# NAG Library Routine Document

## 1Purpose

d03nef computes average values of a continuous function of time over the remaining life of an option. It is used together with d03ndf to value options with time-dependent arguments.

## 2Specification

Fortran Interface
 Subroutine d03nef ( t0, tmat, ntd, td, phid, work,
 Integer, Intent (In) :: ntd, lwork Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: t0, tmat, td(ntd), phid(ntd) Real (Kind=nag_wp), Intent (Out) :: phiav(3), work(lwork)
#include nagmk26.h
 void d03nef_ (const double *t0, const double *tmat, const Integer *ntd, const double td[], const double phid[], double phiav[], double work[], const Integer *lwork, Integer *ifail)

## 3Description

d03nef 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 $\stackrel{-}{\varphi }$ are first and second order averages of $\varphi$ over the remaining life of an option.
The routine may be used in conjunction with d03ndf 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:     $\mathbf{t0}$ – Real (Kind=nag_wp)Input
On entry: the current time ${t}_{0}$.
Constraint: ${\mathbf{td}}\left(1\right)\le {\mathbf{t0}}\le {\mathbf{td}}\left({\mathbf{ntd}}\right)$.
2:     $\mathbf{tmat}$ – Real (Kind=nag_wp)Input
On entry: the maturity time $T$.
Constraint: ${\mathbf{td}}\left(1\right)\le {\mathbf{tmat}}\le {\mathbf{td}}\left({\mathbf{ntd}}\right)$.
3:     $\mathbf{ntd}$ – IntegerInput
On entry: the number of discrete times at which $\varphi$ is given.
Constraint: ${\mathbf{ntd}}\ge 2$.
4:     $\mathbf{td}\left({\mathbf{ntd}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the discrete times at which $\varphi$ is specified.
Constraint: ${\mathbf{td}}\left(1\right)<{\mathbf{td}}\left(2\right)<\cdots <{\mathbf{td}}\left({\mathbf{ntd}}\right)$.
5:     $\mathbf{phid}\left({\mathbf{ntd}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{phid}}\left(\mathit{i}\right)$ must contain the value of $\varphi$ at time ${\mathbf{td}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ntd}}$.
6:     $\mathbf{phiav}\left(3\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{phiav}}\left(1\right)$ contains the value of $\varphi$ interpolated to ${t}_{0}$, ${\mathbf{phiav}}\left(2\right)$ contains the first-order average $\stackrel{^}{\varphi }$ and ${\mathbf{phiav}}\left(3\right)$ contains the second-order average $\stackrel{-}{\varphi }$, where:
 $ϕ^=1T-t0 ∫t0Tϕζdζ , ϕ-= 1T-t0 ∫t0Tϕ2ζdζ 1/2 .$
7:     $\mathbf{work}\left({\mathbf{lwork}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
8:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which d03nef is called.
Constraint: ${\mathbf{lwork}}\ge 9×{\mathbf{ntd}}+24$.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, t0 lies outside the range [${\mathbf{td}}\left(1\right),{\mathbf{td}}\left({\mathbf{ntd}}\right)$], or tmat lies outside the range [${\mathbf{td}}\left(1\right),{\mathbf{td}}\left({\mathbf{ntd}}\right)$], or ${\mathbf{ntd}}<2$, or td badly ordered, or ${\mathbf{lwork}}<9×{\mathbf{ntd}}+24$.
${\mathbf{ifail}}=2$
Unexpected failure in internal call to e01baf or e02bbf.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 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 $\stackrel{-}{\varphi }$ is $\mathit{O}\left({H}^{2}\right)$, and exact for linear functions.

## 8Parallelism and Performance

d03nef is not threaded in any implementation.

### 9.1Timing

The time taken is proportional to ntd.

### 9.2Use with d03ndf

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 d03nef providing these values in phid produces an output array phiav suitable for use as the argument r in a subsequent call to d03ndf.
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 routine e01baf. Evaluation is then performed using e02bbf, 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 routine in conjunction with d03ndf 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 d03nef the solution of the Black–Scholes equation by d03ndf is also exact.
The option is valued at a range of times and stock prices.

### 10.1Program Text

Program Text (d03nefe.f90)

### 10.2Program Data

Program Data (d03nefe.d)

### 10.3Program Results

Program Results (d03nefe.r)

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