NAG Library Routine Document

G05XDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G05XDF computes scaled increments of sample paths of a free or non-free Wiener process, where the sample paths are constructed by a Brownian bridge algorithm. The initialization routine G05XCF must be called prior to the first call to G05XDF.

2 Specification

SUBROUTINE G05XDF (

NPATHS, RCORD, D, A, DIFF, Z, LDZ, C, LDC, B, LDB, RCOMM, IFAIL)

INTEGER	NPATHS, RCORD, D, A, LDZ, LDC, LDB, IFAIL
REAL (KIND=nag_wp)	DIFF(D), Z(LDZ,), C(LDC,), B(LDB,), RCOMM()

3 Description

For details on the Brownian bridge algorithm and the bridge construction order see Section 2.6 in the G05 Chapter Introduction and Section 3 in G05XCF. Recall that the terms Wiener process (or free Wiener process) and Brownian motion are often used interchangeably, while a non-free Wiener process (also known as a Brownian bridge process) refers to a process which is forced to terminate at a given point.

Fix two times

t_{0} < T

, let

{(t_{i})}_{1 \leq i \leq N}

be any set of time points satisfying

t_{0} < t_{1} < t_{2} < \dots < t_{N} < T

, and let

X_{t_{0}}

{(X_{t_{i}})}_{1 \leq i \leq N}

X_{T}

denote a

d

-dimensional Wiener sample path at these time points.

The Brownian bridge increments generator uses the Brownian bridge algorithm to construct sample paths for the (free or non-free) Wiener process

X

, and then uses this to compute the scaled Wiener increments

\frac{X_{t_{1}} - X_{t_{0}}}{t_{1} - t_{0}}, \frac{X_{t_{2}} - X_{t_{1}}}{t_{2} - t_{1}}, \dots, \frac{X_{t_{N}} - X_{t_{N - 1}}}{t_{N} - t_{N - 1}}, \frac{X_{T} - X_{t_{N}}}{T - t_{N}}

The example program in Section 9 shows how these increments can be used to compute a numerical solution to a stochastic differential equation (SDE) driven by a (free or non-free) Wiener process.

4 References

Glasserman P (2004) Monte Carlo Methods in Financial Engineering Springer

5 Parameters

Note: the following variable is used in the parameter descriptions:

N = NTIMES

, the length of the array TIMES passed to the initialization routine G05XCF.

1: NPATHS – INTEGERInput

On entry: the number of Wiener sample paths.

Constraint:

NPATHS \geq 1

2: RCORD – INTEGERInput

On entry: the order in which Normal random numbers are stored in Z and in which the generated values are returned in B.

Constraint:

RCORD = 1

2

3: D – INTEGERInput

On entry: the dimension of each Wiener sample path.

Constraint:

D \geq 1

4: A – INTEGERInput

On entry: if

A = 0

, a free Wiener process is created and DIFF is ignored.

A = 1

, a non-free Wiener process is created where DIFF is the difference between the terminal value and the starting value of the process.

Constraint:

A = 0

1

5: DIFF(D) – REAL (KIND=nag_wp) arrayInput

On entry: the difference between the terminal value and starting value of the Wiener process. If

A = 0

, DIFF is ignored.

6: Z(LDZ, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Z must be at least

NPATHS

RCORD = 1

and at least

D \times (N + 1 - A)

RCORD = 2

On entry: the Normal random numbers used to construct the sample paths.

RCORD = 1

and quasi-random numbers are used, the

D \times (N + 1 - A)

, where

N = nint RCOMM (2)

-dimensional quasi-random points should be stored in successive columns of Z.

RCORD = 2

and quasi-random numbers are used, the

D \times (N + 1 - A)

, where

N = nint RCOMM (2)

-dimensional quasi-random points should be stored in successive rows of Z.

On exit: the Normal random numbers premultiplied by C.

7: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which G05XDF is called.

Constraints:

if $RCORD = 1$ , $LDZ \geq D \times (N + 1 - A)$ ;
if $RCORD = 2$ , $LDZ \geq NPATHS$ .

8: C(LDC, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array C must be at least

D

On entry: the lower triangular Cholesky factorization

C

such that

C C^{T}

gives the covariance matrix of the Wiener process. Elements of C above the diagonal are not referenced.

9: LDC – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which G05XDF is called.

Constraint:

LDC \geq D

10: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array B must be at least

NPATHS

RCORD = 1

and at least

D \times (N + 1)

RCORD = 2

On exit: the scaled Wiener increments.

Let

X_{p, i}^{k}

denote the

k

th dimension of the

i

th point of the

p

th sample path where

1 \leq k \leq D

1 \leq i \leq N + 1

and

1 \leq p \leq NPATHS

RCORD = 1

, the increment

\frac{(X_{p, i}^{k} - X_{p, i - 1}^{k})}{(t_{i} - t_{i - 1})}

will be stored at

B (k + (i - 1) \times D, p)

RCORD = 2

, the increment

\frac{(X_{p, i}^{k} - X_{p, i - 1}^{k})}{(t_{i} - t_{i - 1})}

will be stored at

B (p, k + (i - 1) \times D)

11: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which G05XDF is called.

Constraints:

if $RCORD = 1$ , $LDB \geq D \times (N + 1)$ ;
if $RCORD = 2$ , $LDB \geq NPATHS$ .

12: RCOMM( $*$ ) – REAL (KIND=nag_wp) arrayCommunication Array

On entry: communication array as returned by the last call to G05XCF or G05XDF. This array must not be directly modified.

13: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ 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 parameter, the recommended value is

0

. 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

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, RCOMM was not initialized or has been corrupted. On entry, RCOMM was not initialized or has been corrupted. On entry, RCOMM was not initialized or has been corrupted.

$IFAIL = 2$: On entry, $NPATHS = ⟨value⟩$ .
Constraint: $NPATHS \geq 1$ .

$IFAIL = 3$: On entry, the value of RCORD is invalid.

$IFAIL = 4$: On entry, $D = ⟨value⟩$ .
Constraint: $D \geq 1$ .

$IFAIL = 5$: On entry, $A = ⟨value⟩$ .
Constraint: $A = 0 or 1$ .

$IFAIL = 6$: On entry, $LDZ = ⟨value⟩$ and $D \times (NTIMES + 1 - A) = ⟨value⟩$ .
Constraint: $LDZ \geq D \times (NTIMES + 1 - A)$ .

On entry, $LDZ = ⟨value⟩$ and $NPATHS = ⟨value⟩$ .
Constraint: $LDZ \geq NPATHS$ .

$IFAIL = 7$: On entry, $LDC = ⟨value⟩$ .
Constraint: $LDC \geq ⟨value⟩$ .

$IFAIL = 8$: On entry, $LDB = ⟨value⟩$ and $D \times (NTIMES + 1) = ⟨value⟩$ .
Constraint: $LDB \geq D \times (NTIMES + 1)$ .

On entry, $LDB = ⟨value⟩$ and $NPATHS = ⟨value⟩$ .
Constraint: $LDB \geq NPATHS$ .

$IFAIL = - 999$: Dynamic memory allocation failed.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

The scaled Wiener increments produced by this routine can be used to compute numerical solutions to stochastic differential equations (SDEs) driven by (free or non-free) Wiener processes. Consider an SDE of the form

d Y_{t} = f (t, Y_{t}) dt + σ (t, Y_{t}) d X_{t}

on the interval

[t_{0}, T]

where

{(X_{t})}_{t_{0} \leq t \leq T}

is a (free or non-free) Wiener process and

f

and

σ

are suitable functions. A numerical solution to this SDE can be obtained by the Euler–Maruyama method. For any discretization

t_{0} < t_{1} < t_{2} < \dots < t_{N + 1} = T

[t_{0}, T]

, set

Y_{t_{i + 1}} = Y_{t_{i}} + f (t_{i}, Y_{t_{i}}) (t_{i + 1} - t_{i}) + σ (t_{i}, Y_{t_{i}}) (X_{t_{i + 1}} - X_{t_{i}})

for

i = 1, \dots, N

so that

Y_{t_{N + 1}}

is an approximation to

Y_{T}

. The scaled Wiener increments produced by G05XDF can be used in the Euler–Maruyama scheme outlined above by writing

Y_{t_{i + 1}} = Y_{t_{i}} + (t_{i + 1} - t_{i}) (f (t_{i}, Y_{t_{i}}) + σ (t_{i}, Y_{t_{i}}) (\frac{X_{t_{i + 1}} - X_{t_{i}}}{t_{i + 1} - t_{i}})) .

The following example program uses this method to solve the SDE for geometric Brownian motion

d S_{t} = r S_{t} dt + σ S_{t} d X_{t}

where

X

is a Wiener process, and compares the results against the analytic solution

S_{T} = S_{0} exp ((r - σ^{2} / 2) T + σ X_{T}) .

Quasi-random variates are used to construct the Wiener increments.

9.1 Program Text

Program Text (g05xdfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (g05xdfe.r)

G05XDF (PDF version)

G05 Chapter Contents

G05 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentG05XDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G05XDF