# NAG CL Interfaced03nec (dim1_​blackscholes_​means)

## 1Purpose

d03nec computes average values of a continuous function of time over the remaining life of an option. It is used together with d03ndc to value options with time-dependent arguments.

## 2Specification

 #include
 void d03nec (double t0, double tmat, Integer ntd, const double td[], const double phid[], double phiav[], NagError *fail)
The function may be called by the names: d03nec, nag_pde_dim1_blackscholes_means or nag_pde_bs_1d_means.

## 3Description

d03nec computes the quantities
 $ϕt0, ϕ^=1T-t0 ∫t0Tϕζdζ, ϕ¯= 1T-t0 ∫t0Tϕ2ζdζ 1/2$
from a given set of values phid of a continuous time-dependent function $\varphi \left(t\right)$ at a set of discrete points td, where ${t}_{0}$ is the current time and $T$ is the maturity time. Thus $\stackrel{^}{\varphi }$ and $\overline{\varphi }$ are first and second order averages of $\varphi$ over the remaining life of an option.
The function may be used in conjunction with d03ndc in order to value an option in the case where the risk-free interest rate $r$, the continuous dividend $q$, or the stock volatility $\sigma$ is time-dependent and is described by values at a set of discrete times (see Section 9.2). This is illustrated in Section 10.

None.

## 5Arguments

1: $\mathbf{t0}$double Input
On entry: the current time ${t}_{0}$.
Constraint: ${\mathbf{td}}\left[0\right]\le {\mathbf{t0}}\le {\mathbf{td}}\left[{\mathbf{ntd}}-1\right]$.
2: $\mathbf{tmat}$double Input
On entry: the maturity time $T$.
Constraint: ${\mathbf{td}}\left[0\right]\le {\mathbf{tmat}}\le {\mathbf{td}}\left[{\mathbf{ntd}}-1\right]$.
3: $\mathbf{ntd}$Integer Input
On entry: the number of discrete times at which $\varphi$ is given.
Constraint: ${\mathbf{ntd}}\ge 2$.
4: $\mathbf{td}\left[{\mathbf{ntd}}\right]$const double Input
On entry: the discrete times at which $\varphi$ is specified.
Constraint: ${\mathbf{td}}\left[0\right]<{\mathbf{td}}\left[1\right]<\cdots <{\mathbf{td}}\left[{\mathbf{ntd}}-1\right]$.
5: $\mathbf{phid}\left[{\mathbf{ntd}}\right]$const double Input
On entry: ${\mathbf{phid}}\left[\mathit{i}-1\right]$ must contain the value of $\varphi$ at time ${\mathbf{td}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{ntd}}$.
6: $\mathbf{phiav}\left[3\right]$double Output
On exit: ${\mathbf{phiav}}\left[0\right]$ contains the value of $\varphi$ interpolated to ${t}_{0}$, ${\mathbf{phiav}}\left[1\right]$ contains the first-order average $\stackrel{^}{\varphi }$ and ${\mathbf{phiav}}\left[2\right]$ contains the second-order average $\overline{\varphi }$, where:
 $ϕ^=1T-t0 ∫t0Tϕζdζ , ϕ¯= 1T-t0 ∫t0Tϕ2ζdζ 1/2 .$
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{ntd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ntd}}\ge 2$.
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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING
On entry, ${\mathbf{td}}\left[〈\mathit{\text{value}}〉+1\right]\le {\mathbf{td}}\left[〈\mathit{\text{value}}〉\right]$.
Constraint: ${\mathbf{td}}\left[0\right]<{\mathbf{td}}\left[1\right]<\cdots <{\mathbf{td}}\left[{\mathbf{ntd}}-1\right]$.
NE_REAL_3
On entry, ${\mathbf{t0}}=〈\mathit{\text{value}}〉$, ${\mathbf{td}}\left[0\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{td}}\left[{\mathbf{ntd}}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{td}}\left[0\right]\le {\mathbf{t0}}\le {\mathbf{td}}\left[{\mathbf{ntd}}-1\right]$.
On entry, ${\mathbf{tmat}}=〈\mathit{\text{value}}〉$, ${\mathbf{td}}\left[0\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{td}}\left[{\mathbf{ntd}}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{td}}\left[0\right]\le {\mathbf{tmat}}\le {\mathbf{td}}\left[{\mathbf{ntd}}-1\right]$.

## 7Accuracy

If $\varphi \in {C}^{4}\left[{t}_{0},T\right]$ then the error in the approximation of $\varphi \left({t}_{0}\right)$ and $\stackrel{^}{\varphi }$ is $\mathit{O}\left({H}^{4}\right)$, where $H=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\left(T\left(i+1\right)-T\left(i\right)\right)$, for $i=1,2,\dots ,{\mathbf{ntd}}-1$. The approximation is exact for polynomials of degree up to $3$.
The third quantity $\overline{\varphi }$ is $\mathit{O}\left({H}^{2}\right)$, and exact for linear functions.

## 8Parallelism and Performance

d03nec is not threaded in any implementation.

### 9.1Timing

The time taken is proportional to ntd.

### 9.2Use with d03ndc

Suppose you wish to evaluate the analytic solution of the Black–Scholes equation in the case when the risk-free interest rate $r$ is a known function of time, and is represented as a set of values at discrete times. A call to d03nec providing these values in phid produces an output array phiav suitable for use as the argument r in a subsequent call to d03ndc.
Time-dependent values of the continuous dividend $Q$ and the volatility $\sigma$ may be handled in the same way.

### 9.3Algorithmic Details

The ntd data points are fitted with a cubic B-spline using the function e01bac. Evaluation is then performed using e02bbc, and the definite integrals are computed using direct integration of the cubic splines in each interval. The special case of $T={t}_{o}$ is handled by interpolating $\varphi$ at that point.

## 10Example

This example demonstrates the use of the function in conjunction with d03ndc to solve the Black–Scholes equation for valuation of a $5$-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 d03nec the solution of the Black–Scholes equation by d03ndc is also exact.
The option is valued at a range of times and stock prices.

### 10.1Program Text

Program Text (d03nece.c)

### 10.2Program Data

Program Data (d03nece.d)

### 10.3Program Results

Program Results (d03nece.r)