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where  B  B  B . By the properties of Brownian motion (independent and
t  t t  t t i
i
i
d d
identically increment) B  B  B  B  B   tB where B is standard

t  t t  t t t  t t t  1 1
i i i i i
0,1 .
normal distribution N  
For the model with jumps, we also discretize the model in Eq. (4) and use similar

computation as above. Then, we also have the today price at t 
t
i
     2 N t i  
t 
S S exp     t    B  Y

t 0  i  i
i 2 i
    i 1  
and the next day price at t  t   is
t
i
     2 N t  t  
i

S  S exp     t    t    B t  Y .

t  t 0  2  i t  i
i
i
    i 1  
Then also the return can be obtained by
     2 N t  t  
i

S exp     t    t    B t  Y 
S 0   2   i t  i 1 i  
i
t  t  .
i
S     2 N t i 
t 
t i  
S exp     t    B  Y 
0 2 i i i
    i 1  
We again apply the mathematical manipulation and the property of exponential function. Then

S     2     N t  t N t i  
i 
i
t  t  
i  exp      t   exp  B   B exp   Y  Y 
S   2   t  t t i  i 
i
t     i 1 i 1 
i
.
N t  t N t i N t  t
i
i
As  Y  i  Y  i  Y , then we get
i
i 1 i 1 i N  1

i t
     2 N t  t  
i
 
S S exp      t    B   Y (7)

t  t t   t  t i
i i  2   i 
 i N   t i 1 
where N  N and N has Poisson distribution with parameter  .
t
t i t  t t  t
i
i
We will simulate the USS price by using Eq. (6) and Eq. (7) and all parameters will be
estimate from the historical data of the USS price. To estimate the parameters of the continuous