NAG FL Interface
s30caf (opt_​binary_​con_​price)

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1 Purpose

s30caf computes the price of a binary or digital cash-or-nothing option.

2 Specification

Fortran Interface
Subroutine s30caf ( calput, m, n, x, s, k, 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, k, 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  s30caf_ (const char *calput, const Integer *m, const Integer *n, const double x[], const double *s, const double *k, 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 s30caf or nagf_specfun_opt_binary_con_price.

3 Description

s30caf computes the price of a binary or digital cash-or-nothing option which pays a fixed amount, K, at expiration if the option is in-the-money (see Section 2.4 in the S Chapter Introduction). For a strike price, X, underlying asset price, S, and time to expiry, T, the payoff is, therefore, K, if S>X for a call or S<X for a put. Nothing is paid out when this condition is not met.
The price of a call with volatility, σ, risk-free interest rate, r, and annualised dividend yield, q, is
Pcall = K e-rT Φ(d2)  
and for a put,
Pput = K e-rT Φ(-d2)  
where Φ is the cumulative Normal distribution function,
Φ(x) = 1 2π - x (-y2/2) dy ,  
d2 = ln (S/X) + (r-q-σ2/2) T σT .  
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

Reiner E and Rubinstein M (1991) Unscrambling the binary code Risk 4

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 ​s1.0/z, where z=x02amf(), the safe range parameter.
6: k Real (Kind=nag_wp) Input
On entry: the amount, K, to be paid at expiration if the option is in-the-money, i.e., if s>x(i) when calput='C', or if s<x(i) when calput='P', for i=1,2,,m.
Constraint: k0.0.
7: 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.
8: 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.
9: 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.
10: 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.
11: 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.
12: ldp Integer Input
On entry: the first dimension of the array p as declared in the (sub)program from which s30caf is called.
Constraint: ldpm.
13: 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, k=value.
Constraint: k0.0.
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.
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 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 s15abf and s15adf). An accuracy close to machine precision can generally be expected.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s30caf 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 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 a cash-or-nothing put with a time to expiry of 0.75 years, a stock price of 100 and a strike price of 80. The risk-free interest rate is 6% per year and the volatility is 35% per year. If the option is in-the-money at expiration, i.e., if S>X, the payoff is 10.

10.1 Program Text

Program Text (s30cafe.f90)

10.2 Program Data

Program Data (s30cafe.d)

10.3 Program Results

Program Results (s30cafe.r)