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2

1
n
1

Var R   n  1 n 1 1  R  i R ; R  n  R is average value of the log return.


i
i
1
i

1 n 1 2
Then, Var R      t  n  1   R  R
2
i
i
1
1 n 1  R  R  2

n  1 i
2
  i 1
t 
n
1  1  R   2
n  1 i R
  i 1 (9)
t 
1 n 1  2
where   R  R is the standard deviation of the log return.
n  1 i 1 i
For estimating all parameters of the model with jumps, we apply the method of moment
which is called nonparametric estimation introduced by Johannes (Johannes, 2004) and Valachy
(Valachy, 2004) applied this method to the currencies exchange rate from the Central European

(CE) region. Recall that the jump sizes are assumed to be normally distributed with mean 0 and

2
variance  . Let us assume that all parameters are constants. The formula for diffusion part of
the model is

2
  2   2  (10)
T
where  is the diffusion of the continuous part,  is the total diffusion (or 2 moment),
2
2
nd
T
and  2 is the variance of jump sizes. Under the assumption of constant jump intensity, the
proposed estimation procedure is as follows:

Estimate parametrically the drift 

n   S  
1 t 
E   logS t t  logS  t   n   log      S i t i  t      . (11)
i
1
Estimate  and  based on the calculation of moments, the ratio of the 4 and 6
2
th
th
moments will give us the estimate of  . Consequently, the estimate of  will be
2