nag_amer_bs_price (s30qcc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_amer_bs_price (s30qcc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_amer_bs_price (s30qcc) computes the Bjerksund and Stensland (2002) approximation to the price of an American option.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_amer_bs_price (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double x[], double s, const double t[], double sigma, double r, double q, double p[], NagError *fail)

3  Description

nag_amer_bs_price (s30qcc) 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 .

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:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     optionNag_CallPutInput
On entry: determines whether the option is a call or a put.
option=Nag_Call
A call. The holder has a right to buy.
option=Nag_Put
A put. The holder has a right to sell.
Constraint: option=Nag_Call or Nag_Put.
3:     mIntegerInput
On entry: the number of strike prices to be used.
Constraint: m1.
4:     nIntegerInput
On entry: the number of times to expiry to be used.
Constraint: n1.
5:     x[m]const doubleInput
On entry: x[i-1] must contain Xi, the ith strike price, for i=1,2,,m.
Constraint: x[i-1]z ​ and ​ x[i-1] 1 / z , where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,m.
6:     sdoubleInput
On entry: S, the price of the underlying asset.
Constraint: sz ​ and ​s1z, where z=nag_real_safe_small_number, the safe range parameter and sβ<1z where β is as defined in Section 3.
7:     t[n]const doubleInput
On entry: t[i-1] must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: t[i-1]z, where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,n.
8:     sigmadoubleInput
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.
9:     rdoubleInput
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.
10:   qdoubleInput
On entry: q, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0.
11:   p[m×n]doubleOutput
Note: the i,jth element of the matrix P is stored in
  • p[j-1×m+i-1] when order=Nag_ColMajor;
  • p[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array p contains the computed option prices.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, q=value.
Constraint: q0.0.
On entry, r=value.
Constraint: r0.0.
On entry, s=value.
Constraint: svalue and svalue.
On entry, s=value and β=value.
Constraint: sβ<value.
On entry, sigma=value.
Constraint: sigma>0.0.
NE_REAL_ARRAY
On entry, t[value]=value.
Constraint: t[i]value.
On entry, x[value]=value.
Constraint: x[i]value and x[i]value.

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 nag_bivariate_normal_dist (g01hac), giving a maximum absolute error of less than 5×10-16. The univariate cumulative Normal distribution function also forms part of the evaluation (see nag_cumul_normal (s15abc) and nag_erfc (s15adc)).

8  Further Comments

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 (s30qcce.c)

9.2  Program Data

Program Data (s30qcce.d)

9.3  Program Results

Program Results (s30qcce.r)


nag_amer_bs_price (s30qcc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012