NAG Library Routine Document

s30qcf  (opt_amer_bs_price)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

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

2
Specification

Fortran Interface
Subroutine s30qcf ( calput, m, n, x, s, t, sigma, r, q, p, ldp, ifail)
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
C Header Interface
#include nagmk26.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)

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 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 ,  
where
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.

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
Arguments

1:     calput – Character(1)Input
On entry: determines whether the option is a call or a put.
calput='C'
A call; the holder has a right to buy.
calput='P'
A put; the holder has a right to sell.
Constraint: calput='C' or 'P'.
2:     m – IntegerInput
On entry: the number of strike prices to be used.
Constraint: m1.
3:     n – IntegerInput
On entry: the number of times to expiry to be used.
Constraint: n1.
4:     xm – Real (Kind=nag_wp) arrayInput
On entry: xi must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: xiz ​ and ​ xi 1 / z , where z = x02amf , the safe range parameter, for i=1,2,,m.
5:     s – Real (Kind=nag_wp)Input
On entry: S, the price of the underlying asset.
Constraint: sz ​ and ​s1z, where z=x02amf, the safe range parameter and sβ<1z where β is as defined in Section 3.
6:     tn – Real (Kind=nag_wp) arrayInput
On entry: ti must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: tiz, where z = x02amf , the safe range parameter, for i=1,2,,n.
7:     sigma – Real (Kind=nag_wp)Input
On entry: σ, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: 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: r0.0.
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: q0.0.
10:   pldpn – Real (Kind=nag_wp) arrayOutput
On exit: 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.
11:   ldp – IntegerInput
On entry: the first dimension of the array p as declared in the (sub)program from which s30qcf is called.
Constraint: ldpm.
12:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ 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 argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6
Error Indicators and Warnings

If on entry 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:
ifail=1
On entry, calput=value was an illegal value.
ifail=2
On entry, m=value.
Constraint: m1.
ifail=3
On entry, n=value.
Constraint: n1.
ifail=4
On entry, xvalue=value.
Constraint: xivalue and xivalue.
ifail=5
On entry, s=value.
Constraint: svalue and svalue.
ifail=6
On entry, tvalue=value.
Constraint: tivalue.
ifail=7
On entry, sigma=value.
Constraint: sigma>0.0.
ifail=8
On entry, r=value.
Constraint: r0.0.
ifail=9
On entry, q=value.
Constraint: q0.0.
ifail=11
On entry, ldp=value and m=value.
Constraint: ldpm.
ifail=14
On entry, s=value and β=value.
Constraint: sβ<value.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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).

8
Parallelism 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.

9
Further Comments

None.

10
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.

10.1
Program Text

Program Text (s30qcfe.f90)

10.2
Program Data

Program Data (s30qcfe.d)

10.3
Program Results

Program Results (s30qcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017