  We introduce a family of real random variables $(\beta,\theta)$ arising from
the supersymmetric nonlinear sigma model and containing the family $\beta$
introduced by Sabot, Tarr\`es, and Zeng [STZ17] in the context of the
vertex-reinforced jump process. Using this family we construct an exponential
martingale generalizing the one considered in [DMR17]. Moreover, using the full
supersymmetric nonlinear sigma model we also construct a generalization of the
exponential martingale involving Grassmann variables.
