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DYNAMIC ECONOMETRIC MODELS 
Vol. 10 – Nicolaus Copernicus University – Toruń – 2010 

Piotr Płuciennik  
Adam Mickiewicz University, National Bank of Poland  

Forecasting Financial Processes by Using 
  Diffusion Models  

A b s t r a c t.  Time series forecasting is one of the most important issues in the financial 
econometrics. In the face of growing interest in models with continuous time, as well as rapid 
development of methods of their estimation, we try to use the diffusion models to modeling and 
forecasting time series from various financial markets. We use Monte-Carlo-based method, 
introduced by Cziraky and Kucherenko (2008). Received forecasts are confronted with those 
determined with the commonly applied parametrical time series models.  

K e y w o r d s: diffusion models, ex-post forecasts, Monte-Carlo simulation, the GARCH model, 
the ARIMA model, unit-root. 

1. Introduction 
 Models with continuous time and its particular case – diffusion models are 
exceptionally important class of models. On the developed financial markets 
there are available quotations containing full information about transaction 
prices, so called tick-by-tick data. It provides natural motivation to applying 
diffusion models or jump diffusion models to examination of financial 
instruments price series. Diffusion models were initially used to short-term rate 
modeling (Merton, 1973; Vašiček, 1977; Cox, at al. 1985). They gained in 
importance in the early 70s, when Black and Scholes (1973) introduced 
European call and put option pricing model in which the underlier price was 
modeled with simple diffusion model called Geometric Brownian Motion. 
In the following years many modification of Black-Scholes models were 
introduced. Merton (1974) supposed additionally that the risk-free rate is 
modeled by diffusion model, and Heston (1993) assumed that the volatility 
process is described by mean reverting diffusion models. Forms of Black-
Scholes formula were introduced for american options, term options and even 
volatility index options, which had just appeared on the market. 



Piotr Płuciennik 52

 Other application areas of models with continuous time are issues 
concerning modeling and forecasting interest rates (term structure) and 
valuation of very complicated derivatives based on debt instruments. This 
subject was brought up among others by Jagannathan at al. (2004), which 
applied multidimensional CIR models to caps and swaptions pricing, Tamba 
(2006), which used Hull-White diffusion model to Bermudian swaption pricing, 
and Mannolini, Mari, Renò (2008), which priced caps and floors by extended 
CIR models.  
 The aforementioned derivatives are of outstanding significance. They 
played the crucial role in risk management and in aggressive investment 
strategies. The first hedging strategies were proposed by Black and Scholes 
(1973). Nowadays there are strategies which allow to hedge the positions in 
swaption (Javaheri at al., 2004; Howison at al., 2004) and VIX Options 
(Psychoyios, Skiadopoulos 2006; Sepp 2008; Broadie, Jain 2008). 
 In the following article we use diffusion models to forecast the logarithmic 
levels of DAX, CAC40, NASDAQ and WIG20 indexes. The parameter 
estimates were obtained by modern Phillips and Yu (2009) method and more 
classical, introduced by Hansen (1982), Generalized Method of Moments 
(GMM) with covariance matrix, estimated by using Bartlett kernels, as a weight 
matrix (Newey, West, 1987). We determine the forecasts by using Monte-Carlo 
methods and compare its quality with the forecasts which we obtained by using 
popular parametrical time series models. 

2. Models 
 We use popular diffusion models to describe logarithmic prices of financial 
instruments. The diffusion models were originally used to describe the 
evolution of short-term rates. Their significant feature is the mean reverting 
property. 
 The most simple diffusion model – Vašíček (1977) model – assume that the 
price process is modeled by the following stochastic differential equation 

ttt dBdtXdX σμκ +−= )( , 

with initial condition .00
xXt =  Parameters ,κ  μ  and σ  are strictly positive. 

Parameter μ  can be interpreted as a long term mean level, κ  as a speed of 
reversion, and σ  as a instantaneous volatility.     
 Cox, Ingersoll and Ross (1985) introduced model called CIR, which is an 
extension of Vašíček model. The tX  evolution is described by the formula  

tttt dBXdtXdX σμκ +−= )( , 

with initial condition .00
xXt =  Parameters ,κ  μ  and σ  are strictly positive, 

and have the same interpretation as in Vašíček model. The square root in the 



Forecasting Financial Processes by Using Diffusion Models  53

diffusion function allows to avoid the possibility of nonpositive values of tX , 
provided that the condition 22 σκμ > is met. 

 Chan at al. (1992) introduced model called CKLS. Authors assume that tX  
evolution is described by the following diffusion models 

tttt dBXdtXdX βσμκ +−= )( , 

with initial condition .00
xX t =  Similarly as in Vašíček and CIR model,  ,κ  μ  

and σ  are strictly positive. An additional β  parameter is called the elasticity of 
variance parameter, and ]1,0[∈β . By simply placing the appropriate 
restrictions on the four parameters ,κ  ,μ  σ  and β  we can obtain 7 other 
diffusion models – among others the Vašíček and the CIR model.  
(see Chan at al., 1992). 

3. Determination of One-Day Ex-Post Forecast   
from Diffusion Models 

 Denote the l-step forecast of ltX +  as )(ˆ lX t . Assuming that the minimum 

squared error is the loss function, the forecast )(ˆ lX t  is the random variable 
chosen such that 

,)],...,([min)](ˆ[ 2
1

2 XXgXlXX ttgtlt −≤−+   

where ),...,( 1XXg t  is measurable function towards σ-algebra generated by the 
information available up to time t inclusive. We can show that 

)|()(ˆ
thtt XhX += ( .  

Therefore, if we assume that process is described by diffusion models and 
length of one step equals ,h  then 

,)()ˆ,,()ˆ,,(

)()ˆ,,(+)ˆ,,(

)()ˆ,,(+)ˆ,,()(ˆ

21

21

0

2

0

10

∫ ∫

∫∫

∫∫

+ +

++

++

++=

⎥
⎥

⎦

⎤

⎢
⎢

⎣

⎡
+=

⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢

⎣

⎡
+=

δ δ

δδ

δδ

θσθμ

θσθμ

θσθμ

ht

t

ht

t

sst

ht

t

s

ht

t

st

t

ht

s

ht

st

sdBsXdssXX

sdBsXdssXX

sdBsXdssXXhX



+

+

 

where 1̂θ  and 2̂θ  are parameter estimates vectors of drift and diffusion 
respectively, obtained on the quotations up to time s . As we know, 



Piotr Płuciennik 54

the quotations are available only in discrete intervals. Consequently, we 
approximate the forecast )(ˆ lX t  by the Euler scheme of the form 

.)ˆ,,( ,)ˆ,,()(ˆ
1

0

1

0

)(
1

)(
|1

)(
|

)()( ∑ ∑
−

=

−

=
+++ ++=

h

k

h

k
ktnktnkttt tXtXXhX δδδδδ εδθσδθμ  (1) 

Moreover, for all t holds .]/[
)(

δ
δ

tt XX =  Cziraky i Kucherenko (2008) obtain 

estimates of )(ˆ )( lX
t

δ  by repeating the above recursion using N independent 

realizations of innovations vectors ),,....,( )()(
1

δδ εε knn ++  and for any realization they 

determine the trajectory (1) with initial condition .)(δ
tX  The MC estimator of 

)(ˆ lX t  is then given by the average of the last elements of every trajectory.  
 One-step ex-post forecasts are obtained by fitting the model using data up to 
time T, and then computing the usual fitted equation and residuals for periods 

1+T to ,FT + with additional assumption that the quotation which precede the 
forecast is known. 

4. The Data 
 We take into account daily levels of German DAX, French CAC40, 
American NASDAQ and Polish WIG20 indexes from the period 2. January 
2001 to 29. December 2006. In both series we observe the logarithmic trend. 
During the mentioned period the trend is growing. Therefore, we decide to 
model logarithmic levels of the indexes. The significance of the trend is then 
marginalized.  The descriptive statistics are given in the following Table 1.  

Table 1.  Descriptive statistics of the logarithmic levels of indexes CAC40, DAX, 
NASDAQ and WIG20 from the period 2. January 2001 to 29. December 2006 

Time series obs. 
number 

mean std. dev.  skewness kurtosis min. max. 

CAC40 
DAX 

NASDAQ 
WIG20  

1534 
1528 
1554 
1546 

8.3138 
8.40231 
7.55651 
7.4555 

0.19922 
0.24511 
0.1818 

0.34958 

-0.1759 
-0.3745 
-0.8033 
0.4318 

2.0605 
2.3597 
3.0256 
2.0328 

7.7845 
7.6976 
7.0158 
6.8979 

8.6993 
8.824 
7.9583 
8.1745 

5. Empirical Research 
 In the following section we present results of one day ex-post forecasts 
quality testing of examined time series, which we obtain from diffusion model. 
The model is estimated by using the modern two-stage Phillips and Yu (2009) 
method and the Generalized Method of Moments. We obtain the parameter 
estimation of diffusion models by using our own procedures in Matlab (Phillips-
Yu method) and by using the Matlab libraries by Cliff (2003) (GMM). 
The values of parameters estimations are given in Table 2.  



Forecasting Financial Processes by Using Diffusion Models  55

Table 2.  Parameter estimations of diffusion models obtained by using the Phillips and 
Yu method and the Generalized Method of Moments for logarithmic levels of 
CAC40, DAX, NASDAQ and WIG20 indexes 

Estimation method: Phillips and Yu 
model parameter CAC40 DAX NASDAQ WIG20 

Vašíček κ  
μ  
σ  

-0.10636 
8.1772 

0.16926 

0.29613 
8.4126 
0.21296 

0.0000258 
7.5829 
0.12692 

0.0012536 
7.4681 
0.16780 

CIR κ  
μ  
σ  

-0.10702 
8.1667 

0.058477 

0.29529 
8.4223 

0.072836 

0.081467 
7.5632 

0.045667 

0.00776 
7.3712 

0.052088 
CKLS κ  

μ  
σ  
β  

0.41333 
8.3644 
5.5273 
-4.8234 

0.51308 
8.4174 
5.3016 
-4.2147 

0.081748 
7.5632 
0.72314 
-0.85398 

-0.85398 
6.557 
0.4757 

-0.51843 
Estimation method: GMM 
model parameter CAC40 DAX NASDAQ WIG20 

Vašíček κ  
μ  
σ  

0.034582 
8.3644 

0.11064 

0.042929 
8.4174 
0.12291 

0.91407 
7.5811 
0.26122 

-0.12749 
6.5984 
0.23064 

CIR κ  
μ  
σ  

0.03683 
8.3589 

0.03826 

0.044431 
8.4169 

0.042108 

0.91274 
7.5827 

0.094545 

-0.13399 
6.6412 

0.084197 
CKLS κ  

μ  
σ  
β  

0.033606 
8.3669 

0.17815 
-0.22647 

0.041253 
8.4177 
0.43424 
-0.59029 

0.97411 
7.5632 
4.5291 
-3.7814 

-0.11478 
6.5059 
1.7382 
-1.007 

Starting values: ,0=κ μ  equales the mean of the sample, σ  determines the starting value for σ  by using  
Yoshida (1992) estimator for Vašíček and CIR model. As the starting value for σ  in CKLS model we take 
earlier obtained estimation from CIR model. As the ,β  we take 0.5. 

For examined time series we determine 100 ex-post forecasts, by using 10000 
Monte-Carlo simulations, and to assess the quality of the forecasts we use 
common error measures. Small values of error measures are indicative of good 
quality of the forecasts, and consequently of good model fitting. The good 
quality also implies using diffusion models as the alternative for parametric 
models of time series.  
 Forecast errors were compared with errors obtained from popular time 
series models forecasts. The grade of time series models had been selected by 
using the Schwarz Information Criterion. The occurrence of unit root was 
verified by using Dickey-Fuller (Said, Dickey, 1984) and Phillips-Perron (1988) 
tests. In the case of failure to reject the H0 hypothesis we modeled the 
conditional mean by ARIMA(p,1,q) model. Independently we used two 
innovation distributions – the Student and the Generalized Error Distribution 
developed by Nelson (1991). Moreover, we verified the existence of the ARCH 
effect by using Engle (1982) and McLeod-Li (1983) tests. The latter consists in 
applying Ljung-Box (1978) test to squared residuals of linear model. 



Piotr Płuciennik 56

 As we can observe in Tables 3-6, the forecast errors for WIG20 index are 
bigger than for other indexes. Polish financial market is still a raising market, 
and WIG20 volatility is bigger than volatility of indexes traded in mature 
markets.  
 The type of applied model do not have big influence on the forecast quality. 
For indexes CAC40, DAX and WIG20 the forecast was a little bit better when 
we modeled the logarithmic prices by using diffusion models, but for NASDAQ 
index the forecast errors were smaller for parametric time series models. From 
among diffusion models, the best forecasts we obtained using CIR models. 
Moreover, we can notice that for all examined time series, Phillips and Yu 
method of parameters estimation leads to smaller forecast errors than GMM 
method. 

Table 3.  Values of the forecast errors obtained by using diffusion models. Logarithmic 
levels of CAC40 index  

Error Phillips and Yu method Generalized Method of Moments 
Vašiček CIR CKLS Vašiček CIR CKLS 

MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

6.9069e-5 
2.5575e-5 

0.00077931 
0.0062754 
0.0083108 
0.00072612 
0.00036308 
9.2627e-7 

6.8895e-5 
2.6669e-5 

0.00077539 
0.0062588 
0.0083003 
0.0007242 

0.00036212 
9.2395e-7 

6.902e-5 
2.6053e-5 

0.00078897 
0.0062567 
0.0083078 
7.2396e-6 

3.62e-6 
9.2561e-7 

6.8355e-5 
2.6771e-5 

0.00032694 
0.0062394 
0.0082677 

0.00072199 
0.000361 
9.1675e-7 

6.817e-5 
2.6696e-5 

0.00032105 
0.0062339 
0.0082565 

0.00072135 
0.00036067 
9.1422e-7 

6.9836e-5 
2.7293e-5 
0.0012119 
0.0062989 
0.0083568 

0.00072881 
0.00036444 
9.3652e-7 

 
 
 

Error 

time series parametric models  
ARIMA(0,1,2) 

(Student) 
 

ARIMA(0,1,2) 
(GED) 

ARIMA(0,1,2)- 
GARCH(1,1) 

(Student) 

ARIMA(1,1,1)- 
GARCH(1,1) 

(GED) 
MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

7.0223e-5 
2.6884e-5 

0.00092873 
0.0063371 
0.0083799 

0.00073332 
0.00036668 
9.4187e-7 

- 
- 
- 
- 
- 
- 
- 
- 

7.0214e-5 
2.6873e-5 

0.00092695 
0.0063367 
0.0083794 
0.00073327 
0.00036665 
9.4175e-7 

7.0232e-5 
2.6895e-5 

0.00093054 
0.006338 
0.0083805 
0.00073342 
0.00036673 

9.42e-7 
 

 



Table 4.  Values of the forecast errors obtained by using diffusion models. Logarithmic 
levels of DAX index  

Error Phillips and Yu method Generalized Method of Moments 
Vašiček CIR CKLS Vašiček CIR CKLS 

MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

7.7296e-5 
2.6917e-5 
0.0014902 
0.0066737 
0.0087918 
0.00075446 
0.00037726 
9.8999e-7 

7.7358e-5 
2.7475e-5 
0.0014927 
0.0066797 
0.0087953 
0.00075514 
0.0003776 
9.9078e-7 

7.7691e-5 
2.742e-5 

0.0014958 
0.0066842 
0.0088142 
7.5564e-6 
3.7785e-6 
9.9505e-7 

7.8574e-5 
2.706e-5 

0.0017948 
0.0067379 
0.0088642 
0.00076169 
0.00038089 
1.0063e-6 

7.8449e-5 
2.9142e-5 
0.0017836 
0.0067449 
0.0088571 
0.00076248 
0.00038128 
1.0047e-6 

8.0539e-5 
3.0036e-5 
0.0023101 
0.0068758 
0.0089744 
0.00077721 
0.00038867 
1.0312e-6 

 
 
 

Error 

time series parametric models  
ARIMA(0,1,2) 

(Student) 
 

ARIMA(0,1,2) 
(GED) 

ARIMA(0,1,2)- 
GARCH(1,1) 

(Student) 

ARIMA(1,1,1)- 
GARCH(1,1) 

(GED) 
MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

7.8051e-5 
2.7454e-5 
0.0015387 
0.0067188 
0.0088346 

0.00075963 
0.00037985 
9.9984e-7 

7.8615e-5 
2.7438e-5 
0.0016234 
0.0067622 
0.0088665 
0.00076455 
0.0003823 
1.007e-6 

7.8058e-5 
2.7546e-5 
0.0015409 
0.0067197 
0.0088351 

0.00075973 
0.0003799 
9.9994e-7 

7.8058e-5 
2.7546e-5 
0.0015409 
0.0067197 
0.0088351 

0.00075973 
0.0003799 
9.9994e-7 

Table 5.  Values of the forecast errors obtained by using diffusion models. Logarithmic 
levels of NASDAQ index  

Error Phillips and Yu method Generalized Method of Moments 
Vašiček CIR CKLS Vašiček CIR CKLS 

MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

7.3033e-5 
1.8082e-5 
0.055699 
0.0062114 
0.0085459 
0.00079535 
0.00039763 
1.1983e-6 

7.3078e-5 
1.7335e-5 

0.00064038 
0.0062146 
0.0085486 

0.00079575 
0.00039783 

1.199e-6 

7.3089e-5 
1.8443e-5 
0.055435 
0.0062015 
0.0085492 
7.9409e-6 
3.9699e-6 
1.1992e-6 

7.5079e-5 
2.1237e-5 
0.0013926 
0.0064573 
0.0086648 
0.00082676 
0.00041338 
1.2319e-6 

7.4249e-5 
1.9876e-5 
0.0013442 
0.0064043 
0.0086168 

0.00081998 
0.00040998 
1.2183e-6 

7.5569e-5 
2.1606e-5 
0.0015984 
0.0065172 
0.008693 

0.00083443 
0.00041722 

1.24e-6 
 
 
 

Error 

time series parametric models  
ARIMA(0,1,2) 

(Student) 
 

ARIMA(0,1,2) 
(GED) 

ARIMA(0,1,2)- 
GARCH(1,1) 

(Student) 

ARIMA(1,1,1)- 
GARCH(1,1) 

(GED) 
MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

7.2567e-5 
1.7764e-5 
0.056472 
0.0062011 
0.0085187 

0.00079404 
0.00039697 
1.1907e-6 

7.2567e-5 
1.7769e-5 
0.056466 
0.006201 
0.0085186 
0.00079404 
0.00039697 
1.1907e-6 

7.282e-5 
1.8811e-5 
0.055536 
0.0062328 
0.0085335 

0.00079811 
0.00039901 
1.1949e-6 

7.282e-5 
1.8819e-5 
0.055532 
0.0062328 
0.0085334 

0.00079811 
0.00039901 
1.1948e-6 

 



Piotr Płuciennik 58

Table 6.  Values of the forecast errors obtained by using diffusion models. Logarithmic 
levels of WIG20 index 

Error Phillips and Yu method Generalized Method of Moments 
Vašiček CIR CKLS Vašiček CIR CKLS 

MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

0.00018948 
6.1233e-5 

0.00063017 
0.010608 
0.013765 
0.0013032 

0.00065164 
2.8646e-6 

0.0001892 
6.3602e-5 

0.00063866 
0.010619 
0.013755 
0.0013046 

0.00065232 
2.8603e-6 

0.00018936 
6.5722e-5 
0.058766 
0.010609 
0.013761 
1.3034e-5 
6.5172e-6 
2.8627e-6 

0.00018897 
6.4217e-5 
0.0001788 
0.010582 
0.013747 
0.0013001 

0.00065007 
2.8568e-6 

0.00018938 
6.4506e-5 

0.00032633 
0.010595 
0.013762 
0.0013016 

0.00065083 
2.8629e-6 

0.00018973 
6.7774e-5 
-0.0001169 
0.010586 
0.013774 
0.0013007 

0.00065032 
2.8682e-6 

 
 

Error 

time series parametric models 
ARMA(0,2) 

(GED) 
ARMA(1,1) 
(Student) 

MSE 
MedE 

ME 
MAE 

RMSE 
MAPE 

AMAPE 
LL 

0.01904131 
6.6946E-05 
0.00035112 
1.06889157 
0.13799026 

0.131329 
0.00065666 
2.8789e-06 

0.019061629 
6.87936e-5 

0.000398688 
1.071935667 
0.138063858 
0.1317012 

0.000658529 
2.88198e-6 

Note: MSE – the mean squared error, MedE – the mean median error, ME – the mean error, MAE – the mean 
average error, RMSE – the root of the mean squared error, MAPE – the mean average percentage error, 
AMAPE – corrected average percentage error, and LL – logarithmic loss function (cf. Welfe 1998; Doman, 
Doman, 2004). 

6. Conclusions 
The high quality of the forecast obtained from the diffusion models is indicative 
of good fitting of the diffusion models to the studied time series. The values of 
the forecast errors are often smaller when diffusion models were used.  
It is notable that in diffusion models the volatility depends only on white noise 
and optionally on current value of the process. In ARIMA-GARCH models the 
volatility is described by the second parametric equation.   
 The most surprising fact is that the CKLS model leads to worse quality of 
the forecast than the Vašíček and CIR model. After all, both models are 
special cases of the CKLS model. The reason for that situation lies in very bad 
fitting of the CKLS model to the examined time series. The estimates of β  are 
negative, while the model assumed that ].1,0[∈β  
 It is notable that determining the forecasts by using diffusion models is not 
laborious. The MATLAB procedures used in the conducted research need only 
a fraction of a second to estimate parameters by using two-stage Phillips-Yu 
method and a few seconds if we decide to use the GMM method. The most 
laborious part of the calculation is determining 10000 sample paths, but it takes 
up to two minutes to do this operation. 
 



Forecasting Financial Processes by Using Diffusion Models  59

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Prognozowanie procesów finansowych za pomocą modeli dyfuzji 

Z a r y s  t r e ś c i.  Prognozowanie szeregów czasowych jest jednym z najważniejszych 
zagadnień współczesnej ekonometrii finansowej. W obliczu rosnącego zainteresowania modelami 
z czasem ciągłym i szybkiego rozwoju metod ich estymacji, podejmujemy w pracy próbę 
modelowania i prognozowania szeregów czasowych z różnych rynków finansowych za pomocą 
modeli dyfuzji. Stosujemy w tym celu bazującą na symulacjach Monte-Carlo metodę 
wprowadzoną przez Cziraky i Kucherenko (2008). Jakość otrzymanych prognoz zostaje 
skonfrontowana z jakością prognoz otrzymanych za pomocą powszechnie stosowanych 
parametrycznych modeli szeregów czasowych.  

S ł o w a  k l u c z o w e: model ARIMA, model GARCH, modele dyfuzji, pierwiastek 
jednostkowy, prognozy ex-post, symulacja Monte-Carlo. 




