In earlier work \ref[J. Math. Phys. 42 (2001), no. 3, 1364--1383; Rev. Math. Phys. 13 (2001), no. 10, 1307--1322, given a compact Riemannian manifold $M$, the author considered an algebra of complex functionals of Malliavin type defined over the based loop space $L_x(M)$ endowed with the Brownian bridge measure. He studied the associated complex relative Grothendieck groups in the Malliavin sense and established a complex Bott periodicity theorem. In the present paper the author's main result generalizes this theorem in the stochastic real case, i.e., with an algebra of real functionals of Malliavin type. The proof follows closely an approach developed by M. Karoubi \ref[ $K$-theory, Springer, Berlin, 1978.