NAG Library Function Document
nag_pde_bs_1d_means (d03nec)
1
Purpose
nag_pde_bs_1d_means (d03nec) computes average values of a continuous function of time over the remaining life of an option. It is used together with
nag_pde_bs_1d_analytic (d03ndc) to value options with time-dependent arguments.
2
Specification
#include <nag.h> |
#include <nagd03.h> |
void |
nag_pde_bs_1d_means (double t0,
double tmat,
Integer ntd,
const double td[],
const double phid[],
double phiav[],
NagError *fail) |
|
3
Description
nag_pde_bs_1d_means (d03nec) computes the quantities
from a given set of values
phid of a continuous time-dependent function
at a set of discrete points
td, where
is the current time and
is the maturity time. Thus
and
are first and second order averages of
over the remaining life of an option.
The function may be used in conjunction with
nag_pde_bs_1d_analytic (d03ndc) in order to value an option in the case where the risk-free interest rate
, the continuous dividend
, or the stock volatility
is time-dependent and is described by values at a set of discrete times (see
Section 9.2). This is illustrated in
Section 10.
4
References
None.
5
Arguments
- 1:
– doubleInput
-
On entry: the current time .
Constraint:
.
- 2:
– doubleInput
-
On entry: the maturity time .
Constraint:
.
- 3:
– IntegerInput
-
On entry: the number of discrete times at which is given.
Constraint:
.
- 4:
– const doubleInput
-
On entry: the discrete times at which is specified.
Constraint:
.
- 5:
– const doubleInput
-
On entry: must contain the value of at time , for .
- 6:
– doubleOutput
-
On exit:
contains the value of
interpolated to
,
contains the first-order average
and
contains the second-order average
, where:
- 7:
– 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 had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
- 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.
Unexpected failure in internal call to spline function.
- 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_NOT_STRICTLY_INCREASING
-
On entry, , for .
- NE_REAL_3
-
On entry,
t0 lies outside the range
:
,
and
.
On entry,
tmat lies outside the range
:
,
and
.
7
Accuracy
If then the error in the approximation of and is , where , for . The approximation is exact for polynomials of degree up to .
The third quantity is , and exact for linear functions.
8
Parallelism and Performance
nag_pde_bs_1d_means (d03nec) is not threaded in any implementation.
9.1
Timing
The time taken is proportional to
ntd.
Suppose you wish to evaluate the analytic solution of the Black–Scholes equation in the case when the risk-free interest rate
is a known function of time, and is represented as a set of values at discrete times. A call to
nag_pde_bs_1d_means (d03nec) providing these values in
phid produces an output array
phiav suitable for use as the argument
r in a subsequent call to
nag_pde_bs_1d_analytic (d03ndc).
Time-dependent values of the continuous dividend and the volatility may be handled in the same way.
9.3
Algorithmic Details
The
ntd data points are fitted with a cubic B-spline using the function
nag_1d_spline_interpolant (e01bac). Evaluation is then performed using
nag_1d_spline_evaluate (e02bbc), and the definite integrals are computed using direct integration of the cubic splines in each interval. The special case of
is handled by interpolating
at that point.
10
Example
This example demonstrates the use of the function in conjunction with
nag_pde_bs_1d_analytic (d03ndc) to solve the Black–Scholes equation for valuation of a
-month American call option on a non-dividend-paying stock with an exercise price of $50. The risk-free interest rate varies linearly with time and the stock volatility has a quadratic variation. Since these functions are integrated exactly by
nag_pde_bs_1d_means (d03nec) the solution of the Black–Scholes equation by
nag_pde_bs_1d_analytic (d03ndc) is also exact.
The option is valued at a range of times and stock prices.
10.1
Program Text
Program Text (d03nece.c)
10.2
Program Data
Program Data (d03nece.d)
10.3
Program Results
Program Results (d03nece.r)