Change of measure for BGM (LMM) Model

I've been checking the demos for BGM (LFM) forward rate model. Here's a short reminder to help you follow:

Now, take the following

$$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \sigma_j. dW^i(t)$$

if we consider the same brownians as in the definition of the BGM model, where we particularly have that $$. We would get an inconsistency such that:

$$<\frac{dL_j(t)}{L_j(t)}> = <\sigma_j. dW^j(t)> = \sigma_j^2 dt$$ Whereas, on the other hand, $$<\frac{dL_j(t)}{L_j(t)}> = <\sigma_j. dW^j(t), \mu_{ij} dt + \sigma_j. dW^i(t)> = \sigma_j^2. \rho_{ij} dt$$

This means that $$\rho_{ij} = 1$$ for all $$i$$'s and $$j$$'s!

My question: Is my reasoning false and why plz?

Perhaps the brownians that we define by a change of measure (from measure $$Q^i$$ to $$Q^j$$) are not the ones considered in the definition of BGM.

Thanks

• In the first equation you write there is an indexing error. The left hand side has no index $i$, whereas the right hand side does, and hence this makes no sense. This further means there is no need for an $i$ index in the final bracket process. Ultimately I think the reasoning is false/erroneous because the indices you've been using are erroneous. Being more rigorous with the indices and the bracket process acting on scalars/vectors should help you tidy up your problem. – oliversm Sep 5 '19 at 16:06