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comment What is a real world example of negative forward interest rate?
As for the spreads. The point is you have to treat 3M and 6M forward rates separately, this is typically done when bootstrapping your interest rate curves.
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comment What is a real world example of negative forward interest rate?
The interest future price is just 100 - forward rate (apart from a convexity adjustment). So if the price is greater than 100, the forward rate is negative.
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revised What is a real world example of negative forward interest rate?
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answered What is a real world example of negative forward interest rate?
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comment Setting the r in put-call parity?
Things have changed in the rates world since Hull wrote that. Traders typically use an OIS rate as the risk free rate, not LIBOR. As quant_dev suggests below, the correct rate to use in the put-call parity formula (or indeed for discounting any future cash flow) is the rate you pay/receive on collateral from your counterparties. This is typically the relevant currency's OIS rate (eg EONIA for EUR)
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answered Rationale for OIS discounting for collateralized derivatives?
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revised What SABR $\beta$ to use for EURIBOR swaption smiles
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answered What SABR $\beta$ to use for EURIBOR swaption smiles