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why do short term models are written with respect to Q

February 28th, 2017, 4:32 am

I can not understand why stoch equations for short interest rate are assumed on risk neutral world. If we consider BS option pricing it is clear the sense of the risk neutral equation which is real underlying of the BSE. This real underlying contradicts statement that derivatives take values from its underlying. As far as option does not change its value when real underlying drift changes its value the risk neutral measure can cover this puzzle. 
But short term SDEs is written themselves without its connection to options and actually play the same role as real stocks. Why do they write models with respect to Q? Thanks.
 
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Re: why do short term models are written with respect to Q

February 28th, 2017, 2:59 pm

It is simply a shortcut to avoid repeating standard & careful arguments that begin with a real-world process. To see the careful arguments, read the early literature, for example Vasicek
 
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Re: why do short term models are written with respect to Q

February 28th, 2017, 3:13 pm

It is simply a shortcut to avoid repeating standard & careful arguments that begin with a real-world process. To see the careful arguments, read the early literature, for example Vasicek
Thanks Alan. It seems relevant< i will read.
 
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Re: why do short term models are written with respect to Q

February 28th, 2017, 8:54 pm

It is simply a shortcut to avoid repeating standard & careful arguments that begin with a real-world process. To see the careful arguments, read the early literature, for example Vasicek
Thanks Alan. It seems relevant.  i will read.
The paper of course relevant to original question but I was confused by reading. Starting point is the equation (8)
 
[$]dP = P \mu ( t , s ) dt  +  P \sigma ( t , s ) dz      [$]                                (8)
 
where 
 
[$]\mu ( t , s )  =  \mu ( t , s , r ( t )) , \sigma ( t , s )  =  \sigma ( t , s , r ( t ))[$]       (coeff)
 
are given by (9) and (10). Here t is a current moment and s, s > t is an expiration date.
Here r ( t ) is instant forward rate which is defined by sde (5)
 
[$]dr  = f ( t , r ) dt  + \rho ( t , r ) dz (t)        [$]                                      (5)
 
What I could not comprehend is that interest rate of the bond is r ( t , s ). In stochastic setting it should be defined by equality
 
[$] dr_0 ( t , s )  = \mu ( t , s ) dt  +  \sigma ( t , s ) dz ( t ) [$]
 
 It shown or assumed I am not sure in (coeff) that mu and sigma are depend on forward rate         r ( t ). Actually for continuous r ( t ) either stochastic or not
 
[$]P ( t , s )  =  exp - \int_{t}^{s} r ( u ) du [$]     (a)
 
where r is defined by (5). In other words (a) shows that P ( t , s ) does not depend on r ( t ) at t it depends on r ( u ) where t < u < s. Hence we should apply Ito formula not in the way how it would lead us to (9), (10). In order to specify right formula we should consider the difference
 
[$]P ( t + \Delta , s )  -  P ( t , s )  = [$] 

[$]= exp - \int_{ t  + \Delta }^{s} r ( u ) du  -  exp - \int_{t}^{s} r ( u ) du  =   [$]

[$]=  exp - \int_{t  + \Delta }^{s} r ( u ) du  ( 1 – exp - \int_{t}^{ t  + \Delta } r ( u ) du ) [$] =  

[$]\approx  - exp - \int_{t  + \Delta }^{s} r ( u ) du  ( (f  + \frac{1}{2} \rho^2 ) dt  + \rho dz ) \to  P ( t , s ) (  (f  + \frac{1}{2} \rho^2 ) dt  + \rho dz ) [$]   

 Derivatives of the P with respect to r would become if  P ( t , s , r ( t )) as it stated in the paper. On the other hand formula (a) shows that P at moment t depends on whole path r on [ t , s ] but not at the value of r at moment t. If my point is correct then we could not arrive at the BS type equation (15) which can justify risk neutrality concept. 

 
 
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Re: why do short term models are written with respect to Q

March 1st, 2017, 2:45 pm

It is simply a shortcut to avoid repeating standard & careful arguments that begin with a real-world process. To see the careful arguments, read the early literature, for example Vasicek
Thanks Alan. It seems relevant.  i will read.
The paper of course relevant to original question but I was confused by reading. Starting point is the equation (8)
 
[$]dP = P \mu ( t , s ) dt  +  P \sigma ( t , s ) dz      [$]                                (8)
 
where 
 
[$]\mu ( t , s )  =  \mu ( t , s , r ( t )) , \sigma ( t , s )  =  \sigma ( t , s , r ( t ))[$]       (coeff)
 
are given by (9) and (10). Here t is a current moment and s, s > t is an expiration date.
Here r ( t ) is instant forward rate which is defined by sde (5)
 
[$]dr  = f ( t , r ) dt  + \rho ( t , r ) dz (t)        [$]                                      (5)
 
What I could not comprehend is that interest rate of the bond is r ( t , s ). In stochastic setting it should be defined by equality
 
[$] dr_0 ( t , s )  = \mu ( t , s ) dt  +  \sigma ( t , s ) dz ( t ) [$]
 
 It shown or assumed I am not sure in (coeff) that mu and sigma are depend on forward rate         r ( t ). Actually for continuous r ( t ) either stochastic or not
 
[$]P ( t , s )  =  exp - \int_{t}^{s} r ( u ) du [$]     (a)
 
where r is defined by (5). In other words (a) shows that P ( t , s ) does not depend on r ( t ) at t it depends on r ( u ) where t < u < s. Hence we should apply Ito formula not in the way how it would lead us to (9), (10). In order to specify right formula we should consider the difference
 
[$]P ( t + \Delta , s )  -  P ( t , s )  = [$] 

[$]= exp - \int_{ t  + \Delta }^{s} r ( u ) du  -  exp - \int_{t}^{s} r ( u ) du  =   [$]

[$]=  exp - \int_{t  + \Delta }^{s} r ( u ) du  ( 1 – exp - \int_{t}^{ t  + \Delta } r ( u ) du ) [$] =  

[$]\approx  - exp - \int_{t  + \Delta }^{s} r ( u ) du  ( (f  + \frac{1}{2} \rho^2 ) dt  + \rho dz ) \to  P ( t , s ) (  (f  + \frac{1}{2} \rho^2 ) dt  + \rho dz ) [$]   

 Derivatives of the P with respect to r would become if  P ( t , s , r ( t )) as it stated in the paper. On the other hand formula (a) shows that P at moment t depends on whole path r on [ t , s ] but not at the value of r at moment t. If my point is correct then we could not arrive at the BS type equation (15) which can justify risk neutrality concept.  
This paper presents an attempt to present bond price as a derivative, which underlying is its own short term forward rate. It looks impossible to realize this idea. Generally speaking derivatives pricing begin with an underlying asset which price is independent on derivatives, ie is traded itself regardless whether derivatives exist or not. Derivatives is then defined by its cash flow specified by the underlying security. Bond and its forward rate are equivalent notions. One is fully determines other, ie it looks does not make sense to consider either bond or forward rate as underlying or derivative of the other.
 
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Re: why do short term models are written with respect to Q

March 1st, 2017, 3:18 pm

 
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Re: why do short term models are written with respect to Q

March 3rd, 2017, 7:02 pm

It  think I could be wrong with my conclusion about  Vasicek paper. But something does not clear with the short interest theory. Of course either stochastic model used in randomization bond price should guarantee 0 < B ( t , T ) < 1, B ( T , T ) = 1. In particular stochastic short term rate should be always positive. 
 
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Re: why do short term models are written with respect to Q

March 4th, 2017, 2:15 pm

On the other hand when along with standard formulas
 
[$] B ( t , T )  =  exp - \int_t^T r ( u , T ) du  =  exp - \int _t^T f ( t , v ) dv[$]             (1)
 
we use short term formula
 
[$] B ( t , T ) = exp - \int_t^T r ( v ) dv[$]                                            (2)
 
then r ( u ) here should be implied forward rate , ie r ( v ) = f ( t , v ). Indeed
 
[$]B ( t , T )  =  B ( t , t + \Delta ) B ( t + \Delta , T ; t )  [$] 
 
where  [$] B ( t + \Delta , T ; t )  =  \frac{1}{ 1  +  f ( t + \Delta , T ; t ) ( T – t  – \Delta )} [$]
I used simple compounding interest rate for more transparency. It should be replaced by its continuous version. Next, we should repeat the same with [$] B ( t + \Delta , T ; t ) [$]. Hence we should recognize that r ( v ) = f ( t , v ) in  (2) . On the other hand, we can use other way. We can assume that known value B ( t , T ) is forms by a stochastic short term [$] r ( v , \omega ) [$] which satisfy Vasicek ‘s (6) sde on real prob space
 
[$] dr  = f ( r ) dt  +  \rho ( r ) dZ[$]                                         (3)
 
and
 
[$] B ( t , T ; r )) = E exp - \int_t^T r ( v ; t , r ) dv[$]
 
Here r ( v ; t , r ) , v > t is a solution of eq (3) with initial value  r ( t ; t , r ) = r. Applying BS derivation  leads us to the version BSE (15) in the paper. In this eq risk free constant r comes up in the eq. The situation with classic BSE and its analog (15) is different. Indeed real drift f from (3) is involved. Then in classic BSE risk free is constant on [ 0 , T ] while here r is stoch process (3) on [ 0 , T ]. It does not look easy to comprehend all complexities.  
 
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Re: why do short term models are written with respect to Q

March 4th, 2017, 2:44 pm

Vasicek is a very good model, the short rates -and hence the yield of bonds- is allowed to be negative.
 
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Re: why do short term models are written with respect to Q

March 4th, 2017, 2:59 pm

Vasicek is a very good model, the short rates -and hence the yield of bonds- is allowed to be negative.
My concern is general approach to randomization of the short term ir for bond pricing. I did not talk about a specific model. On the other hand negative rate is still a little bit strange for theory. Are the middle prices  for a bond both bid and ask are larger than its face value?
 
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Re: why do short term models are written with respect to Q

March 4th, 2017, 3:14 pm

Yes, of course with negative yields one can see bid and ask prices larger than the face value.
 
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Re: why do short term models are written with respect to Q

March 4th, 2017, 3:31 pm

Yes, of course with negative yields one can see bid and ask prices larger than the face value.
Then why in pricing bond we should consider face value negative discount factor. The same amount can be considered as lending not borrowing at negative risk free rate.
 
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Re: why do short term models are written with respect to Q

March 4th, 2017, 4:42 pm

Yes, of course with negative yields one can see bid and ask prices larger than the face value.
Then why in pricing bond we should consider face value negative discount factor. The same amount can be considered as lending not borrowing at negative risk free rate.
?? The discount factor is never negative
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Re: why do short term models are written with respect to Q

March 4th, 2017, 9:54 pm

You are right I meant r < 0. 
 
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Re: why do short term models are written with respect to Q

March 4th, 2017, 11:04 pm

Yes, bond can have negative yields. E.g. I posted a link to the bloomberg page with the current price of a 2 year German Govt bond with a negative yield of 0.80%