NAG Library Function Document

nag_bsm_greeks (s30abc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_bsm_greeks (s30abc) computes the European option price given by the Black–Scholes–Merton formula together with its sensitivities (Greeks).

2
Specification

#include <nag.h>
#include <nags.h>
void  nag_bsm_greeks (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[], double delta[], double gamma[], double vega[], double theta[], double rho[], double crho[], double vanna[], double charm[], double speed[], double colour[], double zomma[], double vomma[], NagError *fail)

3
Description

nag_bsm_greeks (s30abc) computes the price of a European call (or put) option together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters, by the Black–Scholes–Merton formula (see Black and Scholes (1973) and Merton (1973)). The annual volatility, σ, risk-free interest rate, r, and dividend yield, q, must be supplied as input. 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 .  
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

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:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     option Nag_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:     m IntegerInput
On entry: the number of strike prices to be used.
Constraint: m1.
4:     n IntegerInput
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:     s doubleInput
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:     sigma doubleInput
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:     r doubleInput
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:   q doubleInput
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: where Pi,j appears in this document, it refers to the array element
  • p[j-1×m+i-1] when order=Nag_ColMajor;
  • p[i-1×n+j-1] when order=Nag_RowMajor.
On exit: Pi,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.
12:   delta[m×n] doubleOutput
Note: the i,jth element of the matrix is stored in
  • delta[j-1×m+i-1] when order=Nag_ColMajor;
  • delta[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array delta contains the sensitivity, PS, of the option price to change in the price of the underlying asset.
13:   gamma[m×n] doubleOutput
Note: the i,jth element of the matrix is stored in
  • gamma[j-1×m+i-1] when order=Nag_ColMajor;
  • gamma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array gamma contains the sensitivity, 2PS2, of delta to change in the price of the underlying asset.
14:   vega[m×n] doubleOutput
Note: where VEGAi,j appears in this document, it refers to the array element
  • vega[j-1×m+i-1] when order=Nag_ColMajor;
  • vega[i-1×n+j-1] when order=Nag_RowMajor.
On exit: VEGAi,j, contains the first-order Greek measuring the sensitivity of the option price Pij to change in the volatility of the underlying asset, i.e., Pij σ , for i=1,2,,m and j=1,2,,n.
15:   theta[m×n] doubleOutput
Note: where THETAi,j appears in this document, it refers to the array element
  • theta[j-1×m+i-1] when order=Nag_ColMajor;
  • theta[i-1×n+j-1] when order=Nag_RowMajor.
On exit: THETAi,j, contains the first-order Greek measuring the sensitivity of the option price Pij to change in time, i.e., - Pij T , for i=1,2,,m and j=1,2,,n, where b=r-q.
16:   rho[m×n] doubleOutput
Note: where RHOi,j appears in this document, it refers to the array element
  • rho[j-1×m+i-1] when order=Nag_ColMajor;
  • rho[i-1×n+j-1] when order=Nag_RowMajor.
On exit: RHOi,j, contains the first-order Greek measuring the sensitivity of the option price Pij to change in the annual risk-free interest rate, i.e., - Pij r , for i=1,2,,m and j=1,2,,n.
17:   crho[m×n] doubleOutput
Note: where CRHOi,j appears in this document, it refers to the array element
  • crho[j-1×m+i-1] when order=Nag_ColMajor;
  • crho[i-1×n+j-1] when order=Nag_RowMajor.
On exit: CRHOi,j, contains the first-order Greek measuring the sensitivity of the option price Pij to change in the annual cost of carry rate, i.e., - Pij b , for i=1,2,,m and j=1,2,,n, where b=r-q.
18:   vanna[m×n] doubleOutput
Note: where VANNAi,j appears in this document, it refers to the array element
  • vanna[j-1×m+i-1] when order=Nag_ColMajor;
  • vanna[i-1×n+j-1] when order=Nag_RowMajor.
On exit: VANNAi,j, contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the volatility of the asset price, i.e., - Δij T = - 2 Pij Sσ , for i=1,2,,m and j=1,2,,n.
19:   charm[m×n] doubleOutput
Note: where CHARMi,j appears in this document, it refers to the array element
  • charm[j-1×m+i-1] when order=Nag_ColMajor;
  • charm[i-1×n+j-1] when order=Nag_RowMajor.
On exit: CHARMi,j, contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the time, i.e., - Δij T = - 2 Pij ST , for i=1,2,,m and j=1,2,,n.
20:   speed[m×n] doubleOutput
Note: where SPEEDi,j appears in this document, it refers to the array element
  • speed[j-1×m+i-1] when order=Nag_ColMajor;
  • speed[i-1×n+j-1] when order=Nag_RowMajor.
On exit: SPEEDi,j, contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the price of the underlying asset, i.e., - Γij S = - 3 Pij S3 , for i=1,2,,m and j=1,2,,n.
21:   colour[m×n] doubleOutput
Note: where COLOURi,j appears in this document, it refers to the array element
  • colour[j-1×m+i-1] when order=Nag_ColMajor;
  • colour[i-1×n+j-1] when order=Nag_RowMajor.
On exit: COLOURi,j, contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the time, i.e., - Γij T = - 3 Pij ST , for i=1,2,,m and j=1,2,,n.
22:   zomma[m×n] doubleOutput
Note: where ZOMMAi,j appears in this document, it refers to the array element
  • zomma[j-1×m+i-1] when order=Nag_ColMajor;
  • zomma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: ZOMMAi,j, contains the third-order Greek measuring the sensitivity of the second-order Greek Γij to change in the volatility of the underlying asset, i.e., - Γij σ = - 3 Pij S2σ , for i=1,2,,m and j=1,2,,n.
23:   vomma[m×n] doubleOutput
Note: where VOMMAi,j appears in this document, it refers to the array element
  • vomma[j-1×m+i-1] when order=Nag_ColMajor;
  • vomma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: VOMMAi,j, contains the second-order Greek measuring the sensitivity of the first-order Greek Δij to change in the volatility of the underlying asset, i.e., - Δij σ = - 2 Pij σ2 , for i=1,2,,m and j=1,2,,n.
24:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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
Parallelism and Performance

nag_bsm_greeks (s30abc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. 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 a European put with a time to expiry of 0.7 years, a stock price of 55 and a strike price of 60. The risk-free interest rate is 10% per year and the volatility is 30% per year.

10.1
Program Text

Program Text (s30abce.c)

10.2
Program Data

Program Data (s30abce.d)

10.3
Program Results

Program Results (s30abce.r)

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