Consequently, under the domestic forward measure, we
The arbitrage-free price of a European call option at time
t
equals
‧
Consider foreign stock and bond under domestic measure
, , 1 2 , , ,
Here, the dynamics SFQ D, under forward measure associated with
,
under the measure associated withGDT*( )T , the dynamics of exchange rate is given blow:
‧
Hence the call as follows
4
‧
The fundamental theorem is based on double martingale proposed by Elliott (1976).
0
The martingale condition under the risk-natural measure
* *
‧
of the jump variable.Then equation (3.A2) implies that 2 *
( )
1 22
J
J Y t J J
.
Also equation (3.A1) implies that
, , ( )
, , ( )
1
, , ( )
, , ( )
) 2By consider Brownian motion only, we reduce the Esscher transform to Girsanov Theorem.
‧
We apply Girsanov Theorem to the term
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Hence, we have a new jump intensity ( ( )) (Y s J*), meaning that the jump intensity
matrix is given by
* *
* 1 1
* *
2 2
0 ( ) 0
0 0 ( )
J
J
of the MMPP.
Q.E.D.
‧
We choose an initial state for the MMPP ( )t . The strong Markov-chain property and the Esscher formula are given as follows: (cf. Dijkstra and Yao (2002))
* *
respectively, the first and second measure space.
Consider the second measure space
* *
Consider the expectations * * * ,( C , J )
Consider the inside expectation with strong Markov chain
* *
‧
is a cumulative normal distribution function.
Consider the first measure space, as was done for the second measure space, then
* * * *
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