nag_bsm_price (s30aac) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_bsm_price (s30aac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_bsm_price (s30aac) computes the European option price given by the Black–Scholes–Merton formula.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_bsm_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_bsm_price (s30aac) computes the price of a European call (or put) option for constant volatility, σ, and risk-free interest rate, r, with a possible dividend yield, q, using the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). For a given strike price, X, the price of a European call with underlying price, S, and time to expiry, T, is
Pcall = Se-qT Φd1 - Xe-rT Φd2
and the corresponding European put price is
Pput = Xe-rT Φ-d2 - Se-qT Φ-d1
and where Φ denotes the cumulative Normal distribution function,
Φx = 12π - x exp -y2/2 dy
and
d1 = ln S/X + r-q+ σ2 / 2 T σT , d2 = d1 - σT .

4  References

Black F and Scholes M (1973) The pricing of options and corporate liabilities Journal of Political Economy 81 637–654
Merton R C (1973) Theory of rational option pricing Bell Journal of Economics and Management Science 4 141–183

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 ​s1.0/z, where z=nag_real_safe_small_number, the safe range parameter.
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, 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 is dependent on the accuracy of the cumulative Normal distribution function, Φ. This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the machine precision (see nag_cumul_normal (s15abc) and nag_erfc (s15adc)). An accuracy close to machine precision can generally be expected.

8  Further Comments

None.

9  Example

This example computes the prices for six European call options using two expiry times and three strike prices as input. The times to expiry are taken as 0.7 and 0.8 years respectively. The stock price is 55, with strike prices, 58, 60 and 62. The risk-free interest rate is 10% per year and the volatility is 30% per year.

9.1  Program Text

Program Text (s30aace.c)

9.2  Program Data

Program Data (s30aace.d)

9.3  Program Results

Program Results (s30aace.r)


nag_bsm_price (s30aac) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

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