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list1
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simple q/ about short rate

February 23rd, 2017, 6:50 pm

 Sorry
 
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Orbit
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Re: simple q/ about short rate

February 23rd, 2017, 10:37 pm

Indeed.
 
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list1
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Re: simple q/ about short rate

February 24th, 2017, 7:59 pm

Formula (37.2) in ' On QF' presents SDE for bond price
[$] dZ ( s , T ) = \mu ( s , T ) Z ( s , T ) ds  +  \sigma ( s , T ) Z ( s , T ) dX ( s ) [$]  
Let coefficients be deterministic and continuous functions on [ 0 , T ].
It is clear that sde should be interpreted in integral form. How does it should be integrated over [ 0 , t ] or over [ t , T ] ? 
 
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Re: simple q/ about short rate

February 25th, 2017, 2:26 pm

Formula (37.2) in ' On QF' presents SDE for bond price
[$] dZ ( s , T ) = \mu ( s , T ) Z ( s , T ) ds  +  \sigma ( s , T ) Z ( s , T ) dX ( s ) [$]                                              (1)
Let coefficients be deterministic and continuous functions on [ 0 , T ].
It is clear that sde should be interpreted in integral form. How does it should be integrated over [ 0 , t ] or over [ t , T ] ? 
We know that eq (1) has sense only as integral eq 
[$]Z ( t , T )  -  Z ( s , T )  =  \int_s^t \mu ( u , T ) Z ( u , T ) du  + \int _s^t \sigma ( u , T ) Z ( u , T ) d X ( u ) [$]
Now we should specify the spot and future dates for dates s and t. Let s be spot and t is a future date. Then putting for example t = T we should  guarantee that 
1) Z ( s , T ) < 1 for s < T , 
2) Z ( T , T )  =  1 , and 3) it is not clear why spot price  Z ( s , T ) at is random while it is known at s. I think 1-2 problems.  probably could be overcome. But 3) is needed to be explained.
Other case when t is spot and s is past date we have similar problems.Unfortunately these technical problems did not discuss in the book but they are effect their practical applications too. What we should conclude in a case when we say that we have a good approximation historical data with model (1) and an opponent say that there is 10% a chance that Z ( t , T ) > 1?