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Assuming $X_t$ and $Y_t$ are diffusion processes and $Z_t=X_t Y_t$, then $$ dZ_t/Z_t = dX_t/X_t+dY_t/Y_t + ... dt $$ So if the diffusive parts of $X_t$, $Y_t$ and $Z_t$ are SABR, then $$ \sigma^Z_t Z_t^{\beta^Z-1} dW^Z_t + ... dt = \sigma^X_t X_t^{\beta^X-1} dW^X_t + \sigma^Y_t Y_t^{\beta^Y-1} dW^Y_t + ... dt $$ and a first necessary condition is $\beta^X = \...


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