}{2}\left(\sqrt{5}-1\right)T$. The two boundary values are defined as $\stackrel{~}{x}=\stackrel{~}{X}\left(t\right)$, $\stackrel{~}{X}=\stackrel{~}{X}\left(T\right)$ with
 $X~(τ) = B0 + (B∞-B0) (1-exp{h(τ)}) ,$
where
 $h(τ) = - (bτ+2σ⁢τ) ( X2 (B∞-B0) B0 ) ,$
 $B∞ ≡ β β-1 X , B0 ≡ max{X, (rr-b) X } ,$
 $β = (12-bσ2) + ( b σ2 -12) 2 + 2 r σ2 .$
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 $\sigma$ is the annual volatility.
The price of an American call option is approximated as
 $Pcall = α(X~) Sβ - α(X~) ϕ (S,t|β,X~,X~)+ ϕ (S,t|1,X~,X~) - ϕ (S,t|1,x~,X~) - X ϕ (S,t|0,X~,X~) + X ϕ (S,t|0,x~,X~) + α (x~) ϕ (S,t|β,x~,X~) - α(x~) Ψ (S,T|β,x~,X~,x~,t) + Ψ (S,T|1,x~,X~,x~,t) - Ψ (S,T|1,X,X~,x~,t) - X Ψ (S,T|0,x~,X~,x~,t) + X Ψ (S,T|0,X,X~,x~,t) ,$
where $\alpha$, $\varphi$ and $\Psi$ are as defined in Bjerksund and Stensland (2002).
The price of a put option is obtained by the put-call transformation,
 $Pput (X,S,T,σ,r,q) = Pcall (S,X,T,σ,q,r) .$
The option price ${P}_{ij}=P\left(X={X}_{i},T={T}_{j}\right)$ is computed for each strike price in a set ${X}_{i}$, $i=1,2,\dots ,m$, and for each expiry time in a set ${T}_{j}$, $j=1,2,\dots ,n$.

## 4References

Bjerksund P and Stensland G (2002) Closed form valuation of American options Discussion Paper 2002/09 NHH Bergen Norway
Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and $t$ probabilities Statistics and Computing 14 151–160

## 5Arguments

1: $\mathbf{calput}$Character(1) Input
On entry: determines whether the option is a call or a put.
${\mathbf{calput}}=\text{'C'}$
A call; the holder has a right to buy.
${\mathbf{calput}}=\text{'P'}$
A put; the holder has a right to sell.
Constraint: ${\mathbf{calput}}=\text{'C'}$ or $\text{'P'}$.
2: $\mathbf{m}$Integer Input
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{m}}\ge 1$.
3: $\mathbf{n}$Integer Input
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{n}}\ge 1$.
4: $\mathbf{x}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{x}}\left(i\right)$ must contain ${X}_{\mathit{i}}$, the $\mathit{i}$th strike price, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)\ge z\text{​ and ​}{\mathbf{x}}\left(\mathit{i}\right)\le 1/z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
5: $\mathbf{s}$Real (Kind=nag_wp) Input
On entry: $S$, the price of the underlying asset.
Constraint: ${\mathbf{s}}\ge z\text{​ and ​}{\mathbf{s}}\le \frac{1}{z}$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter and ${{\mathbf{s}}}^{\beta }<\frac{1}{z}$ where $\beta$ is as defined in Section 3.
6: $\mathbf{t}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{t}}\left(i\right)$ must contain ${T}_{\mathit{i}}$, the $\mathit{i}$th time, in years, to expiry, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{t}}\left(\mathit{i}\right)\ge z$, where $z={\mathbf{x02amf}}\left(\right)$, the safe range parameter, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
7: $\mathbf{sigma}$Real (Kind=nag_wp) Input
On entry: $\sigma$, the volatility of the underlying asset. Note that a rate of 15% should be entered as $0.15$.
Constraint: ${\mathbf{sigma}}>0.0$.
8: $\mathbf{r}$Real (Kind=nag_wp) Input
On entry: $r$, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as $0.05$.
Constraint: ${\mathbf{r}}\ge 0.0$.
9: $\mathbf{q}$Real (Kind=nag_wp) Input
On entry: $q$, the annual continuous yield rate. Note that a rate of 8% should be entered as $0.08$.
Constraint: ${\mathbf{q}}\ge 0.0$.
10: $\mathbf{p}\left({\mathbf{ldp}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{p}}\left(i,j\right)$ contains ${P}_{ij}$, the option price evaluated for the strike price ${{\mathbf{x}}}_{i}$ at expiry ${{\mathbf{t}}}_{j}$ for $i=1,2,\dots ,{\mathbf{m}}$ and $j=1,2,\dots ,{\mathbf{n}}$.
11: $\mathbf{ldp}$Integer Input
On entry: the first dimension of the array p as declared in the (sub)program from which s30qcf is called.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
12: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{calput}}=⟨\mathit{\text{value}}⟩$ was an illegal value.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{x}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\left(i\right)\ge ⟨\mathit{\text{value}}⟩$ and ${\mathbf{x}}\left(i\right)\le ⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{s}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{s}}\ge ⟨\mathit{\text{value}}⟩$ and ${\mathbf{s}}\le ⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{t}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{t}}\left(i\right)\ge ⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{sigma}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{sigma}}>0.0$.
${\mathbf{ifail}}=8$
On entry, ${\mathbf{r}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{r}}\ge 0.0$.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{q}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{q}}\ge 0.0$.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{ldp}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=14$
On entry, ${\mathbf{s}}=⟨\mathit{\text{value}}⟩$ and $\beta =⟨\mathit{\text{value}}⟩$.
Constraint: ${{\mathbf{s}}}^{\beta }<⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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 g01haf, giving a maximum absolute error of less than $5×{10}^{-16}$. The univariate cumulative Normal distribution function also forms part of the evaluation (see s15abf and s15adf).

## 8Parallelism and Performance

s30qcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
s30qcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

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.

### 10.1Program Text

Program Text (s30qcfe.f90)

### 10.2Program Data

Program Data (s30qcfe.d)

### 10.3Program Results

Program Results (s30qcfe.r)