The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 411 Wednesday, February 4, 2004 1:08 PM

EXHIBIT 14.2

Conditional Probabilities

411

PA

11

∩

A

,

{

+

(

,

10

)

P

12

}

(

,

∩

A

10

)

P

34

}

PA

21

,

{

+

PA

11

A

10

)

=

----------------------------------------

=

------------------------------------------

• 
  

=

0.5

)

=

----------------------------------------

=

------------------------------------------

• 
  

=

0.5

(

,

,

PA

21

(

,

A

10 ,

{

+++

PA

10

)

P

1234

}

PA

10

{

+++

(

,

)

P

1234

}

(

,^

(

,

10

)

P

1

PA

12

∩

A

,

{}

∩

A

10

)

P

2

PA

22

,

{}

(

,

PA

12

A

10

)

=

----------------------------------------

=

------------------------------------------

• 
  

=

0.25

A

10

)

=

----------------------------------------

=

------------------------------------------

• 
  

=

0.25

(

,

,

PA

22

(

,

,

{

+++

PA

10

)

P

1234

}

(

,

{

+++

PA

10

)

P

1234

}

(

,^

(

,

10

)

P

3

PA

32

∩

A

,

{}

∩

A

10

)

P

4

PA

42

,

{}

(

,

PA

32

A

10

)

=

----------------------------------------

=

------------------------------------------

• 
  

=

0.25

)

=

----------------------------------------

=

------------------------------------------

• 
  

=

0.25

(

,

,

PA

42

(

,

A

10 ,

{

+++

PA

10

)

P

1234

}

(

,

{

+++

PA

10

)

P

1234

}

(

,^

(

,

11

)

P

1

∩

A

21

)

P

∅

PA

12

∩

A

,

{}

0.25

PA

12

,

{}

(

,

PA

12

A

11

)

=

----------------------------------------

=

-----------------------

=

-----------

=

0.5

)

=

----------------------------------------

=

-----------------------

=

0

(

,

,

PA

12

(

,

A

21 ,

(

,

{

+

PA

11

)

P

12

}

0.5

{

+

(

,

)

P

12

}

PA

21

PA

22

∩

A

,

)

P

2

∩

A

21

)

P

∅

(

,

11

{}

0.25

PA

22

,

{}

(

,

PA

22

A

11

)

=

----------------------------------------

=

-----------------------

=

-----------

=

0.5

)

=

----------------------------------------

=

-----------------------

=

0

(

,

,

PA

22

(

,

A

21 ,

(

,

{

+

PA

11

)

P

12

}

0.5

{

+

(

,

)

P

12

}

PA

21

(

,

11

)

P

∅

PA

32

∩

A

,

{}

∩

A

21

)

P

3

PA

32

,

{}

(

,

PA

32

A

11

)

=

----------------------------------------

=

-----------------------

=

0

)

=

----------------------------------------

=

-----------------------

=

0.5

(

,

,

PA

32

(

,

A

21 ,

{

+

PA

11

)

P

12

}

(

,

{

+

PA

21

)

P

34

}

(

,^

(

,

11

)

P

∅

PA

42

∩

A

,

{}

∩

A

21

)

P

4

PA

42

,

{}

(

,

PA

42

A

11

)

=

----------------------------------------

=

-----------------------

=

0

)

=

----------------------------------------

=

-----------------------

=

0.5

(

,

,

PA

42

(

,

A

21 ,

{

+

PA

11

)

P

12

}

(

,

{

+

PA

21

)

P

34

}

(

,