the diffusion coefficient in the dynamics of the GOP $D^i$ is proportial to $a\sqrt{V(t)}$ (resp. $b/\sqrt{V(t)}$, resp. $(a\sqrt{V(t)} + b/\sqrt{V(t)})$).\
The question we would like to address is the following: Given a specification of the market price of risk process $\lambda$ for the domestic currency denomination of securities, **what are the associated dynamics of the domestic and foreign specifications of the GOP?**

\[lemmacasilambda\] Consider a two-currency model, where the exchange rate model for $S^{1,2}$ is of Heston type . The following statements hold true:

1.  \[lemma1:point1\]If $\lambda(t)=a\sqrt{V(t)}$, $a\in{\mathbb{R}}$, then the GOP denominations $D^1$ and $D^2$ follow both Heston-type models.

2.  \[lemma1:point3\]If $\lambda(t) = \frac{b}{\sqrt{V(t)}}$, $b\in{\mathbb{R}}$, then the GOP denomination $D^1$ follows a $3/2$ model, whereas $D^2$ follows a $4/2$ model.

3.  \[lemma1:point2\]If $\lambda(t) = a\sqrt{V(t)}+ \frac{b}{\sqrt{V(t)}}$, $a,b\in{\mathbb{R}}$, then the GOP denominations $D^1$ and $D^2$ follow both $4/2$ type models.

The proof for this result is given in Appendix \[Appendixlemmacasilambda\].

Note that if we had started in with the volatility $1/\sqrt{V(t)}$, then for $\lambda(t) = \frac{b}{\sqrt{V(t)}}$ we would have always fallen into the world of 4/2 type models.

Lemma \[lemmacasilambda\] highlights an interesting interplay between several well-known financial models. It shows that the $4/2$ model arises naturally from a standard Heston 