In this paper is presented a possibility to reduce execution time involved in evaluating a financial option by means finite difference scheme, using a cyclic odd-even reduction technique on a coarse-grained parallel model.coarse grained multicomputer, parallel algorithm, tridiagonal linear system, band matrices, odd-even cyclic reduction algorithm.

Dumitru FANACHE

Ion SMEUREANU

English

Research Papers in Economics

Revista Informatica Economică nr.3(47)/2008 
 
115
The Cyclic Odd-Even Reduction Method  
Applied in Mathematical Finance 
 
Ion SMEUREANU 
Department of Economy Informatics, Academy of Economic Studies, Bucharest, România 
Dumitru FANACHE 
Mathematics Department, Valahia University Târgovişte, România 
 
In this paper is to give a possibility of reducing execution time involved in evaluat-
ing a financial option by means finite difference scheme, using a cyclic odd-even reduction 
technique on a coarse-grained parallel model. 
Keywords: coarse grained multicomputer, parallel algorithm, tridiagonal linear system, band 
matrices, odd-even cyclic reduction algorithm. 
 
Introduction 
The general strategy for solving Black 
Scholes equation is to use a change of va-
riables to transform  into one that’s more 
tractable, the diffusion equation([7][3]). The 
terminal boundary condition  becomes the in-
itial boundary condition. Once the problem 
has been transformed  we use algebraic me-
thods to solve it numerically.  The Crank-
Nicolson finite difference equation  may be 
used to calculate a discretised solution to dif-
fusion equation. Importantly, initial and both 
boundary conditions must be provided for a 
solution to be calculable.  
We seek the function V(S,t) satisfying the 
PDE 
∂V
∂t
+ rS
∂V
∂S
+
σ
2
2
∂
2V
∂S
2 − rV = 0 ( 1) 
with the constraints: 
[ ) [] ) ( ) , ( ; , 0 ; , 0 S g T S V T t S = ∈ +∞ ∈     (2) 
where  g , the terminal boundary condition, is 
the payoff at maturity of the option whose 
value will be given by V . We will assume 
that it does not pay dividends, and that both 
the risk-free rate( r ) and the underlying’s 
volatility(σ ) are constant over the life of the 
option[7]. 
Analytic solutions to the PDE depend upon 
the observation(see detail in [4]) that it can 
be transformed into the diffusion equation in 
a function u(x,τ): 
∂u
∂τ
=
∂
2u
∂x
2  ( 3) 
with  the constraints 
( ) [ ] ) ( ) 0 , ( ; , 0 ; , x f x u x f = ∈ +∞ ∞ − ∈ τ τ (4) 
where  f , the initial boundary condition, is 
g  from the original formulation of the prob-
lem suitably transformed. Using a change of 
variables that completely transforms (1) and 
(2) into (3) and (4):  
τ τ
σ
τ
b ax e
K
t S V
x u t T
K
S
x
+ = − = ⎟
⎠
⎞
⎜
⎝
⎛ =
) , (
) , ( ); (
2
; ln
2
   ( 5) 
 
where:  
2
2 2
2
2
; 2 a
r
b
r
a + =
−
=
σ σ
σ
 ( 6) 
and  K  is an arbitrary positive constant, 
usually chosen to be the strike price of the 
option. Once the problem has been trans-
formed into (3) and (4), we use algebraic me-
thods to solve it numerically. Using   Crank 
Nicolson method  in (3), we obtain:  
() () τ τ b Au =  ( 7) 
In (7), matrix  A is tridiagonal, symmetric, 
and strictly diagonally dominated, meaning 
that our equation will have a unique solu-
tion([1][4]). 
 
2. The CGM model and  Odd-Even Cyclic 
Reduction method 
Many current application in parallel ma-
chines are restricted to trivially parallelizable 
problem with low communication require-
ments. In real machines communication time 
1 Revista Informatica Economică nr.3(47)/2008 
 
116
is usually much greater than computation 
time.  
The Coarse Grained Multicomputer (CGM) 
model  to be an adequate model of paral-
lelism sufficiently close to existing parallel 
machines. It is a simple model and neverthe-
less intends to give a reasonable prediction of 
performance when parallel algorithms on this 
model are implemented. 
In the CGM model the effort to reduce com-
munication is centered on reducing the num-
ber of communication rounds. Under this 
model, we design a communication efficient 
parallel algorithm for the solution of tridia-
gonal linear systems with n equation and n 
unknowns. This algorithms requires only a 
constant number of communication rounds. 
The amount of data transmitted in each 
communication round is proportional to the 
number of processors and independent of n. 
In addition to showing the theoretical com-
plexity, we have implemented the proposed 
algorithm on a real distributed memory paral-
lel machine. The experimental results ob-
tained, indicate the efficiency and scalability 
of the algorithm. 
The CGM model uses only two parameters, n 
and p, where n is the size of the input and p 
the number of processors each with  ( ) p n O /  
local memory. Each processor is connected 
by a router that can deliver messages in a 
point to point fashion. A CGM algorithm con-
sists of an alternating sequence of computa-
tion round and communication rounds sepa-
rated by barrier synchronizations.  
In the  computation round, we usually the 
best possible sequential algorithm in each 
processor to process its data locally. A com-
munication round consists of a single h-
relation with   p n h / ≤ , that is, each proces-
sor exchanges at most a total of  () p n O /  data 
with other processors in one communication 
round. The proposed algorithm requires the 
transmission of only  ( ) p O  data in each 
communication round. 
In the CGM model the communication cost 
of a parallel algorithm is modeled by the 
number of communication rounds. The ob-
jective is to design algorithms that require a 
small amount of communications rounds. 
Many algorithms for graph an geometric 
problems [2] require only a constant or 
( ) p O log . Contrary to PRAM algorithms that 
frequently are designed for  () n O p =  and 
each processor receives a small number of 
input data, here we consider the more realis-
tic case of  n>>p. The CGM model is par-
ticularly suitable in nowdays parallel ma-
chines where the overall computation speed 
is considerably larger than the overall com-
munication speed. 
A tridiagonal system is one in which all ele-
ments, except possibly those on the main di-
agonal, and the ones just above or below it, 
are 0’s. Instead of the usual notation  ij a  for 
the element in row i, column j, of A, we use 
di to represent the main diagonal element 
ii a , i u  for upper diagonal element  1 , + i i a  and 
i l  for the lower diagonal element  1 , − i i a . For 
the sake of uniformity, we define 
0 1 0 = = − n u l . With the notation defined 
above, a tridiagonal system of linear equation 
can be written as follows, where 
0 1 = = − n x x  are dummy variables that are 
introduced for uniformity. 
 
1 1 1 1 2 1
2 3 3 2 2 1 2
1 2 1 1 1 0 1
0 1 0 0 0 1 0
− − − − − −
−
= + +
= + +
= + +
= + +
n n n n n n n b x u x d x l
b x u x d x l
b x u x d x l
b x u x d x l
M
 ( 8) 
 
Odd-even cyclic reduction ([2],[1]) is a re-
cursive method for solving tridiagonal sys-
tems of size  1 2 − =
m n . This method is di-
vided into two parts: reduction  and  back 
substitution. The first step of reduction is to 
remove each odd-indexed  i x  and create a tri-
diagonal system of size  1 2
1 −
− m . We then do 
the same to this new system and continue on 
in the same manner until we are left with a 
system of size 1. Observe that the ith equa-
tion can be rewritten as 
( )( ) 1 1 / 1 + − − − = i i i i i i i x u x l b d x  ( 9) Revista Informatica Economică nr.3(47)/2008 
 
117
Taking the above equation (9) for each odd i 
and substituing into even-numbered equa-
tions (the ones with even indices for l, d, u 
and  b), we obtain for each even  () n i i ≤ ≤ 0  
an equation of the form: 
1
1
1
1
2
1
1
1
1
1
1
2
1
1
+
+
−
−
+
+
+
+
+
−
−
−
−
− − − = − ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− − + −
i
i i
i
i i
i i
i
i i
i
i
i i
i
i i
i i
i
i i
d
b u
d
b l
b x
d
u u
x
d
l u
d
u l
d x
d
l l
 ( 10) 
In this way, the n equation are reduced to 
⎡⎤ 2 / n  tridiagonal linear equation in the 
even-indexed variables. Applying the same 
method recursively, leads to  4 / n  equations, 
then  8 / n  equations, and, eventually, a single 
equation in  0 x . Solving this last equation to 
obtain the value of  0 x , and substituting 
backwards, allows us to compute the value of 
each of the n variables. Figure 1 shows the 
structure of the odd-even reduction method. 
Forming each new equation requires six mul-
tiplications, six divisions, and four additions, 
but these can all be done in parallel using 
2 / n p =  processors. Assuming unit-time 
arithmetic operations, we obtain the recur-
rence  () ( ) n n T n T 2 log 8 8 2 / ≈ + =  for the to-
tal number of computational steps. The six 
division operations can be replaced with one 
reciprocation per new equation, to find  j d / 1 
for each odd j, plus six multiplications. Ob-
viously, the above odd-even reduction me-
thod is applicable only if none of the   j d  
values obtained in the course of the computa-
tion is 0. 
In the above analysis, interprocessor com-
munication time was not taken into account. 
The analysis is thus valid only for the PRAM 
or for an architecture whose topology 
matches the communication structure shown 
in Figure 1. A binary X-tree architecture 
whose communication structure closely 
matches the needs of the above computation. 
 
*
     x12                 x8                                      x4                  x0 
x0 
x8                                           x0 
 x14         x12               x10           x8               x6         x4          x2      x0 
  x15  x14  x13  x12  x11  x10   x9  x8   x7     x6   x5   x4   x3    x2   x1    x0 
* find x1 in terms of x0 and x2 from eqn. 1; 
substitute x1 in eqn. 0 and 2. 
 
Fig.1. The structure of odd-even reduction for solving tridiagonal system of linear equation 
 
To perform odd-even reduction on linear ar-
ray of  n p =  processors, we can assume that 
each processor initially  holds one of the n 
equations. Direct one-step communication 
between neighboring processors leads to 
even-numbered processors obtaining the re-
duced set of  2 / n equations as discussed 
above. The next reduction phase requires 
two-step communication, then four-step, and 
eventualy  () 2 / n step, leading to linear run-
ning time(of the same order as sequential 
time). On an n-processor 2D mesh, odd-even 
reduction can be easily organized to require 
( ) p Θ  time. It is worth noting that solving a 
tridiagonal system of linear equation can be 
converted to a parallel prefix problem as fol-
lows([2][6]).  
We propose here a CGM algorithm(see Fig-
ure 4) that requires a constant number of 
communication rounds. By using the CGM 
paradigm, however, the algorithm proposed 
in this paper has been conceived indepen-Revista Informatica Economică nr.3(47)/2008 
 
118
dently in a relatively natural way, following 
the CGM principles, namely, minimizing 
communication rounds and using as much 
local processing as possible. Furthermore, we 
have implemented this algorithm on a distri-
buted memory parallel machine to verify its 
efficiency in practice.  
Consider a distributed memory parallel com-
puter of p processors  1 2 0 , , , − p P P P L  with 
p n >> . Assume that each processor has suf-
ficient local memory to store  () p n O /  ele-
ments. (see Figure 2). We subdivide matrix 
A and the vector b  into horizontal blocks or 
submatrices of  p n/  consecutive rows each. 
Each processor stores a submatrix of  A and 
b .  
The proposed algorithm makes use of a mod-
ified version of the odd-even reduction algo-
rithm([1],[2],[6]). Each processor applies 
odd-even reduction to its  p n/  equations and 
eliminates all equation but the first and the 
last ones. Thus each processor will have only 
four unknowns. Each processor then send the 
two remaining equations to processor 0. Pro-
cessor 0 applies odd-even reduction locally 
and solves for unknowns. Each processor 
receives the solved unknowns from processor 
0 and solves for the remaining unknowns lo-
cally. The algorithm consists of the following 
five phases (see Figure 4) alternating be-
tween local processing and communication. 
 
… 
local memory processor
P0  P1  P2 Pp-1
 
() p n O /
 
() p n O /
 
() p n O /  
 
 
() p n O /
 
 
Fig.2. A parallel computer with distributed memory 
 
Theorem 1. A tridiagonal linear system with 
n equations and n unknowns can be solved 
on a CGM with p processors and  ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
p
n
O l o -
cal memory per processor using  () 1 O  com-
munication  rounds with the transmission of  
() p O  data per round. 
Proof: Step 1, 3 and 5 relate to local 
processing only. Step 2 and 4 require one 
communication round each. In the first com-
munication round (step 2) each processor 
send a constant amount of data to processor 
0, which in turn receives a total of  () p O  da-
ta. In the second communication round (step 
4) processor 0 sends a total of  () p O  data and 
each of remaining processors receive a con-
stant amount of data. 
The sequential time was obtained with an op-
timized sequential algorithm run in a single 
processor (not the parallel algorithm run time 
one processor). For the experiment we use 
the following system:  1 − = = i i u l  for all 
1 , , 1 − = n i L  and  2 = i d  for all 
1 , , 0 − = n i L  (with the solution of all   1 = i x  
for   1 , , 1 , 0 − = n i L ). 
We obtain an almost linear speedup for large 
n, regardless of the communication protocol 
utilized, as shown by the numerical re-
sults(see  Figure 3 for the time curves).The 
times are given in units of clock ticks of the 
machine(1 clock tick =10
-6 seconds). 
 
3. Conclusions 
Under the CGM model we have designed a 
communication-efficient  parallel algorithm 
for the solution of tridiagonal linear systems. 
This algorithm requires only a constant num-
ber of communication round with of  ( ) p O  
data per round. In addition to showing its 
theoretical complexity, we have implemented 
this algorithm on a real distributed memory 
parallel machine. The experimental results 
show an almost linear speedup for large n. 
This is a very significant result since the par-
ticular machine we used presents a consider-Revista Informatica Economică nr.3(47)/2008 
 
119
able communication latency and low commu-
nication bandwidth. It indicate the efficiency 
and scalability of the proposed algorithm
 
 
Fig.3. Total time of the tridiagonal algorithm in MPI 
 
 
S1. 
Each processor applies the odd-even algorithm locally to eliminate all rows except the first and the 
last rows in its submatrix. With this, each processor  i P eliminates  2 / − p n   equations and  2 / − p n  
unknowns, namely  ()
1
1
3 2
, , ,
−
+
+ +
p
i n
p
ni
p
ni x x x L . 
 
 
S2. 
Each processor  i P  sends its two remaining equation (with 4 unknowns   
() ()
1
1 1
1
, , ,
+
+ +
+
p
i n
p
i n
p
ni
p
ni x x x x  ) to a same  processor, say  0 P . This processor thus obtains a system 
with  p 2 equations and  p 2  unknowns. Notice that this resulting system also consists of a tridiagonal 
matrix. 
 
S3. 
Processor  0 P  solves the system locally by od-even reduction or any other sequential method. It thus 
obtains the solution for the  p 2  unknowns. 
S4.  Processor  0 P  sends to each processor  i P  the computed value for 4 unknowns in the respective equa-
tions received in step 2. 
S5.  Each processor performs the inverse process of odd-even reduction used in step 1, by using the solu-
tion received for its two equations to solve the remaining equations. 
Fig.4.  Parallel algorithm odd-even reduction 
 
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Dumitru Fanache, A Linear Algorithm for Black Scholes Economic Model,

On the Eigenvalues of Some Tridiagonal Matrices,

Optimal and Eficient Paralel Tridiagonal Solvers Using Direct Methods ,

Parallelised Numeric Solutions for Implicit-Explicit Domain Decomposition,

Parhani,Introducing to Parallel Processing(Algorihms and Architectures (electronic format),

The Cyclic Odd-Even Reduction Method Applied in Mathematical Finance

http://www.revistaie.ase.ro/content/47/21Smeureanu,%20Fanache.pdf

The Cyclic Odd-Even Reduction Method Applied in Mathematical Finance

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