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.