NAG Library Routine Document

d03ncf (dim1_blackscholes_fd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{kopt}$ – IntegerInput

On entry: specifies the kind of option to be valued.

$\mathbf{kopt}=1$

A European call option.

$\mathbf{kopt}=2$

An American call option.

$\mathbf{kopt}=3$

A European put option.

$\mathbf{kopt}=4$

An American put option.

Constraint: $\mathbf{kopt}=1$, $2$, $3$ or $4$.

2:     $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the exercise price $X$.

3:     $\mathbf{mesh}$ – Character(1)Input

On entry: indicates the type of finite difference mesh to be used:

$\mathbf{mesh}=\text{'U'}$

Uniform mesh.

$\mathbf{mesh}=\text{'C'}$

Custom mesh supplied by you.

Constraint: $\mathbf{mesh}=\text{'U'}$ or $\text{'C'}$.

4:     $\mathbf{ns}$ – IntegerInput

On entry: the number of stock prices to be used in the finite difference mesh.

Constraint: $\mathbf{ns}\ge 2$.

5:     $\mathbf{s}\left(\mathbf{ns}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if $\mathbf{mesh}=\text{'C'}$, $\mathbf{s}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th stock price in the mesh, for $\mathit{i}=1,2,\dots ,\mathbf{ns}$. These values should be in increasing order, with $\mathbf{s}\left(1\right)=S_{\min }$ and $\mathbf{s}\left(\mathbf{ns}\right)=S_{\max }$.

If $\mathbf{mesh}=\text{'U'}$, $\mathbf{s}\left(1\right)$ must be set to $S_{\min }$ and $\mathbf{s}\left(\mathbf{ns}\right)$ to $S_{\max }$, but $\mathbf{s}\left(2\right),\mathbf{s}\left(3\right),\dots ,\mathbf{s}\left(\mathbf{ns}-1\right)$ need not be initialized, as they will be set internally by the routine in order to define a uniform mesh.

On exit: if $\mathbf{mesh}=\text{'U'}$, the elements of s define a uniform mesh over $\left[S_{\min },S_{\max }\right]$.

If $\mathbf{mesh}=\text{'C'}$, the elements of s are unchanged.

Constraints:

• if $\mathbf{mesh}=\text{'C'}$, $\mathbf{s}\left(1\right)\ge 0.0$ and $\mathbf{s}\left(\mathit{i}\right)<\mathbf{s}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ns}-1$;
• if $\mathbf{mesh}=\text{'U'}$, $0.0\le \mathbf{s}\left(1\right)<\mathbf{s}\left(\mathbf{ns}\right)$.

6:     $\mathbf{nt}$ – IntegerInput

On entry: the number of time-steps to be used in the finite difference method.

Constraint: $\mathbf{nt}\ge 2$.

7:     $\mathbf{t}\left(\mathbf{nt}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if $\mathbf{mesh}=\text{'C'}$ then $\mathbf{t}\left(\mathit{j}\right)$ must contain the $\mathit{j}$th time in the mesh, for $\mathit{j}=1,2,\dots ,\mathbf{nt}$. These values should be in increasing order, with $\mathbf{t}\left(1\right)=t_{\min }$ and $\mathbf{t}\left(\mathbf{nt}\right)=t_{\max }$.

If $\mathbf{mesh}=\text{'U'}$ then $\mathbf{t}\left(1\right)$ must be set to $t_{\min }$ and $\mathbf{t}\left(\mathbf{nt}\right)$ to $t_{\max }$, but $\mathbf{t}\left(2\right),\mathbf{t}\left(3\right),\dots ,\mathbf{t}\left(\mathbf{nt}-1\right)$ need not be initialized, as they will be set internally by the routine in order to define a uniform mesh.

On exit: if $\mathbf{mesh}=\text{'U'}$, the elements of t define a uniform mesh over $\left[t_{\min },t_{\max }\right]$.

If $\mathbf{mesh}=\text{'C'}$, the elements of t are unchanged.

Constraints:

• if $\mathbf{mesh}=\text{'C'}$, $\mathbf{t}\left(1\right)\ge 0.0$ and $\mathbf{t}\left(\mathit{j}\right)<\mathbf{t}\left(\mathit{j}+1\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{nt}-1$;
• if $\mathbf{mesh}=\text{'U'}$, $0.0\le \mathbf{t}\left(1\right)<\mathbf{t}\left(\mathbf{nt}\right)$.

8:     $\mathbf{tdpar}\left(3\right)$ – Logical arrayInput

On entry: specifies whether or not various arguments are time-dependent. More precisely, $r$ is time-dependent if $\mathbf{tdpar}\left(1\right)=\mathrm{.TRUE.}$ and constant otherwise. Similarly, $\mathbf{tdpar}\left(2\right)$ specifies whether $q$ is time-dependent and $\mathbf{tdpar}\left(3\right)$ specifies whether $\sigma$ is time-dependent.

9:     $\mathbf{r}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array r must be at least $\mathbf{nt}$ if $\mathbf{tdpar}\left(1\right)=\mathrm{.TRUE.}$, and at least $1$ otherwise.

On entry: if $\mathbf{tdpar}\left(1\right)=\mathrm{.TRUE.}$ then $\mathbf{r}\left(\mathit{j}\right)$ must contain the value of the risk-free interest rate $r\left(t\right)$ at the $\mathit{j}$th time in the mesh, for $\mathit{j}=1,2,\dots ,\mathbf{nt}$.

If $\mathbf{tdpar}\left(1\right)=\mathrm{.FALSE.}$ then $\mathbf{r}\left(1\right)$ must contain the constant value of the risk-free interest rate $r$. The remaining elements need not be set.

10:   $\mathbf{q}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array q must be at least $\mathbf{nt}$ if $\mathbf{tdpar}\left(2\right)=\mathrm{.TRUE.}$, and at least $1$ otherwise.

On entry: if $\mathbf{tdpar}\left(2\right)=\mathrm{.TRUE.}$ then $\mathbf{q}\left(\mathit{j}\right)$ must contain the value of the continuous dividend $q\left(t\right)$ at the $\mathit{j}$th time in the mesh, for $\mathit{j}=1,2,\dots ,\mathbf{nt}$.

If $\mathbf{tdpar}\left(2\right)=\mathrm{.FALSE.}$ then $\mathbf{q}\left(1\right)$ must contain the constant value of the continuous dividend $q$. The remaining elements need not be set.

11:   $\mathbf{sigma}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array sigma must be at least $\mathbf{nt}$ if $\mathbf{tdpar}\left(3\right)=\mathrm{.TRUE.}$, and at least $1$ otherwise.

On entry: if $\mathbf{tdpar}\left(3\right)=\mathrm{.TRUE.}$ then $\mathbf{sigma}\left(\mathit{j}\right)$ must contain the value of the volatility $\sigma \left(t\right)$ at the $\mathit{j}$th time in the mesh, for $\mathit{j}=1,2,\dots ,\mathbf{nt}$.

If $\mathbf{tdpar}\left(3\right)=\mathrm{.FALSE.}$ then $\mathbf{sigma}\left(1\right)$ must contain the constant value of the volatility $\sigma$. The remaining elements need not be set.

12:   $\mathbf{alpha}$ – Real (Kind=nag_wp)Input

On entry: the value of $\lambda$ to be used in the time-stepping scheme. Typical values include:

$\mathbf{alpha}=0.0$

Explicit forward Euler scheme.

$\mathbf{alpha}=0.5$

Implicit Crank–Nicolson scheme.

$\mathbf{alpha}=1.0$

Implicit backward Euler scheme.

The value $0.5$ gives second-order accuracy in time. Values greater than $0.5$ give unconditional stability. Since $0.5$ is at the limit of unconditional stability this value does not damp oscillations.

Suggested value: $\mathbf{alpha}=0.55$.

Constraint: $0.0\le \mathbf{alpha}\le 1.0$.

13:   $\mathbf{ntkeep}$ – IntegerInput

On entry: the number of solutions to be stored in the time direction. The routine calculates the solution backwards from $\mathbf{t}\left(\mathbf{nt}\right)$ to $\mathbf{t}\left(1\right)$ at all times in the mesh. These time solutions and the corresponding Greeks will be stored at times $\mathbf{t}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ntkeep}$, in the arrays f, theta, delta, gamma, lambda and rho. Other time solutions will be discarded. To store all time solutions set $\mathbf{ntkeep}=\mathbf{nt}$.

Constraint: $1\le \mathbf{ntkeep}\le \mathbf{nt}$.

14:   $\mathbf{f}\left(\mathbf{ldf},\mathbf{ntkeep}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{f}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ns}$ and $\mathit{j}=1,2,\dots ,\mathbf{ntkeep}$, contains the value $f$ of the option at the $\mathit{i}$th mesh point $\mathbf{s}\left(\mathit{i}\right)$ at time $\mathbf{t}\left(\mathit{j}\right)$.

15:   $\mathbf{theta}\left(\mathbf{ldf},\mathbf{ntkeep}\right)$ – Real (Kind=nag_wp) arrayOutput

16:   $\mathbf{delta}\left(\mathbf{ldf},\mathbf{ntkeep}\right)$ – Real (Kind=nag_wp) arrayOutput

17:   $\mathbf{gamma}\left(\mathbf{ldf},\mathbf{ntkeep}\right)$ – Real (Kind=nag_wp) arrayOutput

18:   $\mathbf{lambda}\left(\mathbf{ldf},\mathbf{ntkeep}\right)$ – Real (Kind=nag_wp) arrayOutput

19:   $\mathbf{rho}\left(\mathbf{ldf},\mathbf{ntkeep}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the values of various Greeks at the $i$th mesh point $\mathbf{s}\left(i\right)$ at time $\mathbf{t}\left(j\right)$, as follows:

$$\begin{array}{lll}\mathbf{theta}\left(i,j\right)=\frac{\partial f}{\partial t}\text{,}& \mathbf{delta}\left(i,j\right)=\frac{\partial f}{\partial S}\text{,}& \mathbf{gamma}\left(i,j\right)=\frac{{\partial }^{2}f}{\partial S^2}\text{,}\\ \\ \mathbf{lambda}\left(i,j\right)=\frac{\partial f}{\partial \sigma }\text{,}& \mathbf{rho}\left(i,j\right)=\frac{\partial f}{\partial r}\text{.}\end{array}$$

20:   $\mathbf{ldf}$ – IntegerInput

On entry: the first dimension of the arrays f, theta, delta, gamma, lambda and rho as declared in the (sub)program from which d03ncf is called.

Constraint: $\mathbf{ldf}\ge \mathbf{ns}$.

21:   $\mathbf{work}\left(4\times \mathbf{ns}\right)$ – Real (Kind=nag_wp) arrayWorkspace

22:   $\mathbf{iwork}\left(\mathbf{ns}\right)$ – Integer arrayWorkspace

23:   $\mathbf{ifail}$ – 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).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{alpha}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{alpha}\le 1.0$.

On entry, $\mathbf{alpha}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{alpha}\ge 0.0$.

On entry, $\mathbf{kopt}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{kopt}=1$, $2$, $3$ or $4$.

On entry, $\mathbf{ldf}=〈\mathit{\text{value}}〉$ and $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldf}\ge \mathbf{ns}$.

On entry, $\mathbf{mesh}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{mesh}=\text{'U'}$ or $\text{'C'}$.

On entry, $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ns}\ge 2$.

On entry, $\mathbf{nt}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nt}\ge 2$.

On entry, $\mathbf{ntkeep}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ntkeep}\ge 1$.

On entry, $\mathbf{ntkeep}=〈\mathit{\text{value}}〉$ and $\mathbf{nt}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ntkeep}\le \mathbf{nt}$.

On entry, $\mathbf{s}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{s}\left(1\right)\ge 0.0$.

On entry, $\mathbf{t}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{t}\left(1\right)\ge 0.0$.

$\mathbf{ifail}=2$

On entry, $\mathbf{s}\left(\mathbf{ns}\right)=〈\mathit{\text{value}}〉$ and $\mathbf{s}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{s}\left(\mathbf{ns}\right)>\mathbf{s}\left(1\right)$.

On entry, $\mathbf{t}\left(\mathbf{nt}\right)=〈\mathit{\text{value}}〉$ and $\mathbf{t}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{t}\left(\mathbf{nt}\right)>\mathbf{t}\left(1\right)$.

$\mathbf{ifail}=3$

On entry, $\mathbf{s}\left(〈\mathit{\text{value}}〉+1\right)\le \mathbf{s}\left(〈\mathit{\text{value}}〉\right)$.
Constraint: when $\mathbf{mesh}=\text{'C'}$, $\mathbf{s}\left(\mathit{i}\right)<\mathbf{s}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ns}-1$.

On entry, $\mathbf{t}\left(〈\mathit{\text{value}}〉+1\right)\le \mathbf{t}\left(〈\mathit{\text{value}}〉\right)$.
Constraint: when $\mathbf{mesh}=\text{'C'}$, $\mathbf{t}\left(\mathit{i}\right)<\mathbf{t}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nt}-1$.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

9.1

Timing

9.2

Algorithmic Details

10

Example

10.1

Program Text

Program Text (d03ncfe.f90)

10.2

Program Data

Program Data (d03ncfe.d)

10.3

Program Results

Program Results (d03ncfe.r)