The model determines a stochastic continuous process as continuous limit of a stochastic discrete process so to show that the stochastic continuous process converges to the stochastic discrete process such that we can integrate it. Furthermore, the model determines the expected volatility and the expected mean so to show that the volatility and the mean are increasing function of the time

Giandomenico, Rossano

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MPRA
Munich Personal RePEc Archive
Martingale Model
Rossano Giandomenico
2006
Online at https://mpra.ub.uni-muenchen.de/21973/
MPRA Paper No. 21973, posted 12. April 2010 02:03 UTC
Graduate at University of Bologna (Italy) 
e-mail: rossano1976@libero.it 
1 
Martingale Model 
 
 
 
Rossano Giandomenico                                    
 
 
Abstract: The model determines a stochastic continuous process as continuous limit of a stochastic discrete process 
so to show that the stochastic continuous process converges to the stochastic discrete process such that we can 
integrate it. Furthermore, the model determines the expected volatility and the expected mean so to show that the 
volatility and the mean are increasing function of the time. 
  
 
Introduction  
 
In the 1827 Robert Brown was the first to observe and to describe the motion of a small particle 
suspended in a liquid as result of the successive and casual impacts of the near particle so to 
note that its variance is an increasing function of the time, from this the term Brownian motion.  
In the 1905 Albert Einstein proposed on this a mathematics theory that was developed and 
presented in a more rigorous way by Norbert Wiener in the 1923. 
The first stochastic model of the market price that used a Brownian motion was developed in 
the 1900 by Louis Bachelier in its extraordinary thesis at Sorbone of Paris. He has modelled the 
market price in the continuous time such that the uncertainty on the immediately future was 
preserved. This process usually called Martingale has the problem to permit to the market price 
to assume a negative value but as we will see this problem is easy to solve. In fact, if we 
consider that it models the rate of return, instead the arithmetic change of the market price, we 
get a stochastic continuous process that follows a lognormal distribution in each finite interval 
known as geometric Brown process. Many authors have given formal mathematics assumptions 
to a generalized stochastic equation but we show that it can be get as continuous limit of a 
discrete casual variable trying to explicit the economics assumptions implicit in the mathematics 
assumptions. The first implication is the continuous negotiations that can seem an abstraction 
from the reality but if the interval of the time is very short or indeterminate small the continuous 
solution is a good approximation of the discrete solution. 
 
 
Martingale 
 
The influences that can determine the fluctuations of the market are many, past events, present 
events and future events, which are incorporated in the market price. 
Thus, the current fluctuations are function not only of the precedent fluctuations but either of the 
market expectation that depends from an infinite number of factors such that it is impossible a 
mathematics prediction. 
Furthermore, we can note that speculators of the market use two kind of probability: 
 
1) Mathematics probability determinable at priori (objective) 
 
2) Probability of the future events impossible to predict in a mathematics way (subjective) 
 
The last one is the probability that speculators try to predict. In every instant the sellers believe 
in a decrease of the market price, instead, the buyers believe in an increase of it. Thus, the 
market believes neither in an increasing neither in a decreasing of the market price because the 
number of sellers are exactly equal to the number of buyers. 
 2 
But if the market believes neither in an increasing neither in a decreasing of the market price we 
can assume a fluctuation of a given width with a given probability at priori. 
Formally, in the discrete time we have: 
 
S(τ)  =  S(t)  +  ε(t,τ) 
 
Where  ε(t,τ)  is a casual variable with normal distribution, mean zero and variance  σ2(t,τ)  such 
that  Cov [ε(t,τ) , ε(τ,s)] = 0   ∀   τ < s  , the process is independent from the others on each 
finite interval. 
We can note that the conditional expected value is: 
 
E [ S(τ)S(t) ]  =  S(t)                ∀           τ > t 
 
The process looks just one step back. This is what is called Martingale and its interpretation is 
easy enough, the conditional expected value is influenced neither from the past value neither 
from any current information, but depends only from the current price. 
This is equal to assume that the market is efficient such that any available information on the 
past, present and future, is incorporated quickly in the current price. As such, the past prices 
don’t have provisional value. In fact, if this is not the case the speculators can beat the market 
by using the technical analysis of historical series. 
We have a Supermartingale when: 
 
E [ S(τ)S(t) ]  ≤  S(t) 
 
We have Submartingale when: 
 
E [ S(τ)S(t) ]  ≥  S(t) 
 
We call this either Martingale with drift because  E [ε(t,τ)]  =  µ(t,τ)  such that  µ(t,τ)  is the drift 
of the stochastic process. 
At this point, we have to construct a continuous process as limit of the discrete process with the 
same characteristics such that it is independent from the width of the time interval  ∆t . 
If we take an interval of time  [ t , τ ]  and we share it in  n  interval of the same width we have:  
 
∆t  = [(τ – t) / n ] . 
 
From this we can write the discrete time model as follows: 
 
                                                                         n-1                                     
                                                 S(τ)  –  S(t)  =  ∑  S(tk +∆t)  –  S(tk)                                           1.1 
                                                                         k=0                                              
Where: 
 
S(tk +∆t)  −  S(tk)   =   S(tk+1)  –  S(tk)  –  Etk [S(tk+1) – S(tk)]  =   ε(k) 
 
Etk[S(tk +∆t) – S(tk)] = 0     
  
Cov [ ε(k) , ε(k+1) ]  =  0 
 
 
Now we put: 
σk ε(k) √∆t 
                             S(tk+∆t) – S(tk)    =                                                                                        1.2 
√σk2∆t 
 
 3 
Where: 
σk2  =  Etk [ε2(k)]/∆t 
 
We can note that as  n → ∞  and  ∆t  becomes infinitesimal, we have for the theorem of the 
central limit: 
dS(t) = σ(t) N[0,1]√dt 
 
Where:      
σ(t)  =  σ(t,τ) / √(τ – t)        denotes the instantaneous volatility 
 
This shows that the continuous negotiations bring to have a normal distribution for the dynamic 
of the price. We can write it as follows: 
 
dS(t) = σ(t) dW(t) 
 
Where: 
dW(t)  =  N[0,1]√dt        denotes a Wiener process 
 
We can note that the Wiener process is not derivable with respect to the time. In fact, the 
derivative becomes infinite as  dt   tends to zero:    
 
dW / dt  =  N[0,1](1/2)(dt –1/2) 
 
If a Wiener process is not derivable with respect to the time we can’t integrate it. In fact, either 
if the Wiener process is continuous, it is a function of an infinite variation. Thus, we can’t 
compute the integral for each single trajectory, but given our assumption that the stochastic 
continuous process is the limit toward the discrete time process converges as the interval of time 
becomes infinitesimal, we can assume that the Wiener process is given and that the diffusion 
process  σ(t)  is deterministic, because the stochastic continuous process as the time increase 
converges to the stochastic discrete process. Thus we have: 
 
 
 ∆W(t) = N[0,1]√∆t 
 
That is equal to: 
                                             τ                    
∫  dW(t)  =  W(τ)  –  W(t)  =  N[0,1]√τ – t 
                                           t 
 
We remind that an integral is a sum. Thus, we have: 
 
                                                                  τ                    
E [ ∫ dW(t)2 ]=  (τ – t) 
                                                                 t 
                                                                  τ                    
E [ ∫ σ(t) dW(t) ]  =  0 
                                                                t 
                                                       τ                              τ                   
E {[ ∫ σ(t) dW(t)]2} =  ∫ E [ σ (t)2 ] dt 
                                                     t                               t 
 
 
 
 4 
We can note that 1.2 becomes at limit the stochastic differential equation: 
 
dS(t)  =  σ(t) dW(t) 
 
S(t)  =  S(o) 
Instead, the 1.1 becomes: 
                                                                                 τ   
S(τ)   =   S(t)  +  ∫  σ(t) dW(t) 
                                                                                t     
 
We have the following characteristics for the stochastic differential equation: 
 
E [ dS ]  =  0 
 
Var [ dS ]  =  σ2(t) dt 
 
We can note that the variance is an increasing function of the time and the expected value is 
zero. Moreover, we can note that we kept the same characteristics of the discrete time model. 
 
E [ S(τ) ]  =  S(t) 
 
Var [ S(τ) ]  =  σ2(t) (τ – t)  =  σ2(t,τ) 
 
Along the same line, we have for a Martingale with drift the following stochastic differential 
equation: 
 
dS(t)  =   µ(t) dt  +  σ(t) dW(t) 
 
S(t)  =  S(o) 
 
Where:               
µ(t)  =  µ(t,τ) / (τ – t)      denotes the instantaneous mean 
                       
Again we have: 
                                                                      τ                   τ 
S(τ)  =  S(t)  +  ∫  µ(t) dt  +  ∫ σ(t) dW(t) 
                                                                     t                    t       
 
We can note that : 
 
E [ dS ]  =  µ(t) dt 
 
Var [ dS ]  =  σ2(t) dt 
 
Thus: 
E [ S(τ) ]  =  S(t)  +  µ(t) (τ – t)   =  S(t)  +  µ(t,τ) 
                                       
Var [ S(τ) ]  =  σ2(t) (τ – t)  =  σ2(t,τ) 
 
At this point, we have to note that the instantaneous volatility and the drift of the stochastic 
continuous process are forward processes, in the sense that they are based on the effective 
volatility and mean of the discrete time process. As such, they are deterministic, but in the 
reality we can’t observe the future volatility and the future mean. As result, the volatility and the 
drift of the process are stochastic; in this case we can approximate them by using the expected 
 5 
value. We have to note that the drift can be stochastic either with respect to a deterministic value 
of the expected mean of the discrete time process so that it will revert to it. This is what we 
usually call mean-reverting process such that the stochastic continuous time process goes from a 
Supermartingale to a Submartingale in continuous time. 
At this point, we have to note that the price can assume negative value, but as we said we can 
solve this problem by assuming that the Martingale models the rate of return, instead, the 
arithmetic change of the market price. Thus, we get a stochastic continuous process that follows 
a lognormal distribution in each finite interval known as geometric Brown process. In fact, we 
can see that if we put the following process: 
 
dS(t) / S(t)  =  µ(t) dt 
 
We have the following solution: 
S(τ)  =  S(t) e µ(t) (τ  –  t ) 
 
As result, the geometric Brown process follows a lognormal distribution in each finite interval. 
Thus, we have the following: 
 
dS(t) / S(t)  =  µ(t) dt  +  σ(t) dW(t) 
 
We can note that: 
E [ S(τ) ]  =  S(t) e µ(t) (τ  –  t ) 
 
 
 
 
 
 
Expected Volatility and Expected Mean 
 
 
Now we have to note that we showed that the volatility is an increasing function of the time 
with respect to a finite interval of time. In fact, the stochastic continuous process converges to 
the stochastic discrete process. This doesn’t mean that the volatility is an increasing function of 
time. In fact, we can have a greater finite interval with the same volatility. Thus, we need a 
model that permits us to get the expected volatility and its behaviour. 
We can construct a stochastic continuous process with the instantaneous historical variance like 
drift. This permits us to get the expected volatility and to observe the behaviour of the volatility 
from one step to another. However, we can find a probability solution of a stochastic continuous 
process just if we use a geometric Brown process. 
Thus, we have: 
 
 
    σ(o,t)2  =  σ2(o)(t – o)                    denotes the total historical variance 
 
       σ2(o) =  σ(o,t)2 / (t – o)                 denotes the instantaneous historical variance 
 
 
We can put the following geometric Brown process: 
 
 
dσ(t,t)2 / σ(t,t)2  =  µ(t) dt  +  δ(t) dW (t) 
 
σ(t,t)2  =  σ(o,t)2 
 6 
Where: 
µ(t)  =  σ2(o)/ σ(o,t)2  =  1 / (t – o)                              
 
δ(t)   denotes the instantaneous volatility of the variance 
 
Where we have assumed that the initial time is in  t > o . 
We can note that the expected value of the process is an increasing function of time. 
Thus, we have: 
 
E [ σ(t,τ)2 ]  =  σ(t,t)2 e {[1 / (t – o)] (τ – t)} 
 
Therefore, the expected instantaneous volatility is: 
 
E [ σ(t) ] =  √{E [ σ(t,τ)2 ]  −  σ(o,t)2} / (τ – t)  =  √ σ2(o)(e − 1) 
 
Where: 
(τ – t)  =  (t – o) 
 
We got the expected instantaneous volatility in the interval (τ – t) that is an increasing function 
of the number of steps, this shows that the expected volatility is an increasing function of the 
time. The limit of the model is that the expected instantaneous volatility can be negative, but for 
rational value of the parameter δ(t) the probability that this can happen is very low. 
We conclude by noting that this is an expected value and that the effective instantaneous 
volatility can be different from it. At this point, we can construct the same process for the mean 
so to show that the expected mean is an increasing function of time. 
 
Conclusion 
 
The model has showed that a stochastic continuous process is the limit of a discrete time 
process such that we can integrate it either if it is a function of an infinite variation. 
Furthermore, we have determined the expected volatility and the expected mean on the base of 
the historical volatility and the historical mean so to show that they are an increasing function 
of the time. 
 
 
 
 
References 
 
Bjòrk,T. :                                               Arbitrage Theory in Continuous Time 
Oxford University Press (1998)  
 
Øksendal,B. :                                      Stochastic Differential Equations (4th edn) 
Springer Verlag,Berlin Heidelberg (1995) 
 
Louis Bachelier                                              Theory of Speculation (1900) 
Cootner: The Random Character of Stock Market Prices 
Cambridge, Mass, MIT. (1964) 
 
Malliaris,A.G. :                                  Ito’s Calculus in Financial Decision Making 
S.I.A.M. Review, Vol. 25 n° 4 (October1983) 
 
Malliaris,Brock :                             Stochastic Methods in Economics and Finance 
North-Holland (1982) 
 
Merton,R. :              On the Mathematics and Economics Assumptions of Continuous-Time Models 
Massachusetts Institute of Technology (1982) 


Martingale Model

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