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We always regard precious metal as a FX rate e.g XAU/USD.

Is there any concept of spot rate for precious metal? If yes, then we know the dynamic of FX spot rate under risk neutral measure can be written as: $$\dfrac{dS}{S} = (r^d-r^f)dt + \sigma dW.$$ Here $r^d/r^f$ is domestic/foreign risk free interest rate. But for the precious metal e.g. XAU/USD, $r^d$ can be risk free interest rate of USD, what's $r^f$ here? Can anyone tell me the meaning in real market?

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There is no interest rate associated with spot PMs. The reason why FX is evaluated in this way is because you can invest in short-term sovereigns (risk free rate) and get cash flow. So the FX rates are partially determined by the differentials between rates. PMs are quite different. If anything spot PMs are a negative yielding asset because of storage costs. There is a yield associated with the futures due to convenience yield / time value / demand for different tenors. It might be more useful from a valuation context to view gold as a commodity, not a currency.

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