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One way to try to model stochastic volatility of the gold-synthetic spread $X_t - Y_t$ is to use a stochastic volatility model with exogenous variables, which I had been pursuing for the past weeks.  There is a simpler way that might be more practical:  one fits the regression firsts over a period (to construct the synthetic portfolio) and then one fits a stochastic volatility model to the residuals.  This approach can be used to make one-step predictions of a stochastic volatility in order for pair trading positions.