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NAG Library Manual

# NAG Library Routine DocumentS30QCF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S30QCF computes the Bjerksund and Stensland (2002) approximation to the price of an American option.

## 2  Specification

 SUBROUTINE S30QCF ( CALPUT, M, N, X, S, T, SIGMA, R, Q, P, LDP, IFAIL)
 INTEGER M, N, LDP, IFAIL REAL (KIND=nag_wp) X(M), S, T(N), SIGMA, R, Q, P(LDP,N) CHARACTER(1) CALPUT

## 3  Description

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
 $X~τ = B0 + B∞ - B0 1 - exp hτ ,$
where
 $hτ = - bτ+2σ⁢τ X2 B∞ - B0 B0 ,$
 $B∞ ≡ β β-1 X , B0 ≡ maxX, rr-b X ,$
 $β = 12 - bσ2 + b σ2 - 12 2 + 2 r σ2 .$
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 $\sigma$ 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 $\alpha$, $\varphi$ and $\Psi$ are as defined in Bjerksund and Stensland (2002).
The price of a put option is obtained by the put-call transformation,
 $Pput X,S,T,σ,r,q = Pcall S,X,T,σ,q,r .$

## 4  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 $t$ probabilities Statistics and Computing 14 151–160

## 5  Parameters

1:     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:     M – INTEGERInput
On entry: the number of strike prices to be used.
Constraint: ${\mathbf{M}}\ge 1$.
3:     N – INTEGERInput
On entry: the number of times to expiry to be used.
Constraint: ${\mathbf{N}}\ge 1$.
4:     X(M) – REAL (KIND=nag_wp) arrayInput
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:     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:     T(N) – REAL (KIND=nag_wp) arrayInput
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:     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:     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:     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:   P(LDP,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the leading ${\mathbf{M}}×{\mathbf{N}}$ part of the array P contains the computed option prices.
11:   LDP – INTEGERInput
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:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{CALPUT}}\ne \text{'C'}$ or $\text{'P'}$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{M}}\le 0$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{N}}\le 0$.
${\mathbf{IFAIL}}=4$
On entry, ${\mathbf{X}}\left(\mathit{i}\right) or ${\mathbf{X}}\left(\mathit{i}\right)>1/z$, where $z={\mathbf{X02AMF}}\left(\right)$, the safe range parameter.
${\mathbf{IFAIL}}=5$
On entry, ${\mathbf{S}} or ${\mathbf{S}}>1.0/z$, where $z={\mathbf{X02AMF}}\left(\right)$, the safe range parameter.
${\mathbf{IFAIL}}=6$
On entry, ${\mathbf{T}}\left(\mathit{i}\right), where $z={\mathbf{X02AMF}}\left(\right)$, the safe range parameter.
${\mathbf{IFAIL}}=7$
On entry, ${\mathbf{SIGMA}}\le 0.0$.
${\mathbf{IFAIL}}=8$
On entry, ${\mathbf{R}}<0.0$.
${\mathbf{IFAIL}}=9$
On entry, ${\mathbf{Q}}<0.0$.
${\mathbf{IFAIL}}=11$
On entry, ${\mathbf{LDP}}<{\mathbf{M}}$.
${\mathbf{IFAIL}}=14$
On entry, $\beta \ge \frac{1}{z}$, where $z={\mathbf{X02AMF}}\left(\right)$, the safe range parameter (see Section 3).
${\mathbf{IFAIL}}=15$
Internal memory allocation failed.

## 7  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 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).

None.

## 9  Example

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.

### 9.1  Program Text

Program Text (s30qcfe.f90)

### 9.2  Program Data

Program Data (s30qcfe.d)

### 9.3  Program Results

Program Results (s30qcfe.r)