NAG FL Interface
s30qcf (opt_​amer_​bs_​price)

Settings help

FL Name Style:

FL Specification Language:

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 <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.

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{h(τ)}) ,  
h(τ) = - (bτ+2στ) ( 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 σ 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 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=P(X=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
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.
A call; the holder has a right to buy.
A put; the holder has a right to sell.
Constraint: calput='C' or 'P'.
2: m Integer Input
On entry: the number of strike prices to be used.
Constraint: m1.
3: n Integer Input
On entry: the number of times to expiry to be used.
Constraint: n1.
4: x(m) Real (Kind=nag_wp) array Input
On entry: x(i) must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: x(i)z ​ and ​ x(i) 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: t(n) Real (Kind=nag_wp) array Input
On entry: t(i) must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: t(i)z, 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: p(ldp,n) Real (Kind=nag_wp) array Output
On exit: p(i,j) 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 Integer Input
On entry: the first dimension of the array p as declared in the (sub)program from which s30qcf is called.
Constraint: ldpm.
12: 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 -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:
On entry, calput=value was an illegal value.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
On entry, x(value)=value.
Constraint: x(i)value and x(i)value.
On entry, s=value.
Constraint: svalue and svalue.
On entry, t(value)=value.
Constraint: t(i)value.
On entry, sigma=value.
Constraint: sigma>0.0.
On entry, r=value.
Constraint: r0.0.
On entry, q=value.
Constraint: q0.0.
On entry, ldp=value and m=value.
Constraint: ldpm.
On entry, s=value and β=value.
Constraint: sβ<value.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

Background information to multithreading can be found in the Multithreading documentation.
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


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)